https://mec.etseib.upc.edu/en/api.php?action=feedcontributions&feedformat=atom&user=Apons 2026-06-10T02:45:19Z User contributions MediaWiki 1.37.1 https://mec.etseib.upc.edu/en/index.php?title=D5._Inertia_tensor&diff=841 2024-11-09T09:10:35Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ps}{\textrm{p}}<br /> \newcommand{\hs}{\textrm{h}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\Is}{\textrm{I}}<br /> \newcommand{\ks}{\textrm{k}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\ss}{\textrm{s}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\P}{\textrm{P}}<br /> \newcommand{\Q}{\textrm{Q}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\matriz}[9]{<br /> \begin{bmatrix}<br /> {#1} &amp; {#2} &amp; {#3}\\<br /> {#4} &amp; {#5} &amp; {#6}\\<br /> {#7} &amp; {#8} &amp; {#9}<br /> \end{bmatrix}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> \newcommand{\fvec}[2]{\overline{\mathbf{F}}_{\textrm{#1}\rightarrow\textrm{#2}}}<br /> \newcommand{\mvec}[2]{\overline{\mathbf{M}}_{\textrm{#1}\rightarrow\textrm{#2}}}<br /> &lt;/math&gt;<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D4. Vectorial theorems|&#039;&#039;&#039;Vector Theorems&#039;&#039;&#039;]]&lt;/span&gt; relate the external interaction torsor on a system (&lt;math&gt;\sum\overline{\mathbf{F}}_\mathrm{ext}&lt;/math&gt;, &lt;math&gt;\sum\overline{\mathbf{M}}_\mathrm{ext}(\Qs)&lt;/math&gt;) to the change in time of vectors that depend on how the mass is distributed in the system (&#039;&#039;&#039;mass geometry&#039;&#039;&#039;) and on its motion. In the LMT, this vector is the linear momentum of the system, while in the AMT it its angular momentum (or kinetic momentum). This unit provides the tools necessary to describe the mass geometry of a rigid body and to calculate these two vectors.<br /> <br /> ------------<br /> ------------<br /> ==D5.1 Centre of masses==<br /> <br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.2 Inertia tensor==<br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.3 Some relevant properties of the inertia tensor==<br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.4 Steiner’s Theorem==<br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.5 Change of vector basis==<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[D4. Vectorial theorems|&lt;&lt;&lt; D4. Vectorial theorems]]<br /> <br /> [[D6. Examples of 2D dynamics|D6. Examples of 2D dynamics &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D5._Inertia_tensor&diff=840 2024-11-09T09:10:23Z

<p>Apons: Created page with &quot;&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt; &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}} \newcommand{\vvec}{\overline{\textbf{v}}} \newcommand{\evec}{\overline{\textbf{e}}} \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}} \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}} \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\ds}{\textrm{d}} \newcommand{\ps}{\textrm{p}} \newcommand{\hs}{\text...&quot;</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ps}{\textrm{p}}<br /> \newcommand{\hs}{\textrm{h}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\Is}{\textrm{I}}<br /> \newcommand{\ks}{\textrm{k}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\ss}{\textrm{s}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\P}{\textrm{P}}<br /> \newcommand{\Q}{\textrm{Q}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\matriz}[9]{<br /> \begin{bmatrix}<br /> {#1} &amp; {#2} &amp; {#3}\\<br /> {#4} &amp; {#5} &amp; {#6}\\<br /> {#7} &amp; {#8} &amp; {#9}<br /> \end{bmatrix}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> \newcommand{\fvec}[2]{\overline{\mathbf{F}}_{\textrm{#1}\rightarrow\textrm{#2}}}<br /> \newcommand{\mvec}[2]{\overline{\mathbf{M}}_{\textrm{#1}\rightarrow\textrm{#2}}}<br /> &lt;/math&gt;<br /> <br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D4. Vectorial theorems|&#039;&#039;&#039;Vector Theorems&#039;&#039;&#039;]]&lt;/span&gt; relate the external interaction torsor on a system (&lt;math&gt;\sum\overline{\mathbf{F}}_\mathrm{ext}&lt;/math&gt;, &lt;math&gt;\sum\overline{\mathbf{M}}_\mathrm{ext}(\Qs)&lt;/math&gt;) to the change in time of vectors that depend on how the mass is distributed in the system (&#039;&#039;&#039;mass geometry&#039;&#039;&#039;) and on its motion. In the LMT, this vector is the linear momentum of the system, while in the AMT it its angular momentum (or kinetic momentum). This unit provides the tools necessary to describe the mass geometry of a rigid body and to calculate these two vectors.<br /> <br /> ------------<br /> ------------<br /> ==D5.1 Centre of masses==<br /> <br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.2 Inertia tensor==<br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.3 Some relevant properties of the inertia tensor==<br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.4 Steiner’s Theorem==<br /> <br /> <br /> ------------<br /> ------------<br /> ==D5.5 Change of vector basis==<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[D4. Vectorial theorems|&lt;&lt;&lt; D4. Vectorial theorems]]<br /> <br /> [[D6. Examples of 2D dynamics|D6. Examples of 2D dynamics &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D2._Interaction_forces_between_particles&diff=839 2024-11-09T09:02:33Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\cs}{\textrm{c}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|&#039;&#039;&#039;Newton&#039;s second law&#039;&#039;&#039;]]&lt;/span&gt; can be used to predict the acceleration of a particle &#039;&#039;&#039;P&#039;&#039;&#039; when all the interaction forces exerted on P by all the other particles (&#039;&#039;&#039;Q&#039;&#039;&#039;) are known (all &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt;), or to calculate the forces needed to guarantee a predetermined movement (&lt;math&gt;\overline{\mathbf{O}_\Rs\mathbf{P}}(\ts)&lt;/math&gt;).<br /> <br /> This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.<br /> <br /> <br /> --------<br /> --------<br /> ==D2.1 Kinematic dependence of interaction forces==<br /> <br /> According to the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|&#039;&#039;&#039;Principle of Determinacy&#039;&#039;&#039;]]&lt;/span&gt;, the forces of interaction between two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; at each time instant can only depend on the position and velocity of the particles at that time instant:<br /> &lt;center&gt;<br /> &lt;math&gt;\ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) .&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|&#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;]]&lt;/span&gt; (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions and velocities:<br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|&#039;&#039;&#039;homogeneity of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}&lt;/math&gt; separately, but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}})&lt;/math&gt;.&lt;br&gt;<br /> <br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|&#039;&#039;&#039;isotropy of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on the velocities of de &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; separately (as &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) &lt;/math&gt;), but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}&lt;/math&gt;. Vector &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}}&lt;/math&gt; is the same in all Galilean reference frames (this is the meaning of the &#039;&lt;math&gt;\forall&lt;/math&gt;Gal&#039; subscript, that means “for all Gal reference frames”). In general, &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\text {Gal }}&lt;/math&gt; has a component parallel to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;, and another one perpendicular to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;: &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}}&lt;/math&gt;. But the space isotropy does not allow either the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on a direction different from &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt; (as seen in the presentation of &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Newton&#039;s third law&#039;&#039;&#039;]]&lt;/span&gt;). Therefore, we can accept the dependence on vector &lt;math&gt;\Delta \overline{\mathbf{v}}_\rho&lt;/math&gt; but only on the value &lt;math&gt;\left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right|&lt;/math&gt;(though this dependence rarely appears).<br /> <br /> &lt;div&gt;<br /> =====Proof➕=====<br /> &lt;small&gt;<br /> Let us take any pair of Galilean reference frames RGal1 and RGal2. The velocity of particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; relative to RGal2 may be expressed from that in RGal1 through a &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[C3. Composition of movements#C3.1 Composition of velocities|&#039;&#039;&#039;composition of velocities&#039;&#039;&#039;]]&lt;/span&gt;. If AB=RGal2 and REL=RGal1:<br /> &lt;center&gt;<br /> &lt;math&gt;\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q})=\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{P})+\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P} \in \text { RGal1 })-\left[\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{Q})+\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q} \in \text { RGal1 })\right]=&lt;/math&gt;<br /> <br /> &lt;math&gt;=\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{Q})+\left[\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P} \in \text { RGal1 })-\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q} \in \text { RGal1 })\right]&lt;/math&gt;<br /> <br /> As both reference frames are Galilean, their relative motion is a translational one &lt;math&gt;\left(\bar{\Omega}_{\mathrm{RGal1}}^{\mathrm{RGal2}}=\overline{0}\right)&lt;/math&gt;, therefore &lt;math&gt;\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P} \in \text { RGal1 })=\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q} \in \text { RGal1 })&lt;/math&gt;.<br /> <br /> Finally: &lt;math&gt;\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q})=\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{Q})&lt;/math&gt;.<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> &#039;&#039;&#039;Figure D2.1&#039;&#039;&#039; summarizes all these restrictions.<br /> <br /> [[File:D2-1-neut.png|480px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.1&#039;&#039;&#039; Acceptable dependencies of interaction forces on positions and velocities&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> <br /> --------<br /> --------<br /> <br /> ==D2.2 Classification of interaction forces==<br /> <br /> If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” (&lt;math&gt;\rho &gt; 0&lt;/math&gt;). Talking about contact interaction between particles (&lt;math&gt;\rho = 0&lt;/math&gt;) is not possible: if &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space. &lt;br&gt;<br /> <br /> Strictly speaking, it is only possible to talk about contact interactions between a particle &lt;math&gt;\mathbf{P}&lt;/math&gt; and a particle &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belongs to a rigid body &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, or between particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belong to rigid bodies &lt;math&gt;\mathrm{S}_\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, respectively. These interactions are dealt with in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D3. Interaccions entre sòlids rígids#|&#039;&#039;&#039;unitat D3&#039;&#039;&#039;]]&lt;/span&gt;.&lt;br&gt;<br /> <br /> When &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; interact at a distance (&lt;math&gt;\rho &gt; 0&lt;/math&gt;), we talk of &#039;&#039;&#039;direct interaction&#039;&#039;&#039;. If they are not in contact (&lt;math&gt;\rho &gt; 0&lt;/math&gt;) but them there is an element acting as an intermediate between them, we talk of &#039;&#039;&#039;indirect interaction through the element&#039;&#039;&#039;. In dynamics, all elements with negligible mass compared to that of &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; (if they are not part of a rigid body) or to the rigid bodies to which they belong are considered intermediate elements (IE). &lt;br&gt;<br /> <br /> Forces transmitted between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; through those elements fulfil the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Action-Reaction Principle&#039;&#039;&#039;]]&lt;/span&gt;: they have the same value and are parallel to &lt;math&gt;\overline{\mathbf{PQ}}&lt;/math&gt;, though they have opposite directions (&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039;).<br /> <br /> [[File:D2-2-eng.png|500px|thumb|center|link=]] <br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039; Force transmitted through an intermediate element between two particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;&lt;/small&gt;&lt;/center&gt;<br /> Intermediate elements are treated as a black box: the forces introduced between their endpoints come from deformations and phenomena that are outside the scope of the rigid body model considered in this course (for example, phenomena linked to fluid dynamics, thermodynamics or electromagnetics), and therefore what happens inside them is not studied: only a phenomenological description of their consequences on the particles connected through them is sought.&lt;br&gt;<br /> <br /> In this course, we consider four intermediate elements between particles:&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;springs&#039;&#039;&#039;: the forces introduced between two particles come from their deformation; these forces can be attractive or repulsive (in this case, they can be conditioned by the guiding of the element so that they do not bend laterally), and they allow relative movements between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; of any sign.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;dampers&#039;&#039;&#039;: they introduce forces, often associated with the viscosity of fluids, opposed to the relative motion between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;; in the absence of initial relative motion (&lt;math&gt;\dot{\rho}_\mathrm{inic}=0&lt;/math&gt;), these elements do not introduce any force.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;linear actuators&#039;&#039;&#039;: their operation is based on various phenomena, depending on whether they are hydraulic, pneumatic, electric or magnetic; they exert forces that can be predetermined (that is, known along time: &lt;math&gt;\mathrm{F}_\mathrm{acc.lin}(\ts)&lt;/math&gt; they are data in the problem) or forces that are suitable to control a predetermined relative motion &lt;math&gt;\dot{\rho}(\ts)&lt;/math&gt;, either an approaching or a separating motion.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;inextensible threads&#039;&#039;&#039;: they prevent the particles from separating (impedeixen &lt;math&gt;\dot{\rho} &gt; 0&lt;/math&gt; )but not from approaching (they allow &lt;math&gt;\dot{\rho} &lt; 0&lt;/math&gt;). Since they are intermediate elements that forbid movement, the force they introduce is called &#039;&#039;&#039;restriction or constraint force&#039;&#039;&#039;.&lt;br&gt;<br /> <br /> &#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; classifies the interactions between particles considered in this course, according to whether they are direct or indirect, formulable or non-formulable.<br /> <br /> [[File:D2-3-eng.png|500px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; Classification of interactions between two particles.&lt;/small&gt;&lt;/center&gt;<br /> --------<br /> --------<br /> <br /> ==D2.3 Gravitational attraction==<br /> <br /> The gravitational interaction force (law of universal gravitation) was formulated by Newton. It is an attraction force, and is inversely proportional to the square of the distance between particles (&#039;&#039;&#039;Figure D2.4&#039;&#039;&#039;). It is an empirical formulation: it is based on astronomical observations accumulated over many years.<br /> <br /> [[File:D2-4-eng.png|200px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.4&#039;&#039;&#039; Formulation of the gravitational attraction force.&lt;/small&gt;&lt;/center&gt;<br /> &lt;math&gt;\mathrm{G}_0&lt;/math&gt; is the constant of universal gravitation, and its value is &lt;math&gt;\mathrm{G}_0=6,67\cdot 10^{-11} \mathrm{m}^3/(\mathrm{Kg}\cdot \mathrm{s}^2)&lt;/math&gt;&lt;br&gt;<br /> <br /> The distance between particles (&lt;math&gt;\rho&lt;/math&gt;) must be expressed as a function of the coordinates that have been chosen to describe the system configuration.<br /> <br /> <br /> ====✏️ Exemple D2.1: gravitational attraction between two satellites ====<br /> ---------<br /> <br /> &lt;small&gt; <br /> [[File:EX D2-1-eng.png|180px|thumb|left|link=]]<br /> :Two satellites &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;, modelled as particles of mass &lt;math&gt;\ms_\Ps&lt;/math&gt; and &lt;math&gt;\ms_\mathrm{Q}&lt;/math&gt; , describe circular orbits of radii &lt;math&gt;\rs_\Ps&lt;/math&gt; and &lt;math&gt;\rs_\mathrm{Q}&lt;/math&gt;, respectively, around a planet in the same plane. The configuration of the system is described by the angles &lt;math&gt;\theta_\Ps&lt;/math&gt; and &lt;math&gt;\theta_\mathrm{Q}.&lt;/math&gt; &lt;br&gt;<br /> <br /> <br /> <br /> :The gravitational force they exert on each other is:&lt;br&gt;<br /> <br /> <br /> :&lt;math&gt;\rho=|\overline{\mathbf{P Q}}|=|\overline{\mathbf{O Q}}-\overline{\mathbf{O P}}|=\sqrt{\left(\rs_\mathrm{Q} \sin \theta_\mathrm{Q}-\rs_\Ps \sin \theta_\Ps\right)^2+\left(\rs_\mathrm{Q} \cos \theta_\mathrm{Q}-\rs_\Ps \cos \theta_\Ps\right)^2}=\sqrt{\rs_\mathrm{Q}^2+\rs_\Ps^2-2 \rs_\mathrm{Q} \rs_\Ps \sin \left(\theta_\mathrm{Q}+\theta_\Ps\right)} <br /> &lt;/math&gt;&lt;br&gt;<br /> <br /> <br /> :&lt;math&gt;\mathrm{F}_{\Ps \leftrightarrow \mathrm{Q}}^{\text {grav }}=\mathrm{G}_0 \frac{\ms^2}{\rs_\mathrm{Q}^2+\rs_\Ps^2-2 \rs_\mathrm{Q} \rs_\Ps \cos \left(\theta_\Ps-\theta_\mathrm{Q}\right)}&lt;/math&gt;<br /> &lt;/small&gt;<br /> <br /> --------<br /> --------<br /> <br /> ==D2.4 Interaction through springs==<br /> <br /> Springs introduce attractive or repulsive forces between their ends depending on their deformation. From their natural length &lt;math&gt;\rho_\mathrm{nat}&lt;/math&gt; (for which no force is produced between the spring ends), an elongation (&lt;math&gt;\rho-\rho_\mathrm{nat} &gt; 0&lt;/math&gt;) causes attractive forces while a shortening (&lt;math&gt;\rho-\rho_\mathrm{nat} &lt; 0&lt;/math&gt;) causes repulsive forces.&lt;br&gt;<br /> <br /> The mathematical formulation of these forces is obtained empirically from tests that measure the force as a function of the length change. Usually, we start from a static configuration in which the spring length &lt;math&gt;\rho_0&lt;/math&gt; does not have to coincide with the natural one. If &lt;math&gt;\rho_0 &gt; \rho_\mathrm{nat}&lt;/math&gt; , the force &lt;math&gt;\mathrm{F}_0&lt;/math&gt; between the spring ends for that configuration is attractive. Otherwise, (&lt;math&gt;\rho_0 &lt; \rho_\mathrm{nat}&lt;/math&gt;), &lt;math&gt;\mathrm{F}_0&lt;/math&gt; is a repulsive force.&lt;br&gt;<br /> <br /> The linear springs considered in this course have a linear behaviour: The variation in force &lt;math&gt;\Delta\mathrm{F}&lt;/math&gt; from the reference value (&lt;math&gt;\Delta\mathrm{F}=\mathrm{F}-\mathrm{F}_0&lt;/math&gt;) )is proportional to the variation in length &lt;math&gt;\Delta\rho=\rho-\rho_0&lt;/math&gt; through a constant k.&lt;br&gt;<br /> <br /> A spring that is part of a mechanical system can introduce attractive and repulsive forces throughout its operation. Despite this, these forces are drawn with a single criterion (attractive or repulsive), and are formulated so that their value can have a positive or negative sign. In this way, both attractive and repulsive forces can be reproduced with a single drawing (&#039;&#039;&#039;Figure D2.5&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D2-5-eng.png|500px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.5&#039;&#039;&#039; Formulation of the attractive (a) or repulsive (b) force of a spring with linear behaviour.&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ Exemple D2.2: attraction force of a spring with linear behaviour ====<br /> ---------<br /> &lt;small&gt; <br /> [[File:EX D2-2-neut.png|200px|thumb|left|link=]]<br /> :The linear spring acts between particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; moving within two parallel guides. For &lt;math&gt;\mathrm{x}_1=\mathrm{x}_2&lt;/math&gt; , the spring is stretched and the force it exerts between its ends is &lt;math&gt;\mathrm{F}_0&lt;/math&gt;.&lt;br&gt;<br /> <br /> :For &lt;math&gt;\mathrm{x}_1=\mathrm{x}_2&lt;/math&gt; , the spring length is L and the &lt;math&gt;\mathrm{F}_0&lt;/math&gt; force is attractive. For &lt;math&gt;\mathrm{x}_1 \neq \mathrm{x}_2&lt;/math&gt;, the length increases and so does the attraction force.&lt;br&gt;<br /> <br /> <br /> :The expression of the spring force for a general configuration as an attractive force is:&lt;br&gt;<br /> <br /> <br /> &lt;math&gt;\mathrm{F}_\mathrm{at}^\mathrm{molla}=\mathrm{F}_0+k\Delta\rho=\mathrm{F}_0 + k [\rho(\mathrm{x}_1 \neq \mathrm{x}_2)-\rho(\mathrm{x}_1=\mathrm{x}_2)]=\mathrm{F}_0 + k[\sqrt{\mathrm{L}^2+(\mathrm{x}_1-\mathrm{x}_2)^2}-\mathrm{L}]&lt;/math&gt;.<br /> &lt;/small&gt;<br /> <br /> --------<br /> --------<br /> <br /> ==D2.5 Interaction through dampers==<br /> <br /> Linear dampers introduce attractive or repulsive forces between their ends depending on their deformation rate &lt;math&gt;\dot{\rho}&lt;/math&gt;. When the ends of the damper separate, the force is attractive; when they approach, it is repulsive. Unlike springs, dampers do not exert any force between their ends in static situations.&lt;br&gt;<br /> <br /> The force associated with linear dampers with linear behaviour is proportional to that speed &lt;math&gt;\dot{\rho}&lt;/math&gt;:&lt;br&gt;<br /> <br /> &lt;math&gt;\mathrm{F}_\mathrm{at}^\mathrm{amort}=c\dot{\rho}&lt;/math&gt; , &lt;math&gt;\mathrm{F}_\mathrm{rep}^\mathrm{amort}=-c\dot{\rho}&lt;/math&gt;&lt;br&gt;<br /> <br /> In mechanical systems, dampers often appear in parallel with a spring. In that case, the force is formulated according to the criterion that has been chosen for the spring. When they are not part of a spring-damper group, the criterion is set arbitrarily.<br /> <br /> <br /> ====✏️ Exemple D2.3: repulsive force of a damper with linear behaviour ====<br /> ---------<br /> &lt;small&gt; <br /> [[File:EX D2-3-neut.png|200px|thumb|left|link=]]<br /> :The repulsion force of the damper is obtained from the elongation calculated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#✏️ Exemple D2.2: força d’atracció d’una molla de comportament lineal|&#039;&#039;&#039;l’exemple D2.2&#039;&#039;&#039;]]&lt;/span&gt; per derivació:&lt;br&gt;<br /> <br /> <br /> : &lt;math&gt;\rho=\sqrt{\mathrm{L}^2+(\mathrm{x}_1-\mathrm{x}_2)^2}\equiv \sqrt{\mathrm{L}^2+\mathrm{z}^2} \Rightarrow \dot{\rho}= \frac{\mathrm{d}\rho}{\mathrm{d}\mathrm{z}}\frac{\mathrm{d}\mathrm{z}}{\mathrm{d}\mathrm{t}}=\frac{\mathrm{z}\dot{\mathrm{z}}}{\sqrt{\mathrm{L}^2+\mathrm{z}^2}}&lt;/math&gt;&lt;br&gt;<br /> <br /> <br /> :&lt;math&gt;\mathrm{F}_\mathrm{rep}^\mathrm{amort}=-c \frac{\mathrm{z}\dot{\mathrm{z}}}{\sqrt{\mathrm{L}^2+\mathrm{z}^2}}=-c\frac{(\mathrm{x}_1-\mathrm{x}_2)(\dot{\mathrm{x}}_1-\dot{\mathrm{x}}_2)}{\sqrt{\mathrm{L}^2+\mathrm{(\mathrm{x}_1-\mathrm{x}_2})^2}}&lt;/math&gt;<br /> &lt;/small&gt;<br /> <br /> --------<br /> --------<br /> <br /> ==D2.6 Interaction through actuators==<br /> <br /> Linear actuators appear in the vast majority of mechanical systems, and are responsible for controlling their motion. As they are elements based on phenomena that are not strictly mechanical, their formulation in the context of Newtonian dynamics is not possible. The treatment given to them is different from that of other intermediate elements. Two situations are considered:<br /> <br /> * The force introduced between their ends is data of the problem: this means that its value over time is known: &lt;math&gt;\mathrm{F}_\mathrm{lin.ac}=\mathrm{F}_\mathrm{lin.ac}(\ts)&lt;/math&gt;. The movement they produce, in that case, is an unknown of the problem (&#039;&#039;&#039;Figure D2.6a&#039;&#039;&#039;).<br /> <br /> * The force introduced between their ends is the required on to guarantee a predetermined motion. In that case, that force is an unknown (&#039;&#039;&#039;Figure D2.6b&#039;&#039;&#039;).<br /> <br /> [[File:D2-6-eng.png|500px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.6&#039;&#039;&#039; Linear actuator between two particles. (a) the force it introduces is data, and the resulting motion is an unknown; (b) the motion it controls is predetermined, and the force required to produce it is an unknown.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==D2.7 Constraint interactions==<br /> <br /> Constraint forces restrict relative motions between particles, between particles and surfaces, or between rigid bodies. These forces arise from small deformations of the intermediate elements connecting the particles, from local deformations of the surface or of the rigid body, respectively. This course deals with the dynamics of rigid bodies, therefore these deformations (and the associated interaction forces) cannot be formulated: they are unknowns of the dynamic problem.<br /> <br /> Constraint forces adapt themselves to guarantee the restrictions, but always within permitted ranges. Beyond those ranges, we say that the &#039;&#039;&#039;limit condition&#039;&#039;&#039; has been exceeded, the restriction disappears and either the constraint force is replaced by a formulable force or the interaction disappears.<br /> <br /> When a system includes constraints, it is necessary to characterize them. That means investigating the direction the associated forces can have, and the associated limit conditions. That direction is that of the kinematic restrictions to be guaranteed.<br /> <br /> This section deals with the characterization of indirect constraints between particles through a thread, and of direct constraints between a particle and a rigid body. Constraints between rigid bodies (both direct and indirect) are covered in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D3. Interactions between rigid solids#|&#039;&#039;&#039;unit D3&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ===Indirect constraint through inextensible threads===<br /> <br /> Inextensible threads of negligible mass are intermediate elements that prevent particles from separating but not from approaching each other.<br /> <br /> Let us consider two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; connected by a thread. If a force is exerted on each particle by hand in the right direction to try to separate them, the thread introduces a force in the opposite direction to prevent it: that is the constraint force transmitted between &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; through the thread, and it is a traction force. If the forces of the hands on the particles are in the opposite direction, the thread slackens and is not able to guarantee the restriction: through the thread, one particle can pull the other but cannot push it. Therefore, the traction force cannot be negative: if after solving a problem the conclusion is that the force required to maintain the restriction through the thread has to be negative &lt;math&gt;\mathrm{F}_\mathrm{thread}&lt;0&lt;/math&gt;, this indicates that the constraint is not acting any more (the thread has lost tension, and that is equivalent to not having thread, &#039;&#039;&#039;Figure D2.7a&#039;&#039;&#039;). It is a &#039;&#039;&#039;unilateral link&#039;&#039;&#039;.<br /> <br /> On the other hand, the thread allows the &#039;&#039;&#039;P&#039;&#039;&#039; (or &#039;&#039;&#039;Q&#039;&#039;&#039;) motion on a spherical surface with centre &#039;&#039;&#039;Q&#039;&#039;&#039; (or &#039;&#039;&#039;P&#039;&#039;&#039;): in the directions tangent to these surfaces, the thread cannot introduce any force (&#039;&#039;&#039;Figure D2.7b&#039;&#039;&#039;). In other words: the constraint force is orthogonal to the allowed relative motion:<br /> <br /> &lt;math&gt;\overline{\mathbf{F}}_{\rightarrow \mathrm{P}} \cdot \overline{\mathbf{V}}_{\mathrm{RTQ }}(\mathbf{P})=0, \overline{\mathbf{F}}_{\rightarrow \mathrm{Q}} \cdot \overline{\mathbf{V}}_{\mathrm{RTP }}(\mathbf{Q})=0 &lt;/math&gt;<br /> <br /> [[File:D2-7-eng.png|500px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.7&#039;&#039;&#039; Characterization and limit condition of an indirect constraint through an inextensible thread.&lt;/small&gt;&lt;/center&gt;<br /> <br /> The limit condition for this type of constraint is set by the breaking limit of the thread: for each type of material, there is a force (&lt;math&gt;\mathrm{F}_{\mathrm{break}}&lt;/math&gt;) at which the thread breaks. If, when solving a dynamics problem involving a thread, we detect that the force to guarantee the constraint is higher than this limit value (&lt;math&gt;\mathrm{F}_{\mathrm{thread}}&gt;\mathrm{F}_{\mathrm{break}}&lt;/math&gt;), the problem must be solved again without the thread (and that increases the number of DoF of the system under study).<br /> <br /> ===Direct constraint between a particle P and a smooth rigid body S===<br /> <br /> A rigid body S in contact with a particle &#039;&#039;&#039;P&#039;&#039;&#039; is an obstacle for certain movements of &#039;&#039;&#039;P&#039;&#039;&#039;. The constraint force of S on P is the dynamic description of that obstacle. <br /> <br /> The kinematic analysis of &#039;&#039;&#039;P&#039;&#039;&#039; to characterize the constraint force is done from S, which is the element responsible for this force. Thus, the directions of &#039;&#039;&#039;P&#039;&#039;&#039;&#039;s movements for which S constitutes an obstacle are emphasized: they are the directions for which the &#039;&#039;&#039;P&#039;&#039;&#039; velocity is zero.<br /> <br /> &#039;&#039;&#039;Figure D2.8&#039;&#039;&#039; presents the characterization of the contact between &#039;&#039;&#039;P&#039;&#039;&#039; and S when the rigid body is smooth. It is a unilateral constraint: the constraint force on the particle in the direction normal to the surface at the contact point can only be repulsive, since it is not capable of retaining the particle if some other interaction wants to pull it away from the rigid body. As for indirect constraints between particles through &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#D2.7 Constraint interactions|&#039;&#039;&#039;inextensible threads&#039;&#039;&#039;]]&lt;/span&gt;, there is an orthogonality condition between the constraint force and the allowed velocity of &#039;&#039;&#039;P&#039;&#039;&#039; relative to S. The existence of sliding implies that the movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to S is allowed in any direction of the plane tangent to the S at the contact point: the contact does not introduce any force component in those directions.<br /> <br /> [[File:D2-8-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.8&#039;&#039;&#039; Orthogonality between the constraint force and the &#039;&#039;&#039;P&#039;&#039;&#039; velocity relative to the smooth surface of the rigid body S&lt;/small&gt;&lt;/center&gt;<br /> <br /> ===Direct constraint between a particle P and a rough rigid body S===<br /> <br /> When the surface of the rigid body S is rough and the particle &#039;&#039;&#039;P&#039;&#039;&#039; does not slide on it (&#039;&#039;&#039;Figure D2.9&#039;&#039;&#039;), the constraint force can have nonzero components in both tangential directions. Unlike the normal force, these components can have any sign, but their resultant cannot exceed a maximum value &lt;math&gt;\sqrt{\Fs_1^2 + \Fs_2^2}\leq\Fs_{\text{t, max}}^{\text{constraing}}&lt;/math&gt;. In the dry friction model, that value depends on the roughness: the rougher the surface, the higher the maximum value ([[D2. Interaction forces between particles#D2.8 Friction|&#039;&#039;&#039;D2.8 Friction&#039;&#039;&#039;]].<br /> <br /> [[File:D2-9nou-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.9&#039;&#039;&#039; Orthogonality between the constraint force and the &#039;&#039;&#039;P&#039;&#039;&#039; velocity relative to the rough surface of the rigid body S<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> --------<br /> --------<br /> <br /> ==D2.8 Friction==<br /> <br /> When particle &#039;&#039;&#039;P&#039;&#039;&#039; moves relative to the rough surface of the rigid body S, the tangential force is not a constraint force but a friction force, and always opposes the speed of &#039;&#039;&#039;P&#039;&#039;&#039; relative to S:<br /> <br /> &lt;math&gt;\vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction} = |\vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction}|\frac{\vel{P}{S}}{|\vel{P}{S}|}&lt;/math&gt;<br /> <br /> There are several models for formulating the value of &lt;math&gt;\vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction}&lt;/math&gt;, depending on the characteristics of the contact between &#039;&#039;&#039;P&#039;&#039;&#039; and S.<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;Coulomb’s dry friction model&lt;/span&gt;<br /> <br /> The surface roughness of S is described through friction coefficients. When the roughness is isotropic (equal in all directions) two friction coefficients are defined:<br /> <br /> :* static friction coefficient &lt;math&gt;\mu_\es&lt;/math&gt;: it defines the maximum value (limit condition) of the tangential constraint force: &lt;math&gt;\vec{\Fs}_\text{t, max}^\text{constraint} = \mu_\es\Ns&lt;/math&gt;. If a force higher than &lt;math&gt;\mu_\es \Ns&lt;/math&gt; is needed to guarantee that &#039;&#039;&#039;P&#039;&#039;&#039; does not slide on S, sliding occurs and the friction force appears (&#039;&#039;&#039;Figure D2.10a&#039;&#039;&#039;).<br /> <br /> :* kinetic friction coefficient &lt;math&gt;\mu_\ds&lt;/math&gt;: it defines the value of the friction force : &lt;math&gt;|\vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction}| = \mu_\ds\Ns&lt;/math&gt; (&#039;&#039;&#039;Figure D2.10b&#039;&#039;&#039;).<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;Viscous friction model&lt;/span&gt;<br /> <br /> It is a very suitable model when there is some lubrication between P and S. The friction force is formulated as a function of the relative speed between the two. If it is a linear model: &lt;math&gt;|\vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction}| = \cs|\vel{P}{S}|&lt;/math&gt;.<br /> <br /> [[File:D2-10-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.10&#039;&#039;&#039; Limit value of the tangential constraint force between &#039;&#039;&#039;P&#039;&#039;&#039; and the rough surface of S.&lt;/small&gt;&lt;/center&gt;&lt;br&gt;<br /> <br /> <br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> &lt;center&gt;<br /> [[D1. Foundational laws of Newtonian dynamics|&lt;&lt;&lt; D1. Foundational laws of Newtonian dynamics]]<br /> <br /> [[D3. Interactions between rigid solids|D3. Interactions between rigid solids &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:D2-10-eng.png&diff=838 2024-11-09T08:59:48Z

<p>Apons: </p> <hr /> <div>D2-10-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:D2-9nou-eng.png&diff=837 2024-11-09T08:52:00Z

<p>Apons: </p> <hr /> <div>D2-9nou-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:D2-8-eng.png&diff=836 2024-11-09T08:47:12Z

<p>Apons: </p> <hr /> <div>D2-8-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:D2-7-eng.png&diff=835 2024-11-09T08:41:20Z

<p>Apons: </p> <hr /> <div>D2-7-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:D2-6-eng.png&diff=834 2024-11-09T08:32:31Z

<p>Apons: </p> <hr /> <div>D2-6-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:D2-1-neut.png&diff=833 2024-11-09T08:27:36Z

<p>Apons: </p> <hr /> <div>D2-1-neut</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D2._Interaction_forces_between_particles&diff=817 2024-11-07T15:11:59Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\cs}{\textrm{c}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|&#039;&#039;&#039;Newton&#039;s second law&#039;&#039;&#039;]]&lt;/span&gt; can be used to predict the acceleration of a particle &#039;&#039;&#039;P&#039;&#039;&#039; when all the interaction forces exerted on P by all the other particles (&#039;&#039;&#039;Q&#039;&#039;&#039;) are known (all &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt;), or to calculate the forces needed to guarantee a predetermined movement (&lt;math&gt;\overline{\mathbf{O}_\Rs\mathbf{P}}(\ts)&lt;/math&gt;).<br /> <br /> This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.<br /> <br /> <br /> --------<br /> --------<br /> ==D2.1 Kinematic dependence of interaction forces==<br /> <br /> According to the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|&#039;&#039;&#039;Principle of Determinacy&#039;&#039;&#039;]]&lt;/span&gt;, the forces of interaction between two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; at each time instant can only depend on the position and velocity of the particles at that time instant:<br /> &lt;center&gt;<br /> &lt;math&gt;\ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) .&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|&#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;]]&lt;/span&gt; (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions and velocities:<br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|&#039;&#039;&#039;homogeneity of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}&lt;/math&gt; separately, but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}})&lt;/math&gt;.&lt;br&gt;<br /> <br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|&#039;&#039;&#039;isotropy of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on the velocities of de &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; separately (as &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) &lt;/math&gt;), but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}&lt;/math&gt;. Vector &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}}&lt;/math&gt; is the same in all Galilean reference frames (this is the meaning of the &#039;&lt;math&gt;\forall&lt;/math&gt;Gal&#039; subscript, that means “for all Gal reference frames”). In general, &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\text {Gal }}&lt;/math&gt; has a component parallel to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;, and another one perpendicular to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;: &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}}&lt;/math&gt;. But the space isotropy does not allow either the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on a direction different from &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt; (as seen in the presentation of &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Newton&#039;s third law&#039;&#039;&#039;]]&lt;/span&gt;). Therefore, we can accept the dependence on vector &lt;math&gt;\Delta \overline{\mathbf{v}}_\rho&lt;/math&gt; but only on the value &lt;math&gt;\left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right|&lt;/math&gt;(though this dependence rarely appears).<br /> <br /> --------<br /> --------<br /> <br /> ==D2.2 Classification of interaction forces==<br /> <br /> If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” (&lt;math&gt;\rho &gt; 0&lt;/math&gt;). Talking about contact interaction between particles (&lt;math&gt;\rho = 0&lt;/math&gt;) is not possible: if &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space. &lt;br&gt;<br /> <br /> Strictly speaking, it is only possible to talk about contact interactions between a particle &lt;math&gt;\mathbf{P}&lt;/math&gt; and a particle &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belongs to a rigid body &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, or between particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belong to rigid bodies &lt;math&gt;\mathrm{S}_\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, respectively. These interactions are dealt with in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D3. Interaccions entre sòlids rígids#|&#039;&#039;&#039;unitat D3&#039;&#039;&#039;]]&lt;/span&gt;.&lt;br&gt;<br /> <br /> When &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; interact at a distance (&lt;math&gt;\rho &gt; 0&lt;/math&gt;), we talk of &#039;&#039;&#039;direct interaction&#039;&#039;&#039;. If they are not in contact (&lt;math&gt;\rho &gt; 0&lt;/math&gt;) but them there is an element acting as an intermediate between them, we talk of &#039;&#039;&#039;indirect interaction through the element&#039;&#039;&#039;. In dynamics, all elements with negligible mass compared to that of &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; (if they are not part of a rigid body) or to the rigid bodies to which they belong are considered intermediate elements (IE). &lt;br&gt;<br /> <br /> Forces transmitted between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; through those elements fulfil the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Action-Reaction Principle&#039;&#039;&#039;]]&lt;/span&gt;: they have the same value and are parallel to &lt;math&gt;\overline{\mathbf{PQ}}&lt;/math&gt;, though they have opposite directions (&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039;).<br /> <br /> [[File:D2-2-eng.png|500px|thumb|center|link=]] <br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039; Force transmitted through an intermediate element between two particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;&lt;/small&gt;&lt;/center&gt;<br /> Intermediate elements are treated as a black box: the forces introduced between their endpoints come from deformations and phenomena that are outside the scope of the rigid body model considered in this course (for example, phenomena linked to fluid dynamics, thermodynamics or electromagnetics), and therefore what happens inside them is not studied: only a phenomenological description of their consequences on the particles connected through them is sought.&lt;br&gt;<br /> <br /> In this course, we consider four intermediate elements between particles:&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;springs&#039;&#039;&#039;: the forces introduced between two particles come from their deformation; these forces can be attractive or repulsive (in this case, they can be conditioned by the guiding of the element so that they do not bend laterally), and they allow relative movements between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; of any sign.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;dampers&#039;&#039;&#039;: they introduce forces, often associated with the viscosity of fluids, opposed to the relative motion between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;; in the absence of initial relative motion (&lt;math&gt;\dot{\rho}_\mathrm{inic}=0&lt;/math&gt;), these elements do not introduce any force.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;linear actuators&#039;&#039;&#039;: their operation is based on various phenomena, depending on whether they are hydraulic, pneumatic, electric or magnetic; they exert forces that can be predetermined (that is, known along time: &lt;math&gt;\mathrm{F}_\mathrm{acc.lin}(\ts)&lt;/math&gt; they are data in the problem) or forces that are suitable to control a predetermined relative motion &lt;math&gt;\dot{\rho}(\ts)&lt;/math&gt;, either an approaching or a separating motion.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;inextensible threads&#039;&#039;&#039;: they prevent the particles from separating (impedeixen &lt;math&gt;\dot{\rho} &gt; 0&lt;/math&gt; )but not from approaching (they allow &lt;math&gt;\dot{\rho} &lt; 0&lt;/math&gt;). Since they are intermediate elements that forbid movement, the force they introduce is called &#039;&#039;&#039;restriction or constraint force&#039;&#039;&#039;.&lt;br&gt;<br /> <br /> &#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; classifies the interactions between particles considered in this course, according to whether they are direct or indirect, formulable or non-formulable.<br /> <br /> [[File:D2-3-eng.png|500px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; Classification of interactions between two particles.&lt;/small&gt;&lt;/center&gt;<br /> --------<br /> --------<br /> <br /> ==D2.3 Gravitational attraction==<br /> <br /> <br /> <br /> --------<br /> --------<br /> <br /> ==D2.4 Interaction through springs==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.5 Interaction through dampers==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.6 Interaction through actuators==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.7 Constraint interactions==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.8 Friction==<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |Tots els drets reservats]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> &lt;center&gt;<br /> [[D1. Lleis fundacionals de la mecànica newtoniana|&lt;&lt;&lt; D1. Lleis fundacionals de la mecànica newtoniana]]<br /> <br /> [[D3. Interaccions entre sòlids rígids|D3. Interaccions entre sòlids rígids &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D2._Interaction_forces_between_particles&diff=816 2024-11-07T15:02:06Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\cs}{\textrm{c}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|&#039;&#039;&#039;Newton&#039;s second law&#039;&#039;&#039;]]&lt;/span&gt; can be used to predict the acceleration of a particle &#039;&#039;&#039;P&#039;&#039;&#039; when all the interaction forces exerted on P by all the other particles (&#039;&#039;&#039;Q&#039;&#039;&#039;) are known (all &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt;), or to calculate the forces needed to guarantee a predetermined movement (&lt;math&gt;\overline{\mathbf{O}_\Rs\mathbf{P}}(\ts)&lt;/math&gt;).<br /> <br /> This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.<br /> <br /> <br /> --------<br /> --------<br /> ==D2.1 Kinematic dependence of interaction forces==<br /> <br /> According to the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|&#039;&#039;&#039;Principle of Determinacy&#039;&#039;&#039;]]&lt;/span&gt;, the forces of interaction between two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; at each time instant can only depend on the position and velocity of the particles at that time instant:<br /> &lt;center&gt;<br /> &lt;math&gt;\ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) .&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|&#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;]]&lt;/span&gt; (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions and velocities:<br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|&#039;&#039;&#039;homogeneity of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}&lt;/math&gt; separately, but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}})&lt;/math&gt;.&lt;br&gt;<br /> <br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|&#039;&#039;&#039;isotropy of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on the velocities of de &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; separately (as &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) &lt;/math&gt;), but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}&lt;/math&gt;. Vector &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}}&lt;/math&gt; is the same in all Galilean reference frames (this is the meaning of the &#039;&lt;math&gt;\forall&lt;/math&gt;Gal&#039; subscript, that means “for all Gal reference frames”). In general, &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\text {Gal }}&lt;/math&gt; has a component parallel to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;, and another one perpendicular to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;: &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}}&lt;/math&gt;. But the space isotropy does not allow either the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on a direction different from &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt; (as seen in the presentation of &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Newton&#039;s third law&#039;&#039;&#039;]]&lt;/span&gt;). Therefore, we can accept the dependence on vector &lt;math&gt;\Delta \overline{\mathbf{v}}_\rho&lt;/math&gt; but only on the value &lt;math&gt;\left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right|&lt;/math&gt;(though this dependence rarely appears).<br /> <br /> --------<br /> --------<br /> <br /> ==D2.2 Classification of interaction forces==<br /> <br /> If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” (&lt;math&gt;\rho &gt; 0&lt;/math&gt;). Talking about contact interaction between particles (&lt;math&gt;\rho = 0&lt;/math&gt;) is not possible: if &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space. &lt;br&gt;<br /> <br /> Strictly speaking, it is only possible to talk about contact interactions between a particle &lt;math&gt;\mathbf{P}&lt;/math&gt; and a particle &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belongs to a rigid body &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, or between particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belong to rigid bodies &lt;math&gt;\mathrm{S}_\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, respectively. These interactions are dealt with in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D3. Interaccions entre sòlids rígids#|&#039;&#039;&#039;unitat D3&#039;&#039;&#039;]]&lt;/span&gt;.&lt;br&gt;<br /> <br /> When &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; interact at a distance (&lt;math&gt;\rho &gt; 0&lt;/math&gt;), we talk of &#039;&#039;&#039;direct interaction&#039;&#039;&#039;. If they are not in contact (&lt;math&gt;\rho &gt; 0&lt;/math&gt;) but them there is an element acting as an intermediate between them, we talk of &#039;&#039;&#039;indirect interaction through the element&#039;&#039;&#039;. In dynamics, all elements with negligible mass compared to that of &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; (if they are not part of a rigid body) or to the rigid bodies to which they belong are considered intermediate elements (IE). &lt;br&gt;<br /> <br /> Forces transmitted between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; through those elements fulfil the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Action-Reaction Principle&#039;&#039;&#039;]]&lt;/span&gt;: they have the same value and are parallel to &lt;math&gt;\overline{\mathbf{PQ}}&lt;/math&gt;, though they have opposite directions (&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039;).<br /> <br /> [[File:D2-2-eng.png|500px|thumb|center|link=]] <br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039; Force transmitted through an intermediate element between two particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;&lt;/small&gt;&lt;/center&gt;<br /> Intermediate elements are treated as a black box: the forces introduced between their endpoints come from deformations and phenomena that are outside the scope of the rigid body model considered in this course (for example, phenomena linked to fluid dynamics, thermodynamics or electromagnetics), and therefore what happens inside them is not studied: only a phenomenological description of their consequences on the particles connected through them is sought.&lt;br&gt;<br /> <br /> In this course, we consider four intermediate elements between particles:&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;springs&#039;&#039;&#039;: the forces introduced between two particles come from their deformation; these forces can be attractive or repulsive (in this case, they can be conditioned by the guiding of the element so that they do not bend laterally), and they allow relative movements between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; of any sign.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;dampers&#039;&#039;&#039;: they introduce forces, often associated with the viscosity of fluids, opposed to the relative motion between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;; in the absence of initial relative motion (&lt;math&gt;\dot{\rho}_\mathrm{inic}=0&lt;/math&gt;), these elements do not introduce any force.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;linear actuators&#039;&#039;&#039;: their operation is based on various phenomena, depending on whether they are hydraulic, pneumatic, electric or magnetic; they exert forces that can be predetermined (that is, known along time: &lt;math&gt;\mathrm{F}_\mathrm{acc.lin}(\ts)&lt;/math&gt; they are data in the problem) or forces that are suitable to control a predetermined relative motion &lt;math&gt;\dot{\rho}(\ts)&lt;/math&gt;, either an approaching or a separating motion.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;inextensible threads&#039;&#039;&#039;: they prevent the particles from separating (impedeixen &lt;math&gt;\dot{\rho} &gt; 0&lt;/math&gt; )but not from approaching (they allow &lt;math&gt;\dot{\rho} &lt; 0&lt;/math&gt;). Since they are intermediate elements that forbid movement, the force they introduce is called &#039;&#039;&#039;restriction or constraint force&#039;&#039;&#039;.&lt;br&gt;<br /> <br /> &#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; classifies the interactions between particles considered in this course, according to whether they are direct or indirect, formulable or non-formulable.<br /> <br /> [[File:D2-3-eng.png|500px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; Classification of interactions between two particles.&lt;/small&gt;&lt;/center&gt;<br /> --------<br /> --------<br /> <br /> ==D2.3 Gravitational attraction==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.4 Interaction through springs==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.5 Interaction through dampers==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.6 Interaction through actuators==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.7 Constraint interactions==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.8 Friction==<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |Tots els drets reservats]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> &lt;center&gt;<br /> [[D1. Lleis fundacionals de la mecànica newtoniana|&lt;&lt;&lt; D1. Lleis fundacionals de la mecànica newtoniana]]<br /> <br /> [[D3. Interaccions entre sòlids rígids|D3. Interaccions entre sòlids rígids &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D2._Interaction_forces_between_particles&diff=815 2024-11-07T15:00:44Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\cs}{\textrm{c}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|&#039;&#039;&#039;Newton&#039;s second law&#039;&#039;&#039;]]&lt;/span&gt; can be used to predict the acceleration of a particle &#039;&#039;&#039;P&#039;&#039;&#039; when all the interaction forces exerted on P by all the other particles (&#039;&#039;&#039;Q&#039;&#039;&#039;) are known (all &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt;), or to calculate the forces needed to guarantee a predetermined movement (&lt;math&gt;\overline{\mathbf{O}_\Rs\mathbf{P}}(\ts)&lt;/math&gt;).<br /> <br /> This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.<br /> <br /> <br /> --------<br /> --------<br /> ==D2.1 Kinematic dependence of interaction forces==<br /> <br /> According to the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|&#039;&#039;&#039;Principle of Determinacy&#039;&#039;&#039;]]&lt;/span&gt;, the forces of interaction between two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; at each time instant can only depend on the position and velocity of the particles at that time instant:<br /> &lt;center&gt;<br /> &lt;math&gt;\ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) .&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|&#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;]]&lt;/span&gt; (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions and velocities:<br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|&#039;&#039;&#039;homogeneity of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}&lt;/math&gt; separately, but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}})&lt;/math&gt;.&lt;br&gt;<br /> <br /> * The &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamic#D1.1 Galilean reference frames|&#039;&#039;&#039;isotropy of space&#039;&#039;&#039;]]&lt;/span&gt; in Galilean reference frames does not allow the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on the velocities of de &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; separately (as &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) &lt;/math&gt;), but allows the dependence on its difference &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}&lt;/math&gt;. Vector &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}}&lt;/math&gt; is the same in all Galilean reference frames (this is the meaning of the &#039;&lt;math&gt;\forall&lt;/math&gt;Gal&#039; subscript, that means “for all Gal reference frames”). In general, &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\text {Gal }}&lt;/math&gt; has a component parallel to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;, and another one perpendicular to &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;: &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}}&lt;/math&gt;. But the space isotropy does not allow either the dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on a direction different from &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt; (as seen in the presentation of &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamic#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Newton&#039;s third law&#039;&#039;&#039;]]&lt;/span&gt;). Therefore, we can accept the dependence on vector &lt;math&gt;\Delta \overline{\mathbf{v}}_\rho&lt;/math&gt; but only on the value &lt;math&gt;\left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right|&lt;/math&gt;(though this dependence rarely appears).<br /> <br /> --------<br /> --------<br /> <br /> ==D2.2 Classification of interaction forces==<br /> <br /> If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” (&lt;math&gt;\rho &gt; 0&lt;/math&gt;). Talking about contact interaction between particles (&lt;math&gt;\rho = 0&lt;/math&gt;) is not possible: if &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space. &lt;br&gt;<br /> <br /> Strictly speaking, it is only possible to talk about contact interactions between a particle &lt;math&gt;\mathbf{P}&lt;/math&gt; and a particle &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belongs to a rigid body &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, or between particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; that belong to rigid bodies &lt;math&gt;\mathrm{S}_\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathrm{S}_\mathbf{Q}&lt;/math&gt;, respectively. These interactions are dealt with in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D3. Interaccions entre sòlids rígids#|&#039;&#039;&#039;unitat D3&#039;&#039;&#039;]]&lt;/span&gt;.&lt;br&gt;<br /> <br /> When &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; interact at a distance (&lt;math&gt;\rho &gt; 0&lt;/math&gt;), we talk of &#039;&#039;&#039;direct interaction&#039;&#039;&#039;. If they are not in contact (&lt;math&gt;\rho &gt; 0&lt;/math&gt;) but them there is an element acting as an intermediate between them, we talk of &#039;&#039;&#039;indirect interaction through the element&#039;&#039;&#039;. In dynamics, all elements with negligible mass compared to that of &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; (if they are not part of a rigid body) or to the rigid bodies to which they belong are considered intermediate elements (IE). &lt;br&gt;<br /> <br /> Forces transmitted between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; through those elements fulfil the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|&#039;&#039;&#039;Action-Reaction Principle&#039;&#039;&#039;]]&lt;/span&gt;: they have the same value and are parallel to &lt;math&gt;\overline{\mathbf{PQ}}&lt;/math&gt;, though they have opposite directions (&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039;).<br /> <br /> [[File:D2-2-eng.png|500px|thumb|center|link=]] <br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.2&#039;&#039;&#039; Force transmitted through an intermediate element between two particles &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;&lt;/small&gt;&lt;/center&gt;<br /> Intermediate elements are treated as a black box: the forces introduced between their endpoints come from deformations and phenomena that are outside the scope of the rigid body model considered in this course (for example, phenomena linked to fluid dynamics, thermodynamics or electromagnetics), and therefore what happens inside them is not studied: only a phenomenological description of their consequences on the particles connected through them is sought.&lt;br&gt;<br /> <br /> In this course, we consider four intermediate elements between particles:&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;springs&#039;&#039;&#039;: the forces introduced between two particles come from their deformation; these forces can be attractive or repulsive (in this case, they can be conditioned by the guiding of the element so that they do not bend laterally), and they allow relative movements between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt; of any sign.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;dampers&#039;&#039;&#039;: they introduce forces, often associated with the viscosity of fluids, opposed to the relative motion between &lt;math&gt;\mathbf{P}&lt;/math&gt; and &lt;math&gt;\mathbf{Q}&lt;/math&gt;; in the absence of initial relative motion (&lt;math&gt;\dot{\rho}_\mathrm{inic}=0&lt;/math&gt;), these elements do not introduce any force.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;linear actuators&#039;&#039;&#039;: their operation is based on various phenomena, depending on whether they are hydraulic, pneumatic, electric or magnetic; they exert forces that can be predetermined (that is, known along time: &lt;math&gt;\mathrm{F}_\mathrm{acc.lin}(\ts)&lt;/math&gt; they are data in the problem) or forces that are suitable to control a predetermined relative motion &lt;math&gt;\dot{\rho}(\ts)&lt;/math&gt;, either an approaching or a separating motion.&lt;br&gt;<br /> <br /> * &#039;&#039;&#039;inextensible threads&#039;&#039;&#039;: they prevent the particles from separating (impedeixen &lt;math&gt;\dot{\rho} &gt; 0&lt;/math&gt; )but not from approaching (they allow &lt;math&gt;\dot{\rho} &lt; 0&lt;/math&gt;). Since they are intermediate elements that forbid movement, the force they introduce is called &#039;&#039;&#039;restriction or constraint force&#039;&#039;&#039;.&lt;br&gt;<br /> <br /> &#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; classifies the interactions between particles considered in this course, according to whether they are direct or indirect, formulable or non-formulable.<br /> <br /> [[File:D2-3-eng.png|500px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D2.3&#039;&#039;&#039; Classification of interactions between two particles.&lt;/small&gt;&lt;/center&gt;<br /> --------<br /> --------<br /> <br /> ==D2.3 Gravitational attraction==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.4 Interaction through springs==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.5 Interaction through dampers==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.6 Interaction through actuators==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.7 Constraint interactions==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.8 Friction==<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |Tots els drets reservats]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> &lt;center&gt;<br /> [[D1. Lleis fundacionals de la mecànica newtoniana|&lt;&lt;&lt; D1. Lleis fundacionals de la mecànica newtoniana]]<br /> <br /> [[D3. Interaccions entre sòlids rígids|D3. Interaccions entre sòlids rígids &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=Main_Page&diff=800 2024-11-06T08:46:47Z

<p>Apons: /* D2. Interaction forces between particles */</p> <hr /> <div><br /> [[File:Segwaywiki.png|right|top|200px|link=]]<br /> This website is specially built to complement the learning of the &#039;&#039;&#039;Mechanics&#039;&#039;&#039; course in the bachelor&#039;s degrees of &#039;&#039;&#039;[https://etseib.upc.edu/en Barcelona School of Industrial Engineering (ETSEIB)]&#039;&#039;&#039; of the &#039;&#039;&#039;[https://www.upc.edu/en Polytechnic University of Catalonia (UPC) · BarcelonaTech]&#039;&#039;&#039;.<br /> <br /> This aims to be an &#039;&#039;&#039;accessible&#039;&#039;&#039; and &#039;&#039;&#039;interactive&#039;&#039;&#039; &#039;&#039;&#039;open tool&#039;&#039;&#039;. It&#039;s development started on 2022 and it gathers more than 50 years of teaching experience. It&#039;s content is organized in brief units which contain the fundamental concepts and some fully worked-out examples. Some simple mathematical proofs are included, but the longer or complex ones are refered to biblography.<br /> <br /> The contentent is focused on &#039;&#039;&#039;general space movement of rigid bodies and muli-body systems&#039;&#039;&#039;, but &#039;&#039;&#039;particles&#039;&#039;&#039; are also considered. Dynamics formulation is vectorial, due to the relevance of the force vector in mechanical engineering. The last units are an introduction to energetics.<br /> <br /> <br /> &lt;small&gt;<br /> &#039;&#039;&#039;About the status of the site&#039;&#039;&#039;:<br /> <br /> This project has been developed with limited resources, both technical and human. Nowadays, the server presents some issues, so in case any error may appear, we kindly invite the users to refresh the page and continue enjoying the content 😊.<br /> <br /> The best experience will be through a computer or a tablet 📵.<br /> <br /> The site is still under construction and some interactive resources and videos are still to be uploaded. Also, the Dynamics and Energetics blocks are still to be published. Having said that, it already is a good tool to help in the process of learning Mechanics 🎯.<br /> &lt;/small&gt;<br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mechanics:Copyrights |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> __NOTOC__<br /> ------------<br /> ------------<br /> ===[[Introduction]]===<br /> :: [[Introduction#I.1 What is mechanics?|I.1 What is mechanics?]]<br /> :: [[Introduction#I.2 Models for material objects|I.2 Models for material objects]]<br /> :: [[Introduction#I.3 Limitations of Newtonian mechanics|I.3 Limitations of Newtonian mechanics]]<br /> :: [[Introduction#I.4 Reference frame|I.4 Reference frame]]<br /> <br /> ===[[Vector calculus]]===<br /> ::[[Vector calculus#V.1 Geometric representation of a vector|V.1 Geometric representation of a vector]]<br /> ::[[Vector calculus#V.2 Operations between vectors with geometric representation|V.2 Operations between vectors with geometric representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> ::[[Vector calculus#V.3 Analytical representation of a vector|V.3 Analytical representation of a vector]]<br /> ::[[Vector calculus#V.4 Operations between vectors with analytical representation|V.4 Operations between vectors with analytical representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> <br /> ==KINEMATICS==<br /> ===[[C1. Configuration of a mechanical system]]===<br /> :: [[C1. Configuration of a mechanical system#C1.1 Position of a particle|C1.1 Position of a particle]]<br /> :: [[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|C1.2 Configuration of a rigid body]]<br /> :: [[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|C1.3 Orientation of a rigid body with planar motion]]<br /> :: [[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|C1.4 Orientation of a rigid body moving in space]]<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Rotations about fixed axes|Rotations about fixed axes]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Euler rotations|Euler rotations]]&lt;/small&gt;<br /> :: [[C1. Configuration of a mechanical system#C1.5 Independent coordinates|C1.5 Independent coordinates]]<br /> <br /> ===[[C2. Movement of a mechanical system]]===<br /> :: [[C2. Movement of a mechanical system#C2.1 Velocity of a particle|C2.1 Velocity of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.2 Acceleration of a particle|C2.2 Acceleration of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.3 Intrinsic components of the acceleration|C2.3 Intrinsic components of the acceleration]]<br /> :: [[C2. Movement of a mechanical system#C2.4 Angular velocity of a rigid body|C2.4 Angular velocity of a rigid body]]<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Simple rotation|Simple rotation]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Rotation in space|Rotation in space (Rotacions d&#039;Euler)]]&lt;/small&gt;<br /> :: [[C2. Movement of a mechanical system#C2.5 Angular acceleration of a rigid body|C2.5 Angular acceleration of a rigid body]]<br /> :: [[C2. Movement of a mechanical system#C2.6 Particle kinematics versus rigid body kinematicsrígid|C2.6 Particle kinematics versus rigid body kinematics]]<br /> :: [[C2. Movement of a mechanical system#C2.7 Degrees of freedom of a mechanical system|C2.7 Degrees of freedom of a mechanical system]]<br /> :: [[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|C2.8 Usual constraints in mechanical systems]]<br /> :: [[C2. Movement of a mechanical system#C2.E General examples|C2.E General examples]]<br /> <br /> ===[[C3. Composition of movements]]===<br /> :: [[C3. Composition of movements#C3.1 Composition of velocities|C3.1 Composition of velocities]]<br /> :: [[C3. Composition of movements#C3.2 Composition of accelerations|C3.2 Composition of accelerations]]<br /> :: [[C3. Composition of movements#C3.3 Composition versus time derivative|C3.3 Composition versus time derivative]]<br /> :: [[C3. Composition of movements#C3.E General examples|C3.E General examples]]<br /> <br /> ===[[C4. Rigid body kinematics]]===<br /> :: [[C4. Rigid body kinematics#C4.1 Velocity distribution|C4.1 Velocity distribution]]<br /> :: [[C4. Rigid body kinematics#C4.2 Accelerations distribution|C4.2 Accelerations distribution]]<br /> :: [[C4. Rigid body kinematics#C4.3 Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)|Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)]]<br /> ::[[C4. Rigid body kinematics#C4.4 Fixed axode and moving axode|C4.4 Fixed axode and moving axode]]<br /> ::[[C4. Rigid body kinematics#C4.E General examples|C4.E General examples]]<br /> <br /> ===[[C5. Rigid body kinematics: planar motion]]===<br /> :: [[C5. Rigid body kinematics: planar motion#C5.1 Instantaneous Center of Rotation (ICR)|C5.1 Instantaneous Center of Rotation (ICR)]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.2 Examples|C5.2 Examples]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.3 Introduction to vehicle kinematics|C5.3 Introduction to vehicle kinematics]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.E General examples|C5.E General examples]]<br /> <br /> ==DYNAMICS==<br /> <br /> ===[[D1. Foundational laws of Newtonian dynamics#|D1. Foundational laws of Newtonian dynamics]]===<br /> <br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|D1.1 Galilean reference frames]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|D1.2 Galileo’s Principle of Relativity]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|D1.3 Newton’s Principle of Determinacy]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|D1.4 Newton’s first law (inertia law)]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|D1.5 Newton’s second law (fundamental law of dynamics)]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|D1.6 Newton’s third law (action-reaction principle)]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.7 Particle dynamics in non Galilean reference frames|D1.7 Particle dynamics in non Galilean reference frames]]<br /> <br /> ===[[D2. Interaction forces between particles#|D2. Interaction forces between particles]]===<br /> ::[[D2. Interaction forces between particles#D2.1 Kinematic dependence of interaction forces|D2.1 Kinematic dependence of interaction forces]]<br /> ::[[D2. Interaction forces between particles#D2.2 Classification of interaction forces|D2.2 Classification of interaction forces]]<br /> ::[[D2. Interaction forces between particles#D2.3 Gravitational attraction|D2.3 Gravitational attraction]]<br /> ::[[D2. Interaction forces between particles#D2.4 Interaction through springs|D2.4 Interaction through springs]]<br /> ::[[D2. Interaction forces between particles#D2.5 Interaction through dampers|D2.5 Interaction through dampers]]<br /> ::[[D2. Interaction forces between particles#D2.6 Interaction through actuators|D2.6 Interaction through actuators]]<br /> ::[[D2. Interaction forces between particles#D2.7 Constraint interactions|D2.7 Constraint interactions]]<br /> ::[[D2. Interaction forces between particles#D2.8 Friction|D2.8 Friction]]<br /> <br /> ===[[D4. Vectorial theorems#|D4. Vectorial theorems]]===<br /> <br /> ===[[D5. Inertia tensor#|D5. Inertia tensor]]===<br /> ::[[D5. Inertia tensor#D5.1 Centre of masses|D5.1 Centre of masses]]<br /> ::[[D5. Inertia tensor#D5.2 Inertia tensor|D5.2 Inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.3 Some relevant properties of the inertia tensor|D5.3 Some relevant properties of the inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.4 Steiner’s Theorem|D5.4 Steiner’s Theorem]]<br /> ::[[D5. Inertia tensor#D5.5 Change of vector basis|D5.5 Change of vector basis]]<br /> <br /> ===[[D7. Examples of 3D dynamics#|D7. Examples of 3D dynamics]]===<br /> ::[[D7. Examples of 3D dynamics#D7.1 Analysis of the equations of motion|D7.1 Analysis of the equations of motion]]<br /> ::[[D7. Examples of 3D dynamics#D7.2 General examples|D7.2 General examples]]<br /> <br /> ===[[D8. Conservation of dynamic magnitudes#|D8. Conservation of dynamic magnitudes]]===<br /> ::[[D8. Conservation of dynamic magnitudes#D8.1 Examples|D8.1 Examples]]<br /> <br /> ==ENERGETICS==<br /> ::&#039;&#039;UNDER CONSTRUCTION&#039;&#039;<br /> <br /> <br /> -----------<br /> -----------<br /> ==Authors==<br /> <br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors Ana.png|alt=Ana Barjau Condomines|Ana Barjau Condomines|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1001000<br /> File:Autors Ernest4.jpg|alt=Ernest Bosch Soldevila|Ernest Bosch Soldevila|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1099864<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Ilustrations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Joaquim.png|alt=Joaquim Agulló i Batlle|Joquim Agulló i Batlle|link=https://www.agullobatlle.cat/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Editing and interactive animations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Arnau.png|alt=Arnau Marzábal Gatell|Arnau Marzábal Gatell|link=https://www.linkedin.com/in/arnau-marzabal/<br /> File:Autors_Berta.png|alt=Berta Ros Blanco|Berta Ros Blanco|link=https://www.linkedin.com/in/berta-ros/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Collaborators:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Daniel.png|alt=Daniel Clos Costa|Daniel Clos Costa|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1002252<br /> File:Autors_Rosa.png|alt=Rosa Pàmies Vilà|Rosa Pàmies Vilà|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1066910<br /> File:Autors_Albert.png|alt=Albert Peiret Giménez|Albert Peiret Giménez|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1115007<br /> File:Autors_Javier.png|alt=Javier Sistiaga Vidal-Ribas|Javier Sistiaga Vidal-Ribas|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1114855<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;<br /> [https://www.youtube.com/channel/UCqWvnHTViRPI1wHlUQXqH-Q &#039;&#039;&#039;Mechanics Lab&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://etseib.upc.edu/en &#039;&#039;&#039;Barcelona School of Industrial Engineering (ETSEIB)&#039;&#039;&#039;]<br /> <br /> [https://em.upc.edu/en &#039;&#039;&#039;Mechanical Engineering Department&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://www.upc.edu/en &#039;&#039;&#039;Polytechic University of Catalonia (UPC) · BarcelonaTech&#039;&#039;&#039;]<br /> &lt;/center&gt;<br /> <br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==Bibliographic references==<br /> Batlle, J. A., Barjau, A. (2020) “&#039;&#039;&#039;Rigid Body Kinematics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-47907-3<br /> <br /> Batlle, J. A., Barjau, A. (2022) “&#039;&#039;&#039;Rigid Body Dynamics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-84213-6<br /> <br /> Agulló, J. (2002) “&#039;&#039;&#039;Mecànica de la partícula i del sòlid rígid&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-6-1 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> Agulló, J. (2000) “&#039;&#039;&#039;Mecánica de la partícula i del sólido rígido&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-5-3 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> <br /> &lt;center&gt;<br /> &lt;gallery&gt;<br /> File:RBK portada.png|alt=Rigid body kinematics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-kinematics?format=HB&amp;isbn=9781108479073<br /> File:RBD portada.png|alt=Rigid body dynamics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136<br /> File:Llibre verd.png|alt=Mecànica de la partícula i del sòlid rígid|link=https://www.agullobatlle.cat/activitat-docent<br /> File:Llibre vermell.jpg|alt=Mecánica de la partícula i del sólido rígido|link=https://www.agullobatlle.cat/activitat-docent<br /> &lt;/gallery&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> <br /> <br /> <br /> [[File:Logo Lab Mec horitzontal.png|thumb|center|500px|link=https://em.upc.edu/en| ]]</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D2._Interaction_forces_between_particles&diff=799 2024-11-06T08:40:38Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\cs}{\textrm{c}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|&#039;&#039;&#039;Newton&#039;s second law&#039;&#039;&#039;]]&lt;/span&gt; can be used to predict the acceleration of a particle &#039;&#039;&#039;P&#039;&#039;&#039; when all the interaction forces exerted on P by all the other particles (&#039;&#039;&#039;Q&#039;&#039;&#039;) are known (all &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt;), or to calculate the forces needed to guarantee a predetermined movement (&lt;math&gt;\overline{\mathbf{O}_\Rs\mathbf{P}}(\ts)&lt;/math&gt;).<br /> <br /> This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.<br /> <br /> <br /> --------<br /> --------<br /> ==D2.1 Kinematic dependence of interaction forces==<br /> <br /> According to the &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|&#039;&#039;&#039;Principle of Determinacy&#039;&#039;&#039;]]&lt;/span&gt;, the forces of interaction between two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; at each time instant can only depend on the position and velocity of the particles at that time instant:<br /> &lt;center&gt;<br /> &lt;math&gt;\ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) .&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|&#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;]]&lt;/span&gt; (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; on positions and velocities:<br /> *<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> El &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.2 Principi de Relativitat de Galileu|&#039;&#039;&#039;Principi de Relativitat de Galileu&#039;&#039;&#039;]]&lt;/span&gt; (equivalència de les referències galileanes per a laformulació de la dinàmica) imposa restriccions al tipus de dependència de &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; en posicions i velocitats:<br /> * La &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Referències galileanes|&#039;&#039;&#039;homogeneitat de l’espai&#039;&#039;&#039;]]&lt;/span&gt; en referències galileanes impedeix que &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; depengui en les posicions &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}&lt;/math&gt; i &lt;math&gt;\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}&lt;/math&gt; per separat, però permet la dependència en la seva diferència &lt;math&gt;\overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}})&lt;/math&gt;&lt;br&gt;<br /> <br /> *La &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Referències galileanes|&#039;&#039;&#039;isotropia de l&#039;espai&#039;&#039;&#039;]]&lt;/span&gt; en referències galileanes impedeix que &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; depengui de les velocitats de &lt;math&gt;\mathbf{P}&lt;/math&gt; i &lt;math&gt;\mathbf{Q}&lt;/math&gt; per separat (doncs &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) &lt;/math&gt;), però permet que depengui de la seva diferència: &lt;math&gt;\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}&lt;/math&gt;. El vector &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}}&lt;/math&gt; és el mateix independentment de quina sigui la referència Gal (per això s&#039;ha posat el subíndex &#039; &lt;math&gt;\forall&lt;/math&gt;Gal&#039;, que vol dir “per a tota referència Gal”). En general, &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\text {Gal }}&lt;/math&gt; té una component parallela a &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt;, i una de perpendicular a &lt;math&gt;\overline{\mathbf{P Q}}&lt;/math&gt; : &lt;math&gt;\Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}}&lt;/math&gt;. Però la isotropia de l&#039;espai tampoc no permet que &lt;math&gt;\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}&lt;/math&gt; depengui d&#039;una direcció que no sigui &lt;math&gt;\overline{\mathbf{P Q}} &lt;/math&gt;(com s&#039;ha vist quan s&#039;ha presentat la &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.6 Tercera llei de Newton (principi d’acció i reacció)|&#039;&#039;&#039;Tercera llei de Newton&#039;&#039;&#039;]]&lt;/span&gt;). Per tant, s&#039;accepta la dependència en el vector &lt;math&gt;\Delta \overline{\mathbf{v}}_\rho&lt;/math&gt; però només en el valor de &lt;math&gt;\left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right|&lt;/math&gt;(tot i que usualment aquesta última dependència no apareix).<br /> <br /> --------<br /> --------<br /> <br /> ==D2.2 Classification of interaction forces==<br /> <br /> <br /> <br /> <br /> --------<br /> --------<br /> <br /> ==D2.3 Gravitational attraction==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.4 Interaction through springs==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.5 Interaction through dampers==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.6 Interaction through actuators==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.7 Constraint interactions==<br /> <br /> --------<br /> --------<br /> <br /> ==D2.8 Friction==<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |Tots els drets reservats]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> &lt;center&gt;<br /> [[D1. Lleis fundacionals de la mecànica newtoniana|&lt;&lt;&lt; D1. Lleis fundacionals de la mecànica newtoniana]]<br /> <br /> [[D3. Interaccions entre sòlids rígids|D3. Interaccions entre sòlids rígids &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=Main_Page&diff=798 2024-11-06T08:32:02Z

<p>Apons: /* D1. Foundational laws of Newtonian dynamics */</p> <hr /> <div><br /> [[File:Segwaywiki.png|right|top|200px|link=]]<br /> This website is specially built to complement the learning of the &#039;&#039;&#039;Mechanics&#039;&#039;&#039; course in the bachelor&#039;s degrees of &#039;&#039;&#039;[https://etseib.upc.edu/en Barcelona School of Industrial Engineering (ETSEIB)]&#039;&#039;&#039; of the &#039;&#039;&#039;[https://www.upc.edu/en Polytechnic University of Catalonia (UPC) · BarcelonaTech]&#039;&#039;&#039;.<br /> <br /> This aims to be an &#039;&#039;&#039;accessible&#039;&#039;&#039; and &#039;&#039;&#039;interactive&#039;&#039;&#039; &#039;&#039;&#039;open tool&#039;&#039;&#039;. It&#039;s development started on 2022 and it gathers more than 50 years of teaching experience. It&#039;s content is organized in brief units which contain the fundamental concepts and some fully worked-out examples. Some simple mathematical proofs are included, but the longer or complex ones are refered to biblography.<br /> <br /> The contentent is focused on &#039;&#039;&#039;general space movement of rigid bodies and muli-body systems&#039;&#039;&#039;, but &#039;&#039;&#039;particles&#039;&#039;&#039; are also considered. Dynamics formulation is vectorial, due to the relevance of the force vector in mechanical engineering. The last units are an introduction to energetics.<br /> <br /> <br /> &lt;small&gt;<br /> &#039;&#039;&#039;About the status of the site&#039;&#039;&#039;:<br /> <br /> This project has been developed with limited resources, both technical and human. Nowadays, the server presents some issues, so in case any error may appear, we kindly invite the users to refresh the page and continue enjoying the content 😊.<br /> <br /> The best experience will be through a computer or a tablet 📵.<br /> <br /> The site is still under construction and some interactive resources and videos are still to be uploaded. Also, the Dynamics and Energetics blocks are still to be published. Having said that, it already is a good tool to help in the process of learning Mechanics 🎯.<br /> &lt;/small&gt;<br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mechanics:Copyrights |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> __NOTOC__<br /> ------------<br /> ------------<br /> ===[[Introduction]]===<br /> :: [[Introduction#I.1 What is mechanics?|I.1 What is mechanics?]]<br /> :: [[Introduction#I.2 Models for material objects|I.2 Models for material objects]]<br /> :: [[Introduction#I.3 Limitations of Newtonian mechanics|I.3 Limitations of Newtonian mechanics]]<br /> :: [[Introduction#I.4 Reference frame|I.4 Reference frame]]<br /> <br /> ===[[Vector calculus]]===<br /> ::[[Vector calculus#V.1 Geometric representation of a vector|V.1 Geometric representation of a vector]]<br /> ::[[Vector calculus#V.2 Operations between vectors with geometric representation|V.2 Operations between vectors with geometric representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> ::[[Vector calculus#V.3 Analytical representation of a vector|V.3 Analytical representation of a vector]]<br /> ::[[Vector calculus#V.4 Operations between vectors with analytical representation|V.4 Operations between vectors with analytical representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> <br /> ==KINEMATICS==<br /> ===[[C1. Configuration of a mechanical system]]===<br /> :: [[C1. Configuration of a mechanical system#C1.1 Position of a particle|C1.1 Position of a particle]]<br /> :: [[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|C1.2 Configuration of a rigid body]]<br /> :: [[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|C1.3 Orientation of a rigid body with planar motion]]<br /> :: [[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|C1.4 Orientation of a rigid body moving in space]]<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Rotations about fixed axes|Rotations about fixed axes]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Euler rotations|Euler rotations]]&lt;/small&gt;<br /> :: [[C1. Configuration of a mechanical system#C1.5 Independent coordinates|C1.5 Independent coordinates]]<br /> <br /> ===[[C2. Movement of a mechanical system]]===<br /> :: [[C2. Movement of a mechanical system#C2.1 Velocity of a particle|C2.1 Velocity of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.2 Acceleration of a particle|C2.2 Acceleration of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.3 Intrinsic components of the acceleration|C2.3 Intrinsic components of the acceleration]]<br /> :: [[C2. Movement of a mechanical system#C2.4 Angular velocity of a rigid body|C2.4 Angular velocity of a rigid body]]<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Simple rotation|Simple rotation]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Rotation in space|Rotation in space (Rotacions d&#039;Euler)]]&lt;/small&gt;<br /> :: [[C2. Movement of a mechanical system#C2.5 Angular acceleration of a rigid body|C2.5 Angular acceleration of a rigid body]]<br /> :: [[C2. Movement of a mechanical system#C2.6 Particle kinematics versus rigid body kinematicsrígid|C2.6 Particle kinematics versus rigid body kinematics]]<br /> :: [[C2. Movement of a mechanical system#C2.7 Degrees of freedom of a mechanical system|C2.7 Degrees of freedom of a mechanical system]]<br /> :: [[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|C2.8 Usual constraints in mechanical systems]]<br /> :: [[C2. Movement of a mechanical system#C2.E General examples|C2.E General examples]]<br /> <br /> ===[[C3. Composition of movements]]===<br /> :: [[C3. Composition of movements#C3.1 Composition of velocities|C3.1 Composition of velocities]]<br /> :: [[C3. Composition of movements#C3.2 Composition of accelerations|C3.2 Composition of accelerations]]<br /> :: [[C3. Composition of movements#C3.3 Composition versus time derivative|C3.3 Composition versus time derivative]]<br /> :: [[C3. Composition of movements#C3.E General examples|C3.E General examples]]<br /> <br /> ===[[C4. Rigid body kinematics]]===<br /> :: [[C4. Rigid body kinematics#C4.1 Velocity distribution|C4.1 Velocity distribution]]<br /> :: [[C4. Rigid body kinematics#C4.2 Accelerations distribution|C4.2 Accelerations distribution]]<br /> :: [[C4. Rigid body kinematics#C4.3 Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)|Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)]]<br /> ::[[C4. Rigid body kinematics#C4.4 Fixed axode and moving axode|C4.4 Fixed axode and moving axode]]<br /> ::[[C4. Rigid body kinematics#C4.E General examples|C4.E General examples]]<br /> <br /> ===[[C5. Rigid body kinematics: planar motion]]===<br /> :: [[C5. Rigid body kinematics: planar motion#C5.1 Instantaneous Center of Rotation (ICR)|C5.1 Instantaneous Center of Rotation (ICR)]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.2 Examples|C5.2 Examples]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.3 Introduction to vehicle kinematics|C5.3 Introduction to vehicle kinematics]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.E General examples|C5.E General examples]]<br /> <br /> ==DYNAMICS==<br /> <br /> ===[[D1. Foundational laws of Newtonian dynamics#|D1. Foundational laws of Newtonian dynamics]]===<br /> <br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|D1.1 Galilean reference frames]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|D1.2 Galileo’s Principle of Relativity]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|D1.3 Newton’s Principle of Determinacy]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|D1.4 Newton’s first law (inertia law)]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|D1.5 Newton’s second law (fundamental law of dynamics)]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|D1.6 Newton’s third law (action-reaction principle)]]<br /> ::[[D1. Foundational laws of Newtonian dynamics#D1.7 Particle dynamics in non Galilean reference frames|D1.7 Particle dynamics in non Galilean reference frames]]<br /> <br /> ===[[D2. Interaction forces between particles#|D2. Interaction forces between particles]]===<br /> <br /> ===[[D4. Vectorial theorems#|D4. Vectorial theorems]]===<br /> <br /> ===[[D5. Inertia tensor#|D5. Inertia tensor]]===<br /> ::[[D5. Inertia tensor#D5.1 Centre of masses|D5.1 Centre of masses]]<br /> ::[[D5. Inertia tensor#D5.2 Inertia tensor|D5.2 Inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.3 Some relevant properties of the inertia tensor|D5.3 Some relevant properties of the inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.4 Steiner’s Theorem|D5.4 Steiner’s Theorem]]<br /> ::[[D5. Inertia tensor#D5.5 Change of vector basis|D5.5 Change of vector basis]]<br /> <br /> ===[[D7. Examples of 3D dynamics#|D7. Examples of 3D dynamics]]===<br /> ::[[D7. Examples of 3D dynamics#D7.1 Analysis of the equations of motion|D7.1 Analysis of the equations of motion]]<br /> ::[[D7. Examples of 3D dynamics#D7.2 General examples|D7.2 General examples]]<br /> <br /> ===[[D8. Conservation of dynamic magnitudes#|D8. Conservation of dynamic magnitudes]]===<br /> ::[[D8. Conservation of dynamic magnitudes#D8.1 Examples|D8.1 Examples]]<br /> <br /> ==ENERGETICS==<br /> ::&#039;&#039;UNDER CONSTRUCTION&#039;&#039;<br /> <br /> <br /> -----------<br /> -----------<br /> ==Authors==<br /> <br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors Ana.png|alt=Ana Barjau Condomines|Ana Barjau Condomines|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1001000<br /> File:Autors Ernest4.jpg|alt=Ernest Bosch Soldevila|Ernest Bosch Soldevila|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1099864<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Ilustrations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Joaquim.png|alt=Joaquim Agulló i Batlle|Joquim Agulló i Batlle|link=https://www.agullobatlle.cat/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Editing and interactive animations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Arnau.png|alt=Arnau Marzábal Gatell|Arnau Marzábal Gatell|link=https://www.linkedin.com/in/arnau-marzabal/<br /> File:Autors_Berta.png|alt=Berta Ros Blanco|Berta Ros Blanco|link=https://www.linkedin.com/in/berta-ros/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Collaborators:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Daniel.png|alt=Daniel Clos Costa|Daniel Clos Costa|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1002252<br /> File:Autors_Rosa.png|alt=Rosa Pàmies Vilà|Rosa Pàmies Vilà|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1066910<br /> File:Autors_Albert.png|alt=Albert Peiret Giménez|Albert Peiret Giménez|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1115007<br /> File:Autors_Javier.png|alt=Javier Sistiaga Vidal-Ribas|Javier Sistiaga Vidal-Ribas|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1114855<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;<br /> [https://www.youtube.com/channel/UCqWvnHTViRPI1wHlUQXqH-Q &#039;&#039;&#039;Mechanics Lab&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://etseib.upc.edu/en &#039;&#039;&#039;Barcelona School of Industrial Engineering (ETSEIB)&#039;&#039;&#039;]<br /> <br /> [https://em.upc.edu/en &#039;&#039;&#039;Mechanical Engineering Department&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://www.upc.edu/en &#039;&#039;&#039;Polytechic University of Catalonia (UPC) · BarcelonaTech&#039;&#039;&#039;]<br /> &lt;/center&gt;<br /> <br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==Bibliographic references==<br /> Batlle, J. A., Barjau, A. (2020) “&#039;&#039;&#039;Rigid Body Kinematics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-47907-3<br /> <br /> Batlle, J. A., Barjau, A. (2022) “&#039;&#039;&#039;Rigid Body Dynamics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-84213-6<br /> <br /> Agulló, J. (2002) “&#039;&#039;&#039;Mecànica de la partícula i del sòlid rígid&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-6-1 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> Agulló, J. (2000) “&#039;&#039;&#039;Mecánica de la partícula i del sólido rígido&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-5-3 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> <br /> &lt;center&gt;<br /> &lt;gallery&gt;<br /> File:RBK portada.png|alt=Rigid body kinematics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-kinematics?format=HB&amp;isbn=9781108479073<br /> File:RBD portada.png|alt=Rigid body dynamics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136<br /> File:Llibre verd.png|alt=Mecànica de la partícula i del sòlid rígid|link=https://www.agullobatlle.cat/activitat-docent<br /> File:Llibre vermell.jpg|alt=Mecánica de la partícula i del sólido rígido|link=https://www.agullobatlle.cat/activitat-docent<br /> &lt;/gallery&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> <br /> <br /> <br /> [[File:Logo Lab Mec horitzontal.png|thumb|center|500px|link=https://em.upc.edu/en| ]]</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D2._Interaction_forces_between_particles&diff=797 2024-11-06T08:19:43Z

<p>Apons: Created page with &quot;&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt; &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}} \newcommand{\vvec}{\overline{\textbf{v}}} \newcommand{\evec}{\overline{\textbf{e}}} \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}} \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}} \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\cs}{\textrm{c}} \newcommand{\ds}{\textrm{d}} \newcommand{\ms}{\text...&quot;</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\cs}{\textrm{c}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Fs}{\textrm{F}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Ns}{\textrm{N}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{lliure}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> <br /> --------<br /> --------<br /> ==D2.1 Dependència cinemàtica de les forces d’interacció==<br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |Tots els drets reservats]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[D1. Lleis fundacionals de la mecànica newtoniana|&lt;&lt;&lt; D1. Lleis fundacionals de la mecànica newtoniana]]<br /> <br /> [[D3. Interaccions entre sòlids rígids|D3. Interaccions entre sòlids rígids &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=Main_Page&diff=796 2024-11-06T08:18:34Z

<p>Apons: /* DYNAMICS */</p> <hr /> <div><br /> [[File:Segwaywiki.png|right|top|200px|link=]]<br /> This website is specially built to complement the learning of the &#039;&#039;&#039;Mechanics&#039;&#039;&#039; course in the bachelor&#039;s degrees of &#039;&#039;&#039;[https://etseib.upc.edu/en Barcelona School of Industrial Engineering (ETSEIB)]&#039;&#039;&#039; of the &#039;&#039;&#039;[https://www.upc.edu/en Polytechnic University of Catalonia (UPC) · BarcelonaTech]&#039;&#039;&#039;.<br /> <br /> This aims to be an &#039;&#039;&#039;accessible&#039;&#039;&#039; and &#039;&#039;&#039;interactive&#039;&#039;&#039; &#039;&#039;&#039;open tool&#039;&#039;&#039;. It&#039;s development started on 2022 and it gathers more than 50 years of teaching experience. It&#039;s content is organized in brief units which contain the fundamental concepts and some fully worked-out examples. Some simple mathematical proofs are included, but the longer or complex ones are refered to biblography.<br /> <br /> The contentent is focused on &#039;&#039;&#039;general space movement of rigid bodies and muli-body systems&#039;&#039;&#039;, but &#039;&#039;&#039;particles&#039;&#039;&#039; are also considered. Dynamics formulation is vectorial, due to the relevance of the force vector in mechanical engineering. The last units are an introduction to energetics.<br /> <br /> <br /> &lt;small&gt;<br /> &#039;&#039;&#039;About the status of the site&#039;&#039;&#039;:<br /> <br /> This project has been developed with limited resources, both technical and human. Nowadays, the server presents some issues, so in case any error may appear, we kindly invite the users to refresh the page and continue enjoying the content 😊.<br /> <br /> The best experience will be through a computer or a tablet 📵.<br /> <br /> The site is still under construction and some interactive resources and videos are still to be uploaded. Also, the Dynamics and Energetics blocks are still to be published. Having said that, it already is a good tool to help in the process of learning Mechanics 🎯.<br /> &lt;/small&gt;<br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mechanics:Copyrights |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> __NOTOC__<br /> ------------<br /> ------------<br /> ===[[Introduction]]===<br /> :: [[Introduction#I.1 What is mechanics?|I.1 What is mechanics?]]<br /> :: [[Introduction#I.2 Models for material objects|I.2 Models for material objects]]<br /> :: [[Introduction#I.3 Limitations of Newtonian mechanics|I.3 Limitations of Newtonian mechanics]]<br /> :: [[Introduction#I.4 Reference frame|I.4 Reference frame]]<br /> <br /> ===[[Vector calculus]]===<br /> ::[[Vector calculus#V.1 Geometric representation of a vector|V.1 Geometric representation of a vector]]<br /> ::[[Vector calculus#V.2 Operations between vectors with geometric representation|V.2 Operations between vectors with geometric representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> ::[[Vector calculus#V.3 Analytical representation of a vector|V.3 Analytical representation of a vector]]<br /> ::[[Vector calculus#V.4 Operations between vectors with analytical representation|V.4 Operations between vectors with analytical representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> <br /> ==KINEMATICS==<br /> ===[[C1. Configuration of a mechanical system]]===<br /> :: [[C1. Configuration of a mechanical system#C1.1 Position of a particle|C1.1 Position of a particle]]<br /> :: [[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|C1.2 Configuration of a rigid body]]<br /> :: [[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|C1.3 Orientation of a rigid body with planar motion]]<br /> :: [[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|C1.4 Orientation of a rigid body moving in space]]<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Rotations about fixed axes|Rotations about fixed axes]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Euler rotations|Euler rotations]]&lt;/small&gt;<br /> :: [[C1. Configuration of a mechanical system#C1.5 Independent coordinates|C1.5 Independent coordinates]]<br /> <br /> ===[[C2. Movement of a mechanical system]]===<br /> :: [[C2. Movement of a mechanical system#C2.1 Velocity of a particle|C2.1 Velocity of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.2 Acceleration of a particle|C2.2 Acceleration of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.3 Intrinsic components of the acceleration|C2.3 Intrinsic components of the acceleration]]<br /> :: [[C2. Movement of a mechanical system#C2.4 Angular velocity of a rigid body|C2.4 Angular velocity of a rigid body]]<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Simple rotation|Simple rotation]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Rotation in space|Rotation in space (Rotacions d&#039;Euler)]]&lt;/small&gt;<br /> :: [[C2. Movement of a mechanical system#C2.5 Angular acceleration of a rigid body|C2.5 Angular acceleration of a rigid body]]<br /> :: [[C2. Movement of a mechanical system#C2.6 Particle kinematics versus rigid body kinematicsrígid|C2.6 Particle kinematics versus rigid body kinematics]]<br /> :: [[C2. Movement of a mechanical system#C2.7 Degrees of freedom of a mechanical system|C2.7 Degrees of freedom of a mechanical system]]<br /> :: [[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|C2.8 Usual constraints in mechanical systems]]<br /> :: [[C2. Movement of a mechanical system#C2.E General examples|C2.E General examples]]<br /> <br /> ===[[C3. Composition of movements]]===<br /> :: [[C3. Composition of movements#C3.1 Composition of velocities|C3.1 Composition of velocities]]<br /> :: [[C3. Composition of movements#C3.2 Composition of accelerations|C3.2 Composition of accelerations]]<br /> :: [[C3. Composition of movements#C3.3 Composition versus time derivative|C3.3 Composition versus time derivative]]<br /> :: [[C3. Composition of movements#C3.E General examples|C3.E General examples]]<br /> <br /> ===[[C4. Rigid body kinematics]]===<br /> :: [[C4. Rigid body kinematics#C4.1 Velocity distribution|C4.1 Velocity distribution]]<br /> :: [[C4. Rigid body kinematics#C4.2 Accelerations distribution|C4.2 Accelerations distribution]]<br /> :: [[C4. Rigid body kinematics#C4.3 Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)|Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)]]<br /> ::[[C4. Rigid body kinematics#C4.4 Fixed axode and moving axode|C4.4 Fixed axode and moving axode]]<br /> ::[[C4. Rigid body kinematics#C4.E General examples|C4.E General examples]]<br /> <br /> ===[[C5. Rigid body kinematics: planar motion]]===<br /> :: [[C5. Rigid body kinematics: planar motion#C5.1 Instantaneous Center of Rotation (ICR)|C5.1 Instantaneous Center of Rotation (ICR)]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.2 Examples|C5.2 Examples]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.3 Introduction to vehicle kinematics|C5.3 Introduction to vehicle kinematics]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.E General examples|C5.E General examples]]<br /> <br /> ==DYNAMICS==<br /> <br /> ===[[D1. Foundational laws of Newtonian dynamics#|D1. Foundational laws of Newtonian dynamics]]===<br /> <br /> ===[[D2. Interaction forces between particles#|D2. Interaction forces between particles]]===<br /> <br /> ===[[D4. Vectorial theorems#|D4. Vectorial theorems]]===<br /> <br /> ===[[D5. Inertia tensor#|D5. Inertia tensor]]===<br /> ::[[D5. Inertia tensor#D5.1 Centre of masses|D5.1 Centre of masses]]<br /> ::[[D5. Inertia tensor#D5.2 Inertia tensor|D5.2 Inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.3 Some relevant properties of the inertia tensor|D5.3 Some relevant properties of the inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.4 Steiner’s Theorem|D5.4 Steiner’s Theorem]]<br /> ::[[D5. Inertia tensor#D5.5 Change of vector basis|D5.5 Change of vector basis]]<br /> <br /> ===[[D7. Examples of 3D dynamics#|D7. Examples of 3D dynamics]]===<br /> ::[[D7. Examples of 3D dynamics#D7.1 Analysis of the equations of motion|D7.1 Analysis of the equations of motion]]<br /> ::[[D7. Examples of 3D dynamics#D7.2 General examples|D7.2 General examples]]<br /> <br /> ===[[D8. Conservation of dynamic magnitudes#|D8. Conservation of dynamic magnitudes]]===<br /> ::[[D8. Conservation of dynamic magnitudes#D8.1 Examples|D8.1 Examples]]<br /> <br /> ==ENERGETICS==<br /> ::&#039;&#039;UNDER CONSTRUCTION&#039;&#039;<br /> <br /> <br /> -----------<br /> -----------<br /> ==Authors==<br /> <br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors Ana.png|alt=Ana Barjau Condomines|Ana Barjau Condomines|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1001000<br /> File:Autors Ernest4.jpg|alt=Ernest Bosch Soldevila|Ernest Bosch Soldevila|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1099864<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Ilustrations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Joaquim.png|alt=Joaquim Agulló i Batlle|Joquim Agulló i Batlle|link=https://www.agullobatlle.cat/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Editing and interactive animations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Arnau.png|alt=Arnau Marzábal Gatell|Arnau Marzábal Gatell|link=https://www.linkedin.com/in/arnau-marzabal/<br /> File:Autors_Berta.png|alt=Berta Ros Blanco|Berta Ros Blanco|link=https://www.linkedin.com/in/berta-ros/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Collaborators:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Daniel.png|alt=Daniel Clos Costa|Daniel Clos Costa|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1002252<br /> File:Autors_Rosa.png|alt=Rosa Pàmies Vilà|Rosa Pàmies Vilà|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1066910<br /> File:Autors_Albert.png|alt=Albert Peiret Giménez|Albert Peiret Giménez|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1115007<br /> File:Autors_Javier.png|alt=Javier Sistiaga Vidal-Ribas|Javier Sistiaga Vidal-Ribas|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1114855<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;<br /> [https://www.youtube.com/channel/UCqWvnHTViRPI1wHlUQXqH-Q &#039;&#039;&#039;Mechanics Lab&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://etseib.upc.edu/en &#039;&#039;&#039;Barcelona School of Industrial Engineering (ETSEIB)&#039;&#039;&#039;]<br /> <br /> [https://em.upc.edu/en &#039;&#039;&#039;Mechanical Engineering Department&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://www.upc.edu/en &#039;&#039;&#039;Polytechic University of Catalonia (UPC) · BarcelonaTech&#039;&#039;&#039;]<br /> &lt;/center&gt;<br /> <br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==Bibliographic references==<br /> Batlle, J. A., Barjau, A. (2020) “&#039;&#039;&#039;Rigid Body Kinematics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-47907-3<br /> <br /> Batlle, J. A., Barjau, A. (2022) “&#039;&#039;&#039;Rigid Body Dynamics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-84213-6<br /> <br /> Agulló, J. (2002) “&#039;&#039;&#039;Mecànica de la partícula i del sòlid rígid&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-6-1 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> Agulló, J. (2000) “&#039;&#039;&#039;Mecánica de la partícula i del sólido rígido&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-5-3 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> <br /> &lt;center&gt;<br /> &lt;gallery&gt;<br /> File:RBK portada.png|alt=Rigid body kinematics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-kinematics?format=HB&amp;isbn=9781108479073<br /> File:RBD portada.png|alt=Rigid body dynamics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136<br /> File:Llibre verd.png|alt=Mecànica de la partícula i del sòlid rígid|link=https://www.agullobatlle.cat/activitat-docent<br /> File:Llibre vermell.jpg|alt=Mecánica de la partícula i del sólido rígido|link=https://www.agullobatlle.cat/activitat-docent<br /> &lt;/gallery&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> <br /> <br /> <br /> [[File:Logo Lab Mec horitzontal.png|thumb|center|500px|link=https://em.upc.edu/en| ]]</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=794 2024-11-05T20:46:28Z

<p>Apons: /* ✏️ Example D1-7.1 */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4a&#039;&#039;&#039;). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4b&#039;&#039;&#039;); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \mathbf{F}_{\mathrm{ar} \rightarrow \mathrm{P}}^{-}+\mathbf{F}_{\text {cor } \rightarrow \mathrm{P}}^{-} &lt;/math&gt;.<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference frames are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> :: If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.&lt;br&gt;<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=793 2024-11-05T20:45:23Z

<p>Apons: /* ✏️ Example D1-7.1 */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4a&#039;&#039;&#039;). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4b&#039;&#039;&#039;); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \mathbf{F}_{\mathrm{ar} \rightarrow \mathrm{P}}^{-}+\mathbf{F}_{\text {cor } \rightarrow \mathrm{P}}^{-} &lt;/math&gt;.<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference frames are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039; with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> :: If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=792 2024-11-05T20:43:59Z

<p>Apons: /* ✏️ Example D1-7.1 */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4a&#039;&#039;&#039;). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4b&#039;&#039;&#039;); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \mathbf{F}_{\mathrm{ar} \rightarrow \mathrm{P}}^{-}+\mathbf{F}_{\text {cor } \rightarrow \mathrm{P}}^{-} &lt;/math&gt;.<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference frames are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> :: If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=791 2024-11-05T20:43:32Z

<p>Apons: /* D1.7 Particle dynamics in non Galilean reference frames */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4a&#039;&#039;&#039;). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4b&#039;&#039;&#039;); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \mathbf{F}_{\mathrm{ar} \rightarrow \mathrm{P}}^{-}+\mathbf{F}_{\text {cor } \rightarrow \mathrm{P}}^{-} &lt;/math&gt;.<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference frames are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> ::If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * ● Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * ● Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=790 2024-11-05T20:42:46Z

<p>Apons: /* D1.7 Particle dynamics in non Galilean reference frames */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4a&#039;&#039;&#039;). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4b&#039;&#039;&#039;); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \mathbf{F}_{\mathrm{ar} \rightarrow \mathrm{P}}^{-}+\mathbf{F}_{\text {cor } \rightarrow \mathrm{P}}^{-} &lt;/math&gt;.<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference framess are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> ::If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * ● Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * ● Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=789 2024-11-05T20:42:11Z

<p>Apons: /* D1.7 Particle dynamics in non Galilean reference frames */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4a&#039;&#039;&#039;). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4b&#039;&#039;&#039;); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \mathbf{F}_{\mathrm{ar} \rightarrow \mathrm{P}}^{-}+\mathbf{F}_{\text {cor } \rightarrow \mathrm{P}}^{-} &lt;/math&gt;.<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference framess are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> ::If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * ● Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * ● Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=788 2024-11-05T20:38:08Z

<p>Apons: /* D1.1 Galilean reference frames */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4a&#039;&#039;&#039;). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4b&#039;&#039;&#039;); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;,<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference framess are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> ::If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * ● Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * ● Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=787 2024-11-05T20:34:13Z

<p>Apons: /* D1.2 Galileo’s Principle of Relativity */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, Figure D1.4a). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, Figure D1.4b); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;,<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference framess are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> ::If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * ● Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * ● Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=786 2024-11-05T20:33:22Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, Figure D1.4a). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, Figure D1.4b); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;,<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference framess are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> ::If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * ● Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * ● Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=D1._Foundational_laws_of_Newtonian_dynamics&diff=785 2024-11-05T20:33:08Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\QPs}{\textrm{QP}}<br /> \newcommand{\ms}{\textrm{m}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\fs}{\textrm{f}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\js}{\textrm{j}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textbf{P}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\QPvec}{\vec{\Qs\Ps}}<br /> \newcommand{\OPvec}{\vec{\Os\textbf{P}}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \newcommand{\Pll}{\textbf{P}_\textrm{free}}<br /> \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)}<br /> \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)}<br /> \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)}<br /> &lt;/math&gt;<br /> <br /> &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Introduction#|&#039;&#039;&#039;Dynamics&#039;&#039;&#039;]]&lt;/span&gt; is the theory that studies the motion of a material body based on the physical factors that affect it. In Newtonian mechanics, the factors that affect the motion of particles are their mass and the forces on them. Neither of these two concepts (mass and force) is simple. But whatever definition we give to them, it is accepted that there is a unique correlation between them and the particle motion.<br /> <br /> The word “force” is often used in everyday life, and was already in use long before Newton formulated his laws of motion. The first notion of “force” is associated with the muscular sensation necessary to prevent or cause the movement of material bodies. By extension, anything that prevents or causes movement is also called “force.”<br /> <br /> There are two types of problems in dynamics, depending on the data and the unknowns: <br /> <br /> :* Unknown motion: based on the knowledge of all the forces acting on a mechanical system, the aim is to deduce the evolution of its movement.<br /> <br /> :* Unknown force: based on a predetermined movement (i.e., given the evolution of the system&#039;s coordinates), the aim is to predict the forces to be exerted on the system to achieve this movement.<br /> <br /> As any scientific theory, Newton&#039;s mechanics is based on a set of principles (laws or axioms) that cannot be demonstrated, and whose plausibility is justified by experimental results, both real (obtained in a laboratory) and conceptual (obtained through strict, purely mental reasoning). These laws are presented in Newton&#039;s main work (&#039;&#039;Philosophiæ Naturalis Principia Mathematica&#039;&#039;), and aimed to solve the dynamics of a particle. The theorems that address the dynamics of more complex material systems (such as multibody systems) are deduced from these laws.<br /> <br /> In this section, the foundational laws of Newtonian mechanics are stated and discussed as they appear in the &#039;&#039;Principia Mathematica&#039;&#039; (although in more modern language). The discussion of the axiomatic problems they present has been the subject of much literature in the history of science. A rigorous and brief exposition is given here. A more extensive exposition, containing the reformulation made by Ernst Mach and other information of interest, can be found in [https://cambridge.org/es/universitypress/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136 &#039;&#039;&#039;Rigid Body Dynamics, Batlle&amp;Barjau, chapter 1&#039;&#039;&#039;].<br /> <br /> ==D1.1 Galilean reference frames==<br /> <br /> The relationship between forces and motion of a particle P can be expressed generically by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{motion}_{\Rs}(\Ps) = f_{\Rs}(\text{forces})&lt;/math&gt;<br /> &lt;/center&gt;<br /> This equation shows that, since the motion of &lt;math&gt;\Ps&lt;/math&gt; depends on the reference frame R from which it is measured, the term on the right (and therefore the forces) can also depend on R. <br /> <br /> The observation of simple mechanical phenomena suggests that the origin of the forces that prevent or cause the motion of &lt;math&gt;\Ps&lt;/math&gt; is not only related to the existence of material objects (as intuition says). A simple example makes this clear.<br /> <br /> Let us consider a small object (a particle &lt;math&gt;\Ps&lt;/math&gt;) attached to a frame by two identical springs. The frame is fixed to a perfectly smooth horizontal surface (such as an icy ground). If we only consider the horizontal motion of P relative to the reference frame attached to that surface (R), the forces associated with material objects can only come from the springs, since there is no interaction with the smooth surface.<br /> <br /> From an initial time instant &lt;math&gt;t_0&lt;/math&gt; in which the particle is at rest with respect to the surface &lt;math&gt;(\vel{P,$\ts_0$}{R} = \vec{0})&lt;/math&gt; and the springs have the same length, we observe the evolution of its movement from the surface (the evolution of its speed relative to R, &lt;math&gt;\vel{P}{R}&lt;/math&gt;, &#039;&#039;&#039;Figure D1.1&#039;&#039;&#039;).<br /> <br /> <br /> [[File:D1-1-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.1&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a horizontal smooth surface.&lt;/small&gt;&lt;/center&gt;<br /> <br /> Different evolutions can be seen, depending on the movement of the surface relative to the ground:<br /> <br /> :* If the surface is that of a wagon moving uniformly relative to the ground (reference frame R1): regardless of the frame location, its orientation relative to the wagon, and the time at which the experiment is performed, particle &lt;math&gt;\Ps&lt;/math&gt; does not move relative to the wagon, and the springs maintain their length (&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039;). In other words: the result is independent from the position and the orientation of the springs frame, and from the time coordinate. This is equivalent to saying that, for this experiment, &lt;u&gt;the space of that reference frame is homogeneous and isotropic, and time is uniform&lt;/u&gt;. Therefore, the reference frame does not play any part in the result.<br /> <br /> [[File:D1-2-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.2&#039;&#039;&#039; A particle between two springs fixed to a frame attached to a wagon with uniform rectilinear motion.<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, and the springs deform with opposite signs (one stretches and the other shortens, &#039;&#039;&#039;Figure D1.3a&#039;&#039;&#039;); if it is oriented in the transverse direction of the wagon, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame forward &lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, and the change of the springs length is the same (&#039;&#039;&#039;Figure D1.3b&#039;&#039;&#039;); if it is oriented in any other direction, the initial velocity of P has two components (&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0&lt;/math&gt;,&lt;math&gt;\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0&lt;/math&gt;, &#039;&#039;&#039;Figure D1.3c&#039;&#039;&#039;). To summarize: the result of this experiment is &lt;math&gt;\vel{P,t}{R} \neq 0&lt;/math&gt;, and it is independent from the position and the time instant, but not from the orientation. For this experiment, &lt;u&gt;the space of that reference frame is homogeneous but not isotropic, and time is uniform&lt;/u&gt;.<br /> <br /> If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.<br /> <br /> [[File:D1-3-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.3&#039;&#039;&#039; A particle between two springs attached to a frame fixed to a wagon with rectilinear motion and constant braking acceleration.&lt;/small&gt;&lt;/center&gt;<br /> <br /> :* If the surface is that of a horizontal platform, with its centre fixed to the ground and rotating with constant angular velocity relative to the ground (reference frame R3), the initial evolution of the &lt;math&gt;\Ps&lt;/math&gt; motion from rest relative to the platform depends on the initial position and the orientation of the frame: if &lt;math&gt;\Ps&lt;/math&gt; is initially at the centre of the platform, it does not move independently of the orientation of the frame (&lt;math&gt;\vel{P,t}{R} = \vec{0}&lt;/math&gt;, Figure D1.4a). If it is placed in a different position and the orientation of the frame is according to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the longitudinal direction of the frame outwards, so that the inner spring is stretched and the outer one is compressed (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)&lt;/math&gt;, Figure D1.4b); if the orientation of the frame is perpendicular to a radius of the platform, &lt;math&gt;\Ps&lt;/math&gt; initially moves in the transverse direction of the frame outwards, but the two springs have the same length (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)&lt;/math&gt;, Figure D1.4b); If the frame is placed in any other orientation relative to the platform, the initial velocity of &lt;math&gt;\Ps&lt;/math&gt; has two components, and the springs have different lengths (&lt;math&gt;(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}} \neq 0)&lt;/math&gt;, &#039;&#039;&#039;Figure D1.4c&#039;&#039;&#039;). The result of this experiment, then, depends on the position and the orientation, but not on the time instant. For this experiment, &lt;u&gt;the space of the reference frame is neither homogeneous nor isotropic, but time is uniform&lt;/u&gt;.<br /> <br /> [[File:D1-4-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.4&#039;&#039;&#039; Evolution of the movement of a particle between two springs attached to a frame fixed to a platform that rotates with constant angular velocity relative to the ground.&lt;/small&gt;&lt;/center&gt;<br /> <br /> In all the experiments described, the material objects that exert forces on the particle are the same (the springs), but the influence of the reference frame where the observations are made is different: the reference frame R1 does not participate in the result, while R2 and R3 participate through the characteristics of their space and time (&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039;).<br /> <br /> [[File:D1-5-eng.png|350px|thumb|center|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.5&#039;&#039;&#039; Space and time characteristics in the reference frames of the wagon and the platform for the experiment of the particle between springs&lt;/small&gt;&lt;/center&gt;<br /> <br /> A reference frame where time is uniform and space is homogeneous and isotropic is called a &#039;&#039;&#039;Galilean&#039;&#039;&#039; or &#039;&#039;&#039;inertial reference frame&#039;&#039;&#039;. Newton states that time and space in this type of reference are “absolute”, they are realities that exist independently of everything else, and they constitute a “neutral” (or passive) scenario. The study of dynamics in non-Galilean (non-inertial) references leads to the introduction of forces – called &#039;&#039;&#039;inertia forces&#039;&#039;&#039; – that come from the reference itself.<br /> <br /> But it is one thing to define a Galilean reference frame, and another to accept that one exists (how can the uniformity of time and the homogeneity and isotropy of space be verified?). In the &#039;&#039;Principia Mathematica&#039;&#039;, before stating his laws, Newton postulates the existence of a Galilean reference frame. <br /> <br /> Newton&#039;s first two laws provide two additional ways of verifying the Galilean character of a reference frame, which will be discussed in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;sections D1.4&#039;&#039;&#039;]]&lt;/span&gt; and &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|&#039;&#039;&#039;D1.5&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.2 Galileo’s Principle of Relativity==<br /> <br /> A &#039;&#039;&#039;principle of relativity&#039;&#039;&#039; establishes the set of reference framess (or space-time frames) of validity of a theory, and it is fundamental in any scientific discipline in the field of physics. A law (or the corresponding mathematical equation) can be fulfilled in a reference frame R but not in a reference frame R’.<br /> <br /> Newtonian mechanics is also based on a principle of relativity: &#039;&#039;&#039;Galileo&#039;s Principle of Relativity&#039;&#039;&#039;. This principle states that all Galilean reference frames are equivalent for the formulation of the laws that govern the dynamics of mechanical systems. In other words, Galilean references frames are indistinguishable when material objects interact with each other. In a generic way, this can be expressed by the qualitative equation:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}{\textrm{Gal}}(\Ps) = f(\text{forces}),&lt;/math&gt;<br /> &lt;/center&gt;<br /> where the forces represent these interactions (and never come from the reference frame), and their formulation must be the same in all Galilean reference frames.<br /> <br /> <br /> -----------<br /> -----------<br /> ==D1.3 Newton’s Principle of Determinacy==<br /> <br /> In Newton’s &#039;&#039;Principia Mathematica&#039;&#039;, the Principle of Determinacy precedes the three laws of motion. Newton does not postulate it as a law but presents it as an observation. One possible formulation of this principle is “the initial positions and velocities (at a certain time instant) of all the particles of an isolated mechanical system determine univocally its future motion.” This is equivalent to saying that the acceleration (relative to a Galilean reference frame) of a particle &lt;math&gt;\Ps&lt;/math&gt; of the system at time &lt;math&gt;t_0&lt;/math&gt; depends exclusively on the mechanical state (positions and velocities) of the system at time &lt;math&gt;t_0&lt;/math&gt;. Mechanical systems, therefore, do not have memory.<br /> <br /> Using this principle, the qualitative equation that relates the motion of a particle to the forces exerted on it can be written in a more precise way:<br /> &lt;center&gt;<br /> &lt;math&gt;\text{movement}_{\Rs}(\Ps_i) = f_{\Rs} (\text{forces}) \implies \acc{$\Ps_\is$,t}{Gal}=f \left[\text{forces}(\vec{\Os_{\textrm{Gal}}\Ps_\js}(\ts_0),\vel{$\Ps_\js$,t}{Gal})\right]&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Based on empirical observations (basically astronomical) accumulated over time, Newton concludes that the knowledge of initial positions and velocities is sufficient to predict the evolution of mechanical systems.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.4 Newton’s first law (inertia law)==<br /> <br /> The &#039;&#039;&#039;law of inertia&#039;&#039;&#039; is the solution to the simplest dynamic problem we can imagine: that of the free particle &lt;math&gt;\Pll&lt;/math&gt; (a particle in an empty universe). This law postulates that the acceleration of the free particle in a Galilean reference frame is zero, regardless of its initial speed: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\text{Newton&#039;s first law (inertia law):} \:\:\:\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In fact, what constitutes a law (and therefore cannot be proved) is that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}=0&lt;/math&gt;. Indeed:<br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; we observe &lt;math&gt;\Pll&lt;/math&gt; from a Galilean reference frame and see that it has zero speed, this state of rest will have to be maintained, since acquiring a speed different from zero would mean starting to move in a specific direction. But this would be privileging one direction, and that is not compatible with the isotropy of space. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}=\vec{0}\Rightarrow\agal{\Pll,\ts_0} = \vec{0}&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6a&#039;&#039;&#039;).<br /> <br /> :* If at time &lt;math&gt;\ts_0&lt;/math&gt; the speed is not zero (&lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}&lt;/math&gt;), its direction will have to be maintained, since otherwise one direction would be privileged again. Therefore, &lt;math&gt;\vgal{\Pll,\ts_0}\neq\vec{0}\Rightarrow\accn{$\Pll,t_0$}{Gal}=0&lt;/math&gt; (&#039;&#039;&#039;Figure D1.6b&#039;&#039;&#039;).<br /> <br /> In this second case, we can only state that the normal component of the acceleration will be zero as a consequence of the isotropy of space. The Galilean nature of the reference frame does not allow us to ensure that the tangential component has to be zero as well. Indeed, it could be that &lt;math&gt;\accs{$\Pll,t_0$}{Gal}\neq0&lt;/math&gt;, and then the value of the speed would change. If at some point the speed were to reach a zero value, the movement could not be restarted.<br /> <br /> <br /> [[File:D1-6-eng.png|thumb|center|350px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.6&#039;&#039;&#039; Newton’s first law<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Some people claim that Newton&#039;s first law is a special case of the second law, but this is not the case. The utility of the first law is not to solve a problem that will never arise (the universe is full of particles!), but to provide an alternative definition of Galilean reference frame: one where &lt;math&gt;\acc{$\Pll$}{R} = \vec{0}&lt;/math&gt; (R=Gal). The references where &lt;math&gt;\acc{$\Pll$}{R} = \fs_\Rs(\text{space}_\Rs,\text{time}_\Rs)\neq\vec{0}&lt;/math&gt; are not Galilean (R=NGal).<br /> <br /> If &lt;math&gt;\acc{$\Pll$}{R}=\vec{0}&lt;/math&gt; is fulfilled in a reference frame R (and therefore R=Gal), then there is an infinite family of reference frames where this equation is also true: all those that have a rectilinear and uniform translational motion relative to R (a simple composition of accelerations proves it). All the reference frames in this family are equivalent (indistinguishable) when it comes to studying the dynamics of the free particle.<br /> <br /> Strictly determining the Galilean (inertial) character of a reference frame is formally impossible: neither can the free particle experiment be performed (we cannot empty the universe!) nor can it be verified whether spacetime is homogeneous, isotropic and uniform (since space and time are infinite!).<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.5 Newton’s second law (fundamental law of dynamics)==<br /> <br /> &#039;&#039;&#039;Newton’s second law&#039;&#039;&#039; formulates the dynamics of the material particle &#039;&#039;&#039;P&#039;&#039;&#039; that interacts with other material particles &#039;&#039;&#039;Q&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;,<br /> &lt;/center&gt;<br /> where &lt;math&gt;\ms_\Ps&lt;/math&gt; is the &#039;&#039;&#039;P&#039;&#039;&#039; mass, and &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; are the forces exerted on &#039;&#039;&#039;P&#039;&#039;&#039; by the &#039;&#039;&#039;Q&#039;&#039;&#039; particles.<br /> <br /> The &lt;math&gt;\ms_\Ps&lt;/math&gt; parameter appears to be an intrinsic characteristic of &#039;&#039;&#039;P&#039;&#039;&#039;, and therefore to be associated exclusively with this particle. Implicitly, Newton accepts that not all particles are equal, and that the only property that differentiates them is their “mass.” Before stating this law, Newton defines “mass” as a constant parameter that corresponds to the “quantity of matter,” but he does not provide any way to measure it.<br /> <br /> In order to understand what this equation is saying, it is useful to consider the simplest dynamical problem (after the free particle problem): a universe with only two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;. Thus, the sum on the left vanishes, and:<br /> <br /> :* Due to the space isotropy in Galilean reference frames, the acceleration of &#039;&#039;&#039;P&#039;&#039;&#039; generated by &#039;&#039;&#039;Q&#039;&#039;&#039; must necessarily have the direction &lt;math&gt;\QPvec&lt;/math&gt;. It is an acceleration of attraction (approach) or repulsion (separation).<br /> <br /> :* Consequently, the &lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}&lt;/math&gt; force that &#039;&#039;&#039;Q&#039;&#039;&#039; exerts on &#039;&#039;&#039;P&#039;&#039;&#039; will also have the &lt;math&gt;\QPvec&lt;/math&gt; direction.<br /> :* If we eliminate the &#039;&#039;&#039;Q&#039;&#039;&#039; particle, the parameter &lt;math&gt;\ms_\Ps&lt;/math&gt; becomes irrelevant: &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}&lt;/math&gt; regardless of its mass. Mass, then, only manifests itself in interaction.<br /> <br /> Let us now go back to the general case of many particles interacting with &#039;&#039;&#039;P&#039;&#039;&#039;. Newton&#039;s second law contains a &#039;&#039;&#039;principle of superposition&#039;&#039;&#039;: the resultant force acting on &#039;&#039;&#039;P&#039;&#039;&#039; is the sum of those that each of the other particles would separately exert on &#039;&#039;&#039;P&#039;&#039;&#039;: the simultaneous existence of several particles does not alter the interactions between them (&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039;).<br /> <br /> [[File:D1-7-eng.png|thumb|center|500px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.7&#039;&#039;&#039; Principle of superposition<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> Newton&#039;s second law can be used to solve both &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Foundational laws of Newtonian dynamics#|&#039;&#039;&#039;types of dynamic problems&#039;&#039;&#039;]]&lt;/span&gt; described in the introduction to this section, depending on whether the data are the forces acting on P or the P acceleration. Since this is a vector equation (three scalar equations), it may happen, for example, that the acceleration is known in one direction (component) but not in the other two. Then, only one component of the resultant force on P is unknown.<br /> <br /> To solve problems where the unknown is acceleration (the three components), it is necessary to be able to formulate the interaction forces between particles. The Principle of Action and Reaction, the Principle of Determinacy and the Galileo’s Principle of Relativity condition these mathematical formulations. The specific formulation for each type of interaction is obtained from experiments, and is presented in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D2. Interaction forces between particles#|&#039;&#039;&#039;unit D2&#039;&#039;&#039;]]&lt;/span&gt;. <br /> <br /> Newton&#039;s second law allows us to evaluate the Galilean character of a reference frame: for practical purposes, a reference frame is accepted as Galilean when the resolution of dynamic problems does not require the inclusion of forces not associated with material objects. Thus, the Galilean character depends on the type of problems being solved. For short-range problems (which are usually those that are dealt with in mechanical engineering), the Earth behaves as a Galilean reference frame. For medium and long-range problems (ballistics, aeronautics, astronomy...), this is not the case.<br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.6 Newton’s third law (action-reaction principle)==<br /> <br /> The second law and the isotropy of space in Galilean reference frames have shown that the interaction forces between pairs of particles (&lt;math&gt;\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}},\overline{\mathbf{F}}_{\mathrm{P} \rightarrow \mathrm{Q}}&lt;/math&gt;) must have the direction &lt;math&gt;\QPvec&lt;/math&gt; defined by the particles. Newton&#039;s third law ensures that they must be attractive or repulsive, and must have the same value (&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039;).<br /> <br /> [[File:D1-8-eng.png|thumb|center|300px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.8&#039;&#039;&#039; Principle of action-reaction<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> This principle provides essential information about the interaction forces between two particles, and for many scientists it is Newton&#039;s most important law because it introduces symmetry into the description of interactions: &lt;u&gt;each interaction is described by a single magnitude&lt;/u&gt;. &quot;Force&quot; is not something possessed by a particle, but is associated with pairs of particles.<br /> <br /> From this principle, it is possible to obtain a more comprehensive definition of “mass” than the one Newton provides at the beginning of the &#039;&#039;&#039;Principia&#039;&#039;&#039;. Given two interacting particles &#039;&#039;&#039;P&#039;&#039;&#039; and &#039;&#039;&#039;Q&#039;&#039;&#039;, since the mutual force they exert on each other has the same value, their &#039;&#039;&#039;mass ratio&#039;&#039;&#039; (&lt;math&gt;\mu_{\QPs}=\ms_\textrm{Q}/\ms_\Ps&lt;/math&gt;) coincides with their acceleration ratio (&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039;). Thus, the greater the mass, the lower the acceleration, and vice versa. The mass of a particle, then, can be interpreted as the difficulty it has in changing speed. This is an &lt;u&gt;inertial interpretation&lt;/u&gt; of mass.<br /> <br /> <br /> [[File:D1-9-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.9&#039;&#039;&#039; Mass ratio: univocally defined for pairs of interacting particles<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> If we choose a mass standard (a particular value of mass for a particular particle, for example &lt;math&gt;\ms_\textrm{Q}=&lt;/math&gt;1Kg), the mass of every other particle is determined (&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039;). Newton postulates that this mass is constant and intrinsic to each particle.<br /> <br /> [[File:D1-10-neut.png|thumb|center|450px|link=]]<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Figure D1.10&#039;&#039;&#039; Assessment of the mass of each particle given a standard<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==D1.7 Particle dynamics in non Galilean reference frames==<br /> <br /> As stated in &lt;span style=&quot;text-decoration: underline;&quot;&gt; [[D1. Lleis fundacionals de la mecànica newtoniana#D1.1 Galilean reference frames|&#039;&#039;&#039;section D1.1&#039;&#039;&#039;]]&lt;/span&gt;, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:<br /> <br /> &lt;center&gt;<br /> Dynamics in Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> Dynamics in non Galilean reference frames: &lt;math&gt;\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;&lt;br&gt;<br /> &lt;/center&gt;<br /> <br /> Subtracting the first equation from the second:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\rightarrow \mathrm{P}}^{-}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> If we do a composition of movements to relate &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})&lt;/math&gt; and &lt;math&gt;\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})&lt;/math&gt;:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \left.\begin{array}{l}<br /> \mathrm{AB}=\mathrm{Gal} \\<br /> \mathrm{REL}=\mathrm{NGal}<br /> \end{array}\right\} \Rightarrow \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})=\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})-\overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-2 \overline{\mathbf{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})<br /> &lt;/math&gt;,<br /> &lt;/center&gt;&lt;br&gt;<br /> <br /> The inertia force may be decomposed into two forces: the &#039;&#039;&#039;inertia transportation force&#039;&#039;&#039; and the &#039;&#039;&#039;Coriolis inertia force&#039;&#039;&#039;:<br /> &lt;center&gt;<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> Both depend on kinematic magnitudes of the non-Galilean reference frame (NGal) with respect to the Galilean reference frame (Gal), and therefore are different for each NGal reference frame (non-Galilean reference framess are distinguishable in Newtonian dynamics!).<br /> <br /> It should be noted that these are not interaction forces. Thus, the transportation force does not come from a physical dragging of the particle by the reference frame.<br /> <br /> In static situations (absence of movement of &#039;&#039;&#039;P&#039;&#039;&#039; relative to NGal, &lt;math&gt;\overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})=\overline{0}&lt;/math&gt;) or when the reference NGal has a translational motion relative to a Galilean reference frame (&lt;math&gt;\overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}}=\overline{0}&lt;/math&gt;), the Coriolis inertia force is zero.<br /> <br /> <br /> ====✏️ Example D1-7.1 ====<br /> ------<br /> <br /> ::[[File:ExD1-7-1-eng.png|thumb|left|250px|link=]]<br /> &lt;small&gt;<br /> :: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively. <br /> <br /> ::If &#039;&#039;&#039;Q&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;P&#039;&#039;&#039; from the ground reference&lt;/u&gt;, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br /> <br /> ::If &#039;&#039;&#039;P&#039;&#039;&#039; studies the &lt;u&gt;dynamics of &#039;&#039;&#039;Q&#039;&#039;&#039; from the rotating platform reference frame&lt;/u&gt; (which has the centre &#039;&#039;&#039;O&#039;&#039;&#039; fixed to the ground and rotates with constant angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:<br /> <br /> * ● Inertia transportation force &lt;math&gt;\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}&lt;/math&gt;: as the Q &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;transportation motion&#039;&#039;&#039;]]&lt;/span&gt; is circular with centre &#039;&#039;&#039;O&#039;&#039;&#039;, radius R (&lt;math&gt;=\mid \OQvec \mid&lt;/math&gt;) and associated angular velocity &lt;math&gt;\Omega_0&lt;/math&gt;, the transportation force will be radial and outwards (centrifugal) with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> * ● Coriolis inertia force : it is obtained from the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C3. Composition of movements#C3.2 Composition of accelerations|&#039;&#039;&#039;Coriolis acceleration&#039;&#039;&#039;]]&lt;/span&gt;:<br /> &lt;math&gt;\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})&lt;/math&gt;&lt;br&gt;<br /> <br /> As the transportation acceleration is orthogonal to &lt;math&gt;\OQvec&lt;/math&gt; and &lt;math&gt;\overline{\boldsymbol{\Omega}}_0&lt;/math&gt; is vertical, the Coriolis force is radial centripetal (towards &#039;&#039;&#039;O&#039;&#039;&#039;), with value &lt;math&gt;\ms\Rs\Omega_0^2&lt;/math&gt;.<br /> <br /> <br /> &lt;center&gt;[[File:ExD1-7-2-eng.png|thumb|center|300px|link=]]&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> <br /> <br /> C5. Rigid body kinematics: planar motion<br /> <br /> &lt;center&gt;<br /> [[C5. Rigid body kinematics: planar motion|&lt;&lt;&lt; C5. Rigid body kinematics: planar motion]]<br /> <br /> [[D2. Interaction forces between particles|D2. Interaction forces between particles &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=732 2024-10-24T00:33:00Z

<p>Apons: /* 3. Find the velocity and the acceleration of point G of the ring relative to the ground. */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\OPvec}{\vec{\Os\Ps}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.3: rotating pendulum with oscillating articulation point====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The ring-shaped pendulum is articulated to the support, which is linked to the guide through a prismatic joint. The guide is articulated to the ceiling, and its angular velocity relative to the ceiling &lt;math&gt;(\vec{\psio})&lt;/math&gt; is constant. The spring between support and guide guarantees that the former does not fall to the ground when the system is at rest.<br /> <br /> [[File:C2-E.Ex3-1-eng.png|thumb|center|600px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The guide may rotate about the vertical axis through &lt;math&gt;\Os &#039;&lt;/math&gt; (simple rotation).<br /> <br /> :Independently form that rotation, the support may have a translational motion along the guide (rectilinear translational motion).<br /> <br /> :Finally, if those two motions are blocked, the ring may still have a simple rotation about the horizontal axis through &lt;math&gt;\Os &#039;&lt;/math&gt;, which is perpendicular to the ring plane and is fixed to the support.<br /> <br /> :Hence, the system has 3 degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the ring relative to the ground. =====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :The angular velocity of the ring is the superposition of &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[File:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{ring}{E}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{ring}{E}=\dert{\velang{ring}{E}}{E}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{E}=\dert{\vec{\psio}}{E}+\dert{\vec{\dot{\theta}}}{E}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{E}+\dert{(\odot\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The angular acceleration comes exclusively from the change of value and direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, as &lt;math&gt;\vec{\psio}&lt;/math&gt; has both constant value and constant direction.<br /> <br /> :&lt;math&gt;\accang{ring}{E} = \dert{\velang{ring}{E}}{E} = \dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> :The time derivative of the angular velocity of the ring may be done analytically. The vector basis where the projection of &lt;math&gt;\velang{ring}{E}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{ring}{E}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{ring}{E}}{B}=\braq{\dert{\velang{ring}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{ring}{E}}{B}+\braq{\velang{B}{E}<br /> }{B}\times\braq{\velang{ring}{E}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the ring relative to the ground.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; is a position vector for &lt;math&gt;\Gs&lt;/math&gt; in the ground frame, as &lt;math&gt;\Os&#039;&lt;/math&gt; is a point fixed to the ground.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{E}=\dert{\vec{\Os&#039;\Gs}}{E}=\dert{\vec{\Os&#039;\Os}}{E}+\dert{\vec{\Os\Gs}}{E}=\dert{(\downarrow \textrm{x})}{E}+\dert{(\searrow \Ls)}{E}&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex3-3-eng.png|thumb|right|250px|link=]]<br /> <br /> :The term &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; has a variable value but a constant orientation, whereas the term &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, with constant value, changes its orientation relative to the ground because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{E}=\dert{(\downarrow \textrm{x})}{E}=[\text{change of value}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{E}=\dert{(\searrow \Ls)}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Thus, &lt;math&gt;\vel{G}{E}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{G}{E}}{E}=\dert{(\downarrow\dot{\text{x}})}{E}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The three terms of the velocity have variable value, but just the last two rotate (change their orientation) relative to the ground. The second one, which is perpendicular to the ring plane, rotates jst because of &lt;math&gt;\vec{\psio}&lt;/math&gt;, whereas the third one rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;.The time derivatives of those terms are:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{E}=[\text{change of value}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\acc{G}{E}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[File:C4-E-Ex3-3-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Analytical calculation: &#039;&#039;&#039; <br /> <br /> :The whole calculation can be done analytically. The first term in &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; is vertical, hence its projection on the vector basis B fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\psio})&lt;/math&gt; is straightforward; however, the second term can be easily projected on the B’ vector basis fixed to the ring &lt;math&gt;(\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Any of these two vector bases is suitable.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Calculation with the B vector basis&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B}=\braq{\dert{\OGvec}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B}=\braq{\dert{\vel{G}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{G}{E}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Calculation in the B’ vector basis&lt;span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTE:&#039;&#039;&#039; In this web (for lack of more precise symbols), though the arrows s &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; and &lt;math&gt;\searrow&lt;/math&gt; seem to indicate that the vectors form a 45° angle with the vertical direction, this does not have to be the case. The arrows must be interpreted qualitatively, observing the figure that is always included when using this type of notation. For instance, in section 3 of exercise C2-E.1, the &lt;math&gt;\OPvec&lt;/math&gt; vector forms a generic &lt;math&gt;\theta&lt;/math&gt; angle with the vertical direction. If the value of &lt;math&gt;\theta&lt;/math&gt; is less than 90° (as in the following figure), the &lt;math&gt;\OPvec&lt;/math&gt; vector has a downward and rightward component.<br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=731 2024-10-24T00:31:47Z

<p>Apons: /* 3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra. */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\OPvec}{\vec{\Os\Ps}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.3: rotating pendulum with oscillating articulation point====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The ring-shaped pendulum is articulated to the support, which is linked to the guide through a prismatic joint. The guide is articulated to the ceiling, and its angular velocity relative to the ceiling &lt;math&gt;(\vec{\psio})&lt;/math&gt; is constant. The spring between support and guide guarantees that the former does not fall to the ground when the system is at rest.<br /> <br /> [[File:C2-E.Ex3-1-eng.png|thumb|center|600px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The guide may rotate about the vertical axis through &lt;math&gt;\Os &#039;&lt;/math&gt; (simple rotation).<br /> <br /> :Independently form that rotation, the support may have a translational motion along the guide (rectilinear translational motion).<br /> <br /> :Finally, if those two motions are blocked, the ring may still have a simple rotation about the horizontal axis through &lt;math&gt;\Os &#039;&lt;/math&gt;, which is perpendicular to the ring plane and is fixed to the support.<br /> <br /> :Hence, the system has 3 degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the ring relative to the ground. =====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :The angular velocity of the ring is the superposition of &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[File:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{ring}{E}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{ring}{E}=\dert{\velang{ring}{E}}{E}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{E}=\dert{\vec{\psio}}{E}+\dert{\vec{\dot{\theta}}}{E}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{E}+\dert{(\odot\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The angular acceleration comes exclusively from the change of value and direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, as &lt;math&gt;\vec{\psio}&lt;/math&gt; has both constant value and constant direction.<br /> <br /> :&lt;math&gt;\accang{ring}{E} = \dert{\velang{ring}{E}}{E} = \dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> :The time derivative of the angular velocity of the ring may be done analytically. The vector basis where the projection of &lt;math&gt;\velang{ring}{E}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{ring}{E}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{ring}{E}}{B}=\braq{\dert{\velang{ring}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{ring}{E}}{B}+\braq{\velang{B}{E}<br /> }{B}\times\braq{\velang{ring}{E}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the ring relative to the ground.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; is a position vector for &lt;math&gt;\Gs&lt;/math&gt; in the ground frame, as &lt;math&gt;\Os&#039;&lt;/math&gt; is a point fixed to the ground.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{E}=\dert{\vec{\Os&#039;\Gs}}{E}=\dert{\vec{\Os&#039;\Os}}{E}+\dert{\vec{\Os\Gs}}{E}=\dert{(\downarrow \textrm{x})}{E}+\dert{(\searrow \Ls)}{E}&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex3-3-eng.png|thumb|right|250px|link=]]<br /> <br /> :The term &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; has a variable value but a constant orientation, whereas the term &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, with constant value, changes its orientation relative to the ground because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{E}=\dert{(\downarrow \textrm{x})}{E}=[\text{change of value}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{E}=\dert{(\searrow \Ls)}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Thus, &lt;math&gt;\vel{G}{E}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{G}{E}}{E}=\dert{(\downarrow\dot{\text{x}})}{E}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The three terms of the velocity have variable value, but just the last two rotate (change their orientation) relative to the ground. The second one, which is perpendicular to the ring plane, rotates jst because of &lt;math&gt;\vec{\psio}&lt;/math&gt;, whereas the third one rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;.The time derivatives of those terms are:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{E}=[\text{change of value}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\acc{G}{E}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C4-E-Ex3-3-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Analytical calculation: &#039;&#039;&#039; <br /> <br /> :The whole calculation can be done analytically. The first term in &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; is vertical, hence its projection on the vector basis B fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\psio})&lt;/math&gt; is straightforward; however, the second term can be easily projected on the B’ vector basis fixed to the ring &lt;math&gt;(\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Any of these two vector bases is suitable.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Calculation with the B vector basis&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B}=\braq{\dert{\OGvec}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B}=\braq{\dert{\vel{G}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{G}{E}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Calculation in the B’ vector basis&lt;span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTE:&#039;&#039;&#039; In this web (for lack of more precise symbols), though the arrows s &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; and &lt;math&gt;\searrow&lt;/math&gt; seem to indicate that the vectors form a 45° angle with the vertical direction, this does not have to be the case. The arrows must be interpreted qualitatively, observing the figure that is always included when using this type of notation. For instance, in section 3 of exercise C2-E.1, the &lt;math&gt;\OPvec&lt;/math&gt; vector forms a generic &lt;math&gt;\theta&lt;/math&gt; angle with the vertical direction. If the value of &lt;math&gt;\theta&lt;/math&gt; is less than 90° (as in the following figure), the &lt;math&gt;\OPvec&lt;/math&gt; vector has a downward and rightward component.<br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:C2-E.Ex3-3-eng.png&diff=730 2024-10-24T00:21:23Z

<p>Apons: </p> <hr /> <div>C2-E.Ex3-3-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=729 2024-10-24T00:18:34Z

<p>Apons: /* 3. Find the velocity and the acceleration of point Q of the plate relative to the ground. */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\OPvec}{\vec{\Os\Ps}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.3: rotating pendulum with oscillating articulation point====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The ring-shaped pendulum is articulated to the support, which is linked to the guide through a prismatic joint. The guide is articulated to the ceiling, and its angular velocity relative to the ceiling &lt;math&gt;(\vec{\psio})&lt;/math&gt; is constant. The spring between support and guide guarantees that the former does not fall to the ground when the system is at rest.<br /> <br /> [[File:C2-E.Ex3-1-eng.png|thumb|center|600px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The guide may rotate about the vertical axis through &lt;math&gt;\Os &#039;&lt;/math&gt; (simple rotation).<br /> <br /> :Independently form that rotation, the support may have a translational motion along the guide (rectilinear translational motion).<br /> <br /> :Finally, if those two motions are blocked, the ring may still have a simple rotation about the horizontal axis through &lt;math&gt;\Os &#039;&lt;/math&gt;, which is perpendicular to the ring plane and is fixed to the support.<br /> <br /> :Hence, the system has 3 degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the ring relative to the ground. =====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :The angular velocity of the ring is the superposition of &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[File:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{ring}{E}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{ring}{E}=\dert{\velang{ring}{E}}{E}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{E}=\dert{\vec{\psio}}{E}+\dert{\vec{\dot{\theta}}}{E}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{E}+\dert{(\odot\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The angular acceleration comes exclusively from the change of value and direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, as &lt;math&gt;\vec{\psio}&lt;/math&gt; has both constant value and constant direction.<br /> <br /> :&lt;math&gt;\accang{ring}{E} = \dert{\velang{ring}{E}}{E} = \dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> :The time derivative of the angular velocity of the ring may be done analytically. The vector basis where the projection of &lt;math&gt;\velang{ring}{E}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{ring}{E}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{ring}{E}}{B}=\braq{\dert{\velang{ring}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{ring}{E}}{B}+\braq{\velang{B}{E}<br /> }{B}\times\braq{\velang{ring}{E}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=728 2024-10-24T00:17:06Z

<p>Apons: /* 🔎 Example C2-E.3: rotating pendulum with oscillating articulation point */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\OPvec}{\vec{\Os\Ps}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.3: rotating pendulum with oscillating articulation point====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The ring-shaped pendulum is articulated to the support, which is linked to the guide through a prismatic joint. The guide is articulated to the ceiling, and its angular velocity relative to the ceiling &lt;math&gt;(\vec{\psio})&lt;/math&gt; is constant. The spring between support and guide guarantees that the former does not fall to the ground when the system is at rest.<br /> <br /> [[File:C2-E.Ex3-1-eng.png|thumb|center|600px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The guide may rotate about the vertical axis through &lt;math&gt;\Os &#039;&lt;/math&gt; (simple rotation).<br /> <br /> :Independently form that rotation, the support may have a translational motion along the guide (rectilinear translational motion).<br /> <br /> :Finally, if those two motions are blocked, the ring may still have a simple rotation about the horizontal axis through &lt;math&gt;\Os &#039;&lt;/math&gt;, which is perpendicular to the ring plane and is fixed to the support.<br /> <br /> :Hence, the system has 3 degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the ring relative to the ground. =====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :The angular velocity of the ring is the superposition of &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[File:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{ring}{E}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{ring}{E}=\dert{\velang{ring}{E}}{E}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{E}=\dert{\vec{\psio}}{E}+\dert{\vec{\dot{\theta}}}{E}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{E}+\dert{(\odot\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The angular acceleration comes exclusively from the change of value and direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, as &lt;math&gt;\vec{\psio}&lt;/math&gt; has both constant value and constant direction.<br /> <br /> :&lt;math&gt;\accang{ring}{E} = \dert{\velang{ring}{E}}{E} = \dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> :The time derivative of the angular velocity of the ring may be done analytically. The vector basis where the projection of &lt;math&gt;\velang{ring}{E}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{ring}{E}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{ring}{E}}{B}=\braq{\dert{\velang{ring}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{ring}{E}}{B}+\braq{\velang{B}{E}<br /> }{B}\times\braq{\velang{ring}{E}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=727 2024-10-24T00:15:36Z

<p>Apons: /* 🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\OPvec}{\vec{\Os\Ps}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.3: rotating pendulum with oscillating articulation point====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The ring-shaped pendulum is articulated to the support, which is linked to the guide through a prismatic joint. The guide is articulated to the ceiling, and its angular velocity relative to the ceiling &lt;math&gt;(\vec{\psio})&lt;/math&gt; is constant. The spring between support and guide guarantees that the former does not fall to the ground when the system is at rest.<br /> <br /> [[Fitxer:C2-E.Ex3-1-eng.png|thumb|center|600px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The guide may rotate about the vertical axis through &lt;math&gt;\Os &#039;&lt;/math&gt; (simple rotation).<br /> <br /> :Independently form that rotation, the support may have a translational motion along the guide (rectilinear translational motion).<br /> <br /> :Finally, if those two motions are blocked, the ring may still have a simple rotation about the horizontal axis through &lt;math&gt;\Os &#039;&lt;/math&gt;, which is perpendicular to the ring plane and is fixed to the support.<br /> <br /> :Hence, the system has 3 degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the ring relative to the ground. =====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :The angular velocity of the ring is the superposition of &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{ring}{E}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{ring}{E}=\dert{\velang{ring}{E}}{E}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{E}=\dert{\vec{\psio}}{E}+\dert{\vec{\dot{\theta}}}{E}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{E}+\dert{(\odot\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The angular acceleration comes exclusively from the change of value and direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, as &lt;math&gt;\vec{\psio}&lt;/math&gt; has both constant value and constant direction.<br /> <br /> :&lt;math&gt;\accang{ring}{E} = \dert{\velang{ring}{E}}{E} = \dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> :The time derivative of the angular velocity of the ring may be done analytically. The vector basis where the projection of &lt;math&gt;\velang{ring}{E}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{ring}{E}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{ring}{E}}{B}=\braq{\dert{\velang{ring}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{ring}{E}}{B}+\braq{\velang{B}{E<br /> }{B}\times\braq{\velang{ring}{E}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:C2-E.Ex3-2-neut.png&diff=726 2024-10-24T00:11:22Z

<p>Apons: </p> <hr /> <div>C2-E.Ex3-2-neut</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:C2-E.Ex3-1-eng.png&diff=725 2024-10-24T00:08:23Z

<p>Apons: </p> <hr /> <div>C2-E.Ex3-1-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=724 2024-10-24T00:06:52Z

<p>Apons: /* 3. Find the velocity and the acceleration of point Q of the plate relative to the ground. */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\OPvec}{\vec{\Os\Ps}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=723 2024-10-24T00:02:12Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\OPvec}{\vec{\Os\Ps}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=722 2024-10-24T00:01:09Z

<p>Apons: /* 🔎 Example C2-E.2: rotating articulated plate */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground.=====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2nd Euler rotation, axis rotation because of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :The angular acceleration is associated to the change of value of &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, and the change of direction of &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{(\Uparrow\dot{\psi})}{E}=\dert{(\otimes\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{E}=[\text{change of value}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{E}=[\text{change of value}]+[\text{change od direction}]_\Es=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{E}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Hence, &lt;math&gt;\accang{plate}{E}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> <br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :The time derivative of the angular velocity can also be done analytically. The vector basis B where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point Q of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OQvec&lt;/math&gt; is a position vector in the ground frame. Its value is &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, , and its direction is always horizontal. The velocity of &lt;math&gt;\Qs&lt;/math&gt;comes both from the change of that value (as &lt;math&gt;\theta&lt;/math&gt; is variable) and the change of its direction relative to the ground (because of the support rotation &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;).<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> [[File:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{E}=\dert{\OQvec}{E}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :The acceleration of &lt;math&gt;\Qs&lt;/math&gt; comes from the change of value and direction (associated with &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) of both terms in the velocity:<br /> :&lt;math&gt;\acc{Q}{E}=\dert{\vel{Q}{E}}{E}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{E}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{E}=[\text{change of value}]+[\text{change of direction]_\Es=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Finally, &lt;math&gt;\acc{Q}{E}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> [[File:C4-E-Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :The tome derivative can also be done analytically. The vector basis B where the &lt;math&gt;\OPvec&lt;/math&gt; projection is straightforward is the one fixed to the support &lt;math&gt;(\velang{B}{E}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{E}}{B}=\braq{\dert{\OQvec}{E}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{E}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{E}}{B}=\braq{\dert{\vel{Q}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\vel{Q}{E}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:C2-E.Ex2-4-neut.png&diff=721 2024-10-23T23:55:21Z

<p>Apons: </p> <hr /> <div>C2-E.Ex2-4-neut</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=720 2024-10-23T23:46:11Z

<p>Apons: </p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OCvec}{\vec{\Os\Cs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt; orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=719 2024-10-23T23:45:17Z

<p>Apons: /* 🔎 Example C2-E.2: rotating articulated plate */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt;<br /> :The support may rotate freely about a vertical axis fixed to the ground (simple rotation). If we block that motion, the system may still move. <br /> :The &lt;math&gt;\OCvec&lt;/math&gt; bars may have a simple rotation, relative to the support, about the horizontal axis through &lt;math&gt;\Os&lt;/math&gt;orthogonal to the bars. If we block the motion of one of those bars relative to the support, the plate, the &lt;math&gt;\OCvec&lt;/math&gt; bars and the bended bar cannot move. Alternatively, if the vended bar is blocked (if ts vertical translational motion relative to the support is blocked), neither the plate nor the &lt;math&gt;\OCvec&lt;/math&gt; bars may move relative to the support.<br /> :Hence, the system has two degrees of freedom.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=718 2024-10-23T23:43:12Z

<p>Apons: /* 🔎 Example C2-E.2: rotating articulated plate */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :El suport pot girar lliurement al voltant de l’eix vertical fix a terra (rotació simple). Si el suport s’atura respecte del terra, el sistema encara es pot moure. <br /> :Respecte del suport, les barres &lt;math&gt;\OCvec&lt;/math&gt; poden fer una rotació simple al voltant de l’eix horitzontal perpendicular a les barres i que passa per &lt;math&gt;\Os&lt;/math&gt;. Si una d’aquestes barres s’atura respecte del suport, ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt;, ni la barra en colze es poden moure. Una anàlisi alternativa d’aquest segon grau de llibertat és veure que si la barra en colze s’atura (si s’atura la seva translació vertical respecte del suport), ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt; es poden moure respecte del suport.<br /> :Per tant, el sistema té dos graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=717 2024-10-23T23:42:56Z

<p>Apons: /* 🔎 Example C2-E.2: rotating articulated plate */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C4-E-Ex2-1-eng.png.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :El suport pot girar lliurement al voltant de l’eix vertical fix a terra (rotació simple). Si el suport s’atura respecte del terra, el sistema encara es pot moure. <br /> :Respecte del suport, les barres &lt;math&gt;\OCvec&lt;/math&gt; poden fer una rotació simple al voltant de l’eix horitzontal perpendicular a les barres i que passa per &lt;math&gt;\Os&lt;/math&gt;. Si una d’aquestes barres s’atura respecte del suport, ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt;, ni la barra en colze es poden moure. Una anàlisi alternativa d’aquest segon grau de llibertat és veure que si la barra en colze s’atura (si s’atura la seva translació vertical respecte del suport), ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt; es poden moure respecte del suport.<br /> :Per tant, el sistema té dos graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=716 2024-10-23T23:42:13Z

<p>Apons: /* 🔎 Exercici C2-E.2: placa articulada giratòria */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Example C2-E.2: rotating articulated plate====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The rectangular plate is joined to a rotation support through two bars with revolute joints at their endpoints. A third bar is joined to the plate through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;spherical joint &#039;&#039;&#039;]]&lt;/span&gt; at &lt;math&gt;\Ps&lt;/math&gt;, and to the support through a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;cylindrical joiny &#039;&#039;&#039;]]&lt;/span&gt;. ). The support rotates with the variable angular velocity &lt;math&gt;\vecdot{\psi}&lt;/math&gt; vrelative to the ground (E).<br /> <br /> [[File:C2-E.Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> [[File:C2-E.Ex2-2-eng.png|thumb|center|450px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :El suport pot girar lliurement al voltant de l’eix vertical fix a terra (rotació simple). Si el suport s’atura respecte del terra, el sistema encara es pot moure. <br /> :Respecte del suport, les barres &lt;math&gt;\OCvec&lt;/math&gt; poden fer una rotació simple al voltant de l’eix horitzontal perpendicular a les barres i que passa per &lt;math&gt;\Os&lt;/math&gt;. Si una d’aquestes barres s’atura respecte del suport, ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt;, ni la barra en colze es poden moure. Una anàlisi alternativa d’aquest segon grau de llibertat és veure que si la barra en colze s’atura (si s’atura la seva translació vertical respecte del suport), ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt; es poden moure respecte del suport.<br /> :Per tant, el sistema té dos graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=File:C2-E.Ex2-2-eng.png&diff=715 2024-10-23T23:42:02Z

<p>Apons: </p> <hr /> <div>C2-E.Ex2-2-eng</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=Main_Page&diff=714 2024-10-23T23:37:09Z

<p>Apons: /* C2. Movement of a mechanical system */</p> <hr /> <div><br /> [[File:Segwaywiki.png|right|top|200px|link=]]<br /> This website is specially built to complement the learning of the &#039;&#039;&#039;Mechanics&#039;&#039;&#039; course in the bachelor&#039;s degrees of &#039;&#039;&#039;[https://etseib.upc.edu/en Barcelona School of Industrial Engineering (ETSEIB)]&#039;&#039;&#039; of the &#039;&#039;&#039;[https://www.upc.edu/en Polytechnic University of Catalonia (UPC) · BarcelonaTech]&#039;&#039;&#039;.<br /> <br /> This aims to be an &#039;&#039;&#039;accessible&#039;&#039;&#039; and &#039;&#039;&#039;interactive&#039;&#039;&#039; &#039;&#039;&#039;open tool&#039;&#039;&#039;. It&#039;s development started on 2022 and it gathers more than 50 years of teaching experience. It&#039;s content is organized in brief units which contain the fundamental concepts and some fully worked-out examples. Some simple mathematical proofs are included, but the longer or complex ones are refered to biblography.<br /> <br /> The contentent is focused on &#039;&#039;&#039;general space movement of rigid bodies and muli-body systems&#039;&#039;&#039;, but &#039;&#039;&#039;particles&#039;&#039;&#039; are also considered. Dynamics formulation is vectorial, due to the relevance of the force vector in mechanical engineering. The last units are an introduction to energetics.<br /> <br /> <br /> &lt;small&gt;<br /> &#039;&#039;&#039;About the status of the site&#039;&#039;&#039;:<br /> <br /> This project has been developed with limited resources, both technical and human. Nowadays, the server presents some issues, so in case any error may appear, we kindly invite the users to refresh the page and continue enjoying the content 😊.<br /> <br /> The best experience will be through a computer or a tablet 📵.<br /> <br /> The site is still under construction and some interactive resources and videos are still to be uploaded. Also, the Dynamics and Energetics blocks are still to be published. Having said that, it already is a good tool to help in the process of learning Mechanics 🎯.<br /> &lt;/small&gt;<br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mechanics:Copyrights |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> __NOTOC__<br /> ------------<br /> ------------<br /> ===[[Introduction]]===<br /> :: [[Introduction#I.1 What is mechanics?|I.1 What is mechanics?]]<br /> :: [[Introduction#I.2 Models for material objects|I.2 Models for material objects]]<br /> :: [[Introduction#I.3 Limitations of Newtonian mechanics|I.3 Limitations of Newtonian mechanics]]<br /> :: [[Introduction#I.4 Reference frame|I.4 Reference frame]]<br /> <br /> ===[[Vector calculus]]===<br /> ::[[Vector calculus#V.1 Geometric representation of a vector|V.1 Geometric representation of a vector]]<br /> ::[[Vector calculus#V.2 Operations between vectors with geometric representation|V.2 Operations between vectors with geometric representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> ::[[Vector calculus#V.3 Analytical representation of a vector|V.3 Analytical representation of a vector]]<br /> ::[[Vector calculus#V.4 Operations between vectors with analytical representation|V.4 Operations between vectors with analytical representation]]<br /> ::::&lt;small&gt;[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]&lt;/small&gt;<br /> <br /> ==KINEMATICS==<br /> ===[[C1. Configuration of a mechanical system]]===<br /> :: [[C1. Configuration of a mechanical system#C1.1 Position of a particle|C1.1 Position of a particle]]<br /> :: [[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|C1.2 Configuration of a rigid body]]<br /> :: [[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|C1.3 Orientation of a rigid body with planar motion]]<br /> :: [[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|C1.4 Orientation of a rigid body moving in space]]<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Rotations about fixed axes|Rotations about fixed axes]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C1. Configuration of a mechanical system#Euler rotations|Euler rotations]]&lt;/small&gt;<br /> :: [[C1. Configuration of a mechanical system#C1.5 Independent coordinates|C1.5 Independent coordinates]]<br /> <br /> ===[[C2. Movement of a mechanical system]]===<br /> :: [[C2. Movement of a mechanical system#C2.1 Velocity of a particle|C2.1 Velocity of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.2 Acceleration of a particle|C2.2 Acceleration of a particle]]<br /> :: [[C2. Movement of a mechanical system#C2.3 Intrinsic components of the acceleration|C2.3 Intrinsic components of the acceleration]]<br /> :: [[C2. Movement of a mechanical system#C2.4 Angular velocity of a rigid body|C2.4 Angular velocity of a rigid body]]<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Simple rotation|Simple rotation]]&lt;/small&gt;<br /> ::::&lt;small&gt;[[C2. Movement of a mechanical system#Rotation in space|Rotation in space (Rotacions d&#039;Euler)]]&lt;/small&gt;<br /> :: [[C2. Movement of a mechanical system#C2.5 Angular acceleration of a rigid body|C2.5 Angular acceleration of a rigid body]]<br /> :: [[C2. Movement of a mechanical system#C2.6 Particle kinematics versus rigid body kinematicsrígid|C2.6 Particle kinematics versus rigid body kinematics]]<br /> :: [[C2. Movement of a mechanical system#C2.7 Degrees of freedom of a mechanical system|C2.7 Degrees of freedom of a mechanical system]]<br /> :: [[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|C2.8 Usual constraints in mechanical systems]]<br /> :: [[C2. Movement of a mechanical system#C2.E General examples|C2.E General examples]]<br /> <br /> ===[[C3. Composition of movements]]===<br /> :: [[C3. Composition of movements#C3.1 Composition of velocities|C3.1 Composition of velocities]]<br /> :: [[C3. Composition of movements#C3.2 Composition of accelerations|C3.2 Composition of accelerations]]<br /> :: [[C3. Composition of movements#C3.3 Composition versus time derivative|C3.3 Composition versus time derivative]]<br /> :: [[C3. Composition of movements#C3.E General examples|C3.E General examples]]<br /> <br /> ===[[C4. Rigid body kinematics]]===<br /> :: [[C4. Rigid body kinematics#C4.1 Velocity distribution|C4.1 Velocity distribution]]<br /> :: [[C4. Rigid body kinematics#C4.2 Accelerations distribution|C4.2 Accelerations distribution]]<br /> :: [[C4. Rigid body kinematics#C4.3 Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)|Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)]]<br /> ::[[C4. Rigid body kinematics#C4.4 Fixed axode and moving axode|C4.4 Fixed axode and moving axode]]<br /> ::[[C4. Rigid body kinematics#C4.E General examples|C4.E General examples]]<br /> <br /> ===[[C5. Rigid body kinematics: planar motion]]===<br /> :: [[C5. Rigid body kinematics: planar motion#C5.1 Instantaneous Center of Rotation (ICR)|C5.1 Instantaneous Center of Rotation (ICR)]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.2 Examples|C5.2 Examples]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.3 Introduction to vehicle kinematics|C5.3 Introduction to vehicle kinematics]]<br /> :: [[C5. Rigid body kinematics: planar motion#C5.E General examples|C5.E General examples]]<br /> <br /> ==DYNAMICS==<br /> <br /> ===[[D5. Inertia tensor#|D5. Inertia tensor]]===<br /> ::[[D5. Inertia tensor#D5.1 Centre of masses|D5.1 Centre of masses]]<br /> ::[[D5. Inertia tensor#D5.2 Inertia tensor|D5.2 Inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.3 Some relevant properties of the inertia tensor|D5.3 Some relevant properties of the inertia tensor]]<br /> ::[[D5. Inertia tensor#D5.4 Steiner’s Theorem|D5.4 Steiner’s Theorem]]<br /> ::[[D5. Inertia tensor#D5.5 Change of vector basis|D5.5 Change of vector basis]]<br /> <br /> ===[[D7. Examples of 3D dynamics#|D7. Examples of 3D dynamics]]===<br /> ::[[D7. Examples of 3D dynamics#D7.1 Analysis of the equations of motion|D7.1 Analysis of the equations of motion]]<br /> ::[[D7. Examples of 3D dynamics#D7.2 General examples|D7.2 General examples]]<br /> <br /> ===[[D8. Conservation of dynamic magnitudes#|D8. Conservation of dynamic magnitudes]]===<br /> ::[[D8. Conservation of dynamic magnitudes#D8.1 Examples|D8.1 Examples]]<br /> <br /> ==ENERGETICS==<br /> ::&#039;&#039;UNDER CONSTRUCTION&#039;&#039;<br /> <br /> <br /> -----------<br /> -----------<br /> ==Authors==<br /> <br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors Ana.png|alt=Ana Barjau Condomines|Ana Barjau Condomines|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1001000<br /> File:Autors Ernest4.jpg|alt=Ernest Bosch Soldevila|Ernest Bosch Soldevila|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1099864<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Ilustrations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Joaquim.png|alt=Joaquim Agulló i Batlle|Joquim Agulló i Batlle|link=https://www.agullobatlle.cat/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Editing and interactive animations:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Arnau.png|alt=Arnau Marzábal Gatell|Arnau Marzábal Gatell|link=https://www.linkedin.com/in/arnau-marzabal/<br /> File:Autors_Berta.png|alt=Berta Ros Blanco|Berta Ros Blanco|link=https://www.linkedin.com/in/berta-ros/<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> ::&#039;&#039;&#039;Collaborators:&#039;&#039;&#039;<br /> &lt;center&gt;&lt;gallery&gt;<br /> File:Autors_Daniel.png|alt=Daniel Clos Costa|Daniel Clos Costa|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1002252<br /> File:Autors_Rosa.png|alt=Rosa Pàmies Vilà|Rosa Pàmies Vilà|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1066910<br /> File:Autors_Albert.png|alt=Albert Peiret Giménez|Albert Peiret Giménez|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1115007<br /> File:Autors_Javier.png|alt=Javier Sistiaga Vidal-Ribas|Javier Sistiaga Vidal-Ribas|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1114855<br /> &lt;/gallery&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;<br /> [https://www.youtube.com/channel/UCqWvnHTViRPI1wHlUQXqH-Q &#039;&#039;&#039;Mechanics Lab&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://etseib.upc.edu/en &#039;&#039;&#039;Barcelona School of Industrial Engineering (ETSEIB)&#039;&#039;&#039;]<br /> <br /> [https://em.upc.edu/en &#039;&#039;&#039;Mechanical Engineering Department&#039;&#039;&#039;] &#039;&#039;&#039; - &#039;&#039;&#039; [https://www.upc.edu/en &#039;&#039;&#039;Polytechic University of Catalonia (UPC) · BarcelonaTech&#039;&#039;&#039;]<br /> &lt;/center&gt;<br /> <br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==Bibliographic references==<br /> Batlle, J. A., Barjau, A. (2020) “&#039;&#039;&#039;Rigid Body Kinematics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-47907-3<br /> <br /> Batlle, J. A., Barjau, A. (2022) “&#039;&#039;&#039;Rigid Body Dynamics&#039;&#039;&#039;” Cambridge Univerity Press. ISBN: 978-1-108-84213-6<br /> <br /> Agulló, J. (2002) “&#039;&#039;&#039;Mecànica de la partícula i del sòlid rígid&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-6-1 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> Agulló, J. (2000) “&#039;&#039;&#039;Mecánica de la partícula i del sólido rígido&#039;&#039;&#039;&quot; Publicacions OK Punt. ISBN: 84-920850-5-3 (&#039;&#039;Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent &#039;&#039;&#039;web de l&#039;autor&#039;&#039;&#039;])<br /> <br /> <br /> &lt;center&gt;<br /> &lt;gallery&gt;<br /> File:RBK portada.png|alt=Rigid body kinematics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-kinematics?format=HB&amp;isbn=9781108479073<br /> File:RBD portada.png|alt=Rigid body dynamics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&amp;isbn=9781108842136<br /> File:Llibre verd.png|alt=Mecànica de la partícula i del sòlid rígid|link=https://www.agullobatlle.cat/activitat-docent<br /> File:Llibre vermell.jpg|alt=Mecánica de la partícula i del sólido rígido|link=https://www.agullobatlle.cat/activitat-docent<br /> &lt;/gallery&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> <br /> <br /> <br /> [[File:Logo Lab Mec horitzontal.png|thumb|center|500px|link=https://em.upc.edu/en| ]]</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=713 2024-10-21T16:28:01Z

<p>Apons: /* 2. Find the angular velocity and the angular acceleration of the plate relative to the ground. */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.2: placa articulada giratòria====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> La placa rectangular està unida a un suport giratori a través de dues barres paral·leles amb articulacions als extrems. Una tercera barra està enllaçada a la placa a través d’una &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;ròtula esfèrica&#039;&#039;&#039;]]&lt;/span&gt; a P i al suport a través d’un &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;enllaç cilíndric&#039;&#039;&#039;]]&lt;/span&gt;. El suport gira amb velocitat angular &lt;math&gt;\vecdot{\psi}&lt;/math&gt; variable respecte del terra (T).<br /> <br /> [[Fitxer:C3-E.Ex2-1-cat-esp.png|thumb|center|450px|link=]]<br /> <br /> [[Fitxer:C2-E.Ex2-2-cat.png|thumb|center|450px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :El suport pot girar lliurement al voltant de l’eix vertical fix a terra (rotació simple). Si el suport s’atura respecte del terra, el sistema encara es pot moure. <br /> :Respecte del suport, les barres &lt;math&gt;\OCvec&lt;/math&gt; poden fer una rotació simple al voltant de l’eix horitzontal perpendicular a les barres i que passa per &lt;math&gt;\Os&lt;/math&gt;. Si una d’aquestes barres s’atura respecte del suport, ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt;, ni la barra en colze es poden moure. Una anàlisi alternativa d’aquest segon grau de llibertat és veure que si la barra en colze s’atura (si s’atura la seva translació vertical respecte del suport), ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt; es poden moure respecte del suport.<br /> :Per tant, el sistema té dos graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=712 2024-10-21T16:27:33Z

<p>Apons: /* 2. Find the angular velocity and the angular acceleration of the plate relative to the ground. */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}==&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.2: placa articulada giratòria====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> La placa rectangular està unida a un suport giratori a través de dues barres paral·leles amb articulacions als extrems. Una tercera barra està enllaçada a la placa a través d’una &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;ròtula esfèrica&#039;&#039;&#039;]]&lt;/span&gt; a P i al suport a través d’un &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;enllaç cilíndric&#039;&#039;&#039;]]&lt;/span&gt;. El suport gira amb velocitat angular &lt;math&gt;\vecdot{\psi}&lt;/math&gt; variable respecte del terra (T).<br /> <br /> [[Fitxer:C3-E.Ex2-1-cat-esp.png|thumb|center|450px|link=]]<br /> <br /> [[Fitxer:C2-E.Ex2-2-cat.png|thumb|center|450px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :El suport pot girar lliurement al voltant de l’eix vertical fix a terra (rotació simple). Si el suport s’atura respecte del terra, el sistema encara es pot moure. <br /> :Respecte del suport, les barres &lt;math&gt;\OCvec&lt;/math&gt; poden fer una rotació simple al voltant de l’eix horitzontal perpendicular a les barres i que passa per &lt;math&gt;\Os&lt;/math&gt;. Si una d’aquestes barres s’atura respecte del suport, ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt;, ni la barra en colze es poden moure. Una anàlisi alternativa d’aquest segon grau de llibertat és veure que si la barra en colze s’atura (si s’atura la seva translació vertical respecte del suport), ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt; es poden moure respecte del suport.<br /> :Per tant, el sistema té dos graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=711 2024-10-21T16:27:10Z

<p>Apons: /* 2. Find the angular velocity and the angular acceleration of the plate relative to the ground. */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}==&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.2: placa articulada giratòria====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> La placa rectangular està unida a un suport giratori a través de dues barres paral·leles amb articulacions als extrems. Una tercera barra està enllaçada a la placa a través d’una &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;ròtula esfèrica&#039;&#039;&#039;]]&lt;/span&gt; a P i al suport a través d’un &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;enllaç cilíndric&#039;&#039;&#039;]]&lt;/span&gt;. El suport gira amb velocitat angular &lt;math&gt;\vecdot{\psi}&lt;/math&gt; variable respecte del terra (T).<br /> <br /> [[Fitxer:C3-E.Ex2-1-cat-esp.png|thumb|center|450px|link=]]<br /> <br /> [[Fitxer:C2-E.Ex2-2-cat.png|thumb|center|450px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :El suport pot girar lliurement al voltant de l’eix vertical fix a terra (rotació simple). Si el suport s’atura respecte del terra, el sistema encara es pot moure. <br /> :Respecte del suport, les barres &lt;math&gt;\OCvec&lt;/math&gt; poden fer una rotació simple al voltant de l’eix horitzontal perpendicular a les barres i que passa per &lt;math&gt;\Os&lt;/math&gt;. Si una d’aquestes barres s’atura respecte del suport, ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt;, ni la barra en colze es poden moure. Una anàlisi alternativa d’aquest segon grau de llibertat és veure que si la barra en colze s’atura (si s’atura la seva translació vertical respecte del suport), ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt; es poden moure respecte del suport.<br /> :Per tant, el sistema té dos graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

Apons https://mec.etseib.upc.edu/en/index.php?title=C2._Movement_of_a_mechanical_system&diff=710 2024-10-21T16:26:35Z

<p>Apons: /* 3. Find the velocity and the acceleration of point G of the plate relative to the ground. . */</p> <hr /> <div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> &lt;math&gt;\newcommand{\uvec}{\overline{\textbf{u}}}<br /> \newcommand{\vvec}{\overline{\textbf{v}}}<br /> \newcommand{\evec}{\overline{\textbf{e}}}<br /> \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}<br /> \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}}<br /> \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}}<br /> \newcommand{\ds}{\textrm{d}}<br /> \newcommand{\ts}{\textrm{t}}<br /> \newcommand{\us}{\textrm{u}}<br /> \newcommand{\vs}{\textrm{v}}<br /> \newcommand{\Rs}{\textrm{R}}<br /> \newcommand{\Ts}{\textrm{T}}<br /> \newcommand{\Ls}{\textrm{L}}<br /> \newcommand{\Bs}{\textrm{B}}<br /> \newcommand{\es}{\textrm{e}}<br /> \newcommand{\is}{\textrm{i}}<br /> \newcommand{\rs}{\textrm{r}}<br /> \newcommand{\Os}{\textbf{O}}<br /> \newcommand{\Cbf}{\textbf{C}}<br /> \newcommand{\Or}{\Os_\Rs}<br /> \newcommand{\Qs}{\textbf{Q}}<br /> \newcommand{\Cs}{\textbf{C}}<br /> \newcommand{\Ps}{\textrm{P}}<br /> \newcommand{\Es}{\textrm{E}}<br /> \newcommand{\Ss}{\textbf{S}}<br /> \newcommand{\Gs}{\textbf{G}}<br /> \newcommand{\deg}{^\textsf{o}}<br /> \newcommand{\xs}{\textsf{x}}<br /> \newcommand{\ys}{\textsf{y}}<br /> \newcommand{\zs}{\textsf{z}}<br /> \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}}<br /> \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}}<br /> \newcommand{\vec}[1]{\overline{#1}}<br /> \newcommand{\vecbf}[1]{\overline{\textbf{#1}}}<br /> \newcommand{\vecdot}[1]{\overline{\dot{#1}}}<br /> \newcommand{\OQvec}{\vec{\Os\Qs}}<br /> \newcommand{\OGvec}{\vec{\Os\Gs}}<br /> \newcommand{\abs}[1]{\left|{#1}\right|}<br /> \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}<br /> \newcommand{\vector}[3]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}\\<br /> {#3}<br /> \end{Bmatrix}}<br /> \newcommand{\vecdosd}[2]{<br /> \begin{Bmatrix}<br /> {#1}\\<br /> {#2}<br /> \end{Bmatrix}}<br /> \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})}<br /> \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})}<br /> \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})}<br /> \newcommand{\velo}[1]{\vvec_{\textrm{#1}}}<br /> \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}}<br /> \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}}<br /> \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})}<br /> \newcommand{\psio}{\dot{\psi}_0}<br /> \definecolor{blau}{RGB}{39, 127, 255}<br /> \definecolor{verd}{RGB}{9, 131, 9}<br /> &lt;/math&gt;<br /> ==C2.1 Velocity of a particle==<br /> <br /> The &#039;&#039;&#039;velocity of a particle &lt;math&gt;\Qs&lt;/math&gt;&#039;&#039;&#039; (or a point that belongs to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\vvec_{\Rs}(\Qs)&lt;/math&gt;, is the rate of change of its position vector with time. Mathematically, it is the time derivative of a position vector (relative to R). The time derivative of two different position vectors (&lt;math&gt;\overline{\Or\Qs}&lt;/math&gt;, &lt;math&gt;\overline{\Os&#039;_\Rs\Qs}&lt;/math&gt; ) yield the same velocity because points &lt;math&gt;\Os_\Rs&lt;/math&gt; and &lt;math&gt;\Os&#039;_\Rs&lt;/math&gt; are mutually fixed and fixed to the reference frame, hence &lt;math&gt;\overline{\Os_\Rs\Os&#039;_\Rs}&lt;/math&gt; is constant in R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vvec_\Rs(\Qs) = \dert{\vec{\Os_{\Rs}\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}\Os_{\Rs}&#039;}}{R} + \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R} =<br /> \dert{\vec{\Os_{\Rs}&#039;\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> One must bear in mind that the time derivative of a vector depends on the reference frame where it is being calculated. For that reason, there is a subscript R in the preceding equations which reminds of that dependency.<br /> <br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus #V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;time derivative of a vector relative to a reference frame R &#039;&#039;&#039;]]&lt;/span&gt; assesses the evolution of the characteristics of that vector (direction and value) between two close time instants, separated by a time differential. Hence, the velocity &lt;math&gt;\vvec_\Rs(\Qs)&lt;/math&gt; is nonzero whenever the value of the position vector, or its direction, or both change. <br /> <br /> ====✏️ EXAMPLE C2-1.1: rotating platform====<br /> ---------<br /> &lt;small&gt;<br /> ::The platform (RP) rotates about an axis perpendicular to the ground (R). The movement of a point &lt;math&gt;\Qs&lt;/math&gt; on the platform periphery depends on whether it is observed from the ground or from the platform.<br /> <br /> <br /> [[File:C2-Ex1-1-1-neut.png|thumb|center|350px|link=]]<br /> <br /> ::The center of the platform (&lt;math&gt;\Os&lt;/math&gt;) is fixed to both reference frames. Hence, &lt;math&gt;\vec{\Os\Qs}&lt;/math&gt; is a position vector for point &lt;math&gt;\Qs&lt;/math&gt; both in R and RP. It is evident that &lt;math&gt;\vvec_\Rs(\Qs)\neq \vec{0}&lt;/math&gt; i &lt;math&gt;\vvec_{\Rs\Ps}(\Qs)= \vec{0}&lt;/math&gt;, though the vector whose time derivative is being calculated is the same.<br /> <br /> ::As &lt;math&gt;\abs{\OQvec}&lt;/math&gt; is the platform radius r, its value is constant. Hence, the time derivative of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; can only be associated with a change of direction.<br /> <br /> ::To assess the change of orientation of &lt;math&gt;\abs{\OQvec}&lt;/math&gt; relative to the ground or to the platform, we have to define an angle between a straight line fixed in the reference frame (“departure” line) and vector &lt;math&gt;\OQvec&lt;/math&gt; (“arrival” line). For the sake of clarity, we have represented the “departure” line as the direction of the arm of an observer located in the reference frame (thus not moving relative to it).<br /> <br /> :[[File:C2-Ex1-1-2-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)\neq\psi(t+dt) \implies \OQvec&lt;/math&gt; changes its direction relative to &lt;span style=&quot;color:rgb(39,127,255);&quot;&gt;&lt;b&gt;R&lt;/b&gt;&lt;/span&gt; &lt;math&gt;\implies \textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{) \neq \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ::As seen in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.1 Geometric representation of a vector|&#039;&#039;&#039;section V.1&#039;&#039;&#039;]]&lt;/span&gt;, &lt;math&gt;\textcolor{blau}{\vvec_\Rs(}\Qs\textcolor{blau}{)}&lt;/math&gt; is perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, and its value is that of &lt;math&gt;\OQvec(\textrm{r})&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\OQvec&lt;/math&gt; relative to R &lt;math&gt;(\dot{\psi})&lt;/math&gt;:<br /> <br /> <br /> [[File:C2-Ex1-1-3-neut.png|thumb|center|450px|link=]]<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the platform (&lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt;):<br /> <br /> [[File:C2-Ex1-1-4-neut.png|thumb|center|450px|link=]]<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\psi(t)=\psi(t+dt) \implies \OQvec&lt;/math&gt; does not change its direction relative to &lt;span style=&quot;color:rgb(9,131,9);&quot;&gt;RP&lt;/span&gt; &lt;math&gt;\textcolor{verd}{\implies \vvec_\Rs(}\Qs\textcolor{verd}{) = \vec{0}}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> <br /> ::The two logical vector bases for the calculation are:<br /> <br /> :::* Basis B (1,2,3) fixed in R (thus moving in RP): &lt;math&gt;\velang{B}{R}=\vec{0},\velang{B}{RP}= \vec{\dot{\psi}}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed in RP (thus moving in R): &lt;math&gt;\velang{B&#039;}{RP}=\vec{0},\velang{B&#039;}{R} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> [[File:C2-Ex1-1-5-neut.png|thumb|200px|right|link=]]<br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases: <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\OQvec}{B}=\vector{rcos\psi}{rsin\psi}{0}, \: \: \braq{\OQvec}{B&#039;}=\vector{r}{0}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B} = \braq{\dert{\OQvec}{R}}{B}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{-r\dot \psi sin\psi}{r\dot{\psi} cos\psi}{0}<br /> &lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \braq{\vvec_\Rs(\Qs)}{B&#039;}=\braq{\dert{\OQvec}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\braq{\velang{B&#039;}{R}\times \OQvec}{B&#039;}=\vector{0}{0}{\dot\psi} \times \vector{r}{0}{0}= \vector{0}{r\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to RP:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B} =\braq{\dert{\OQvec}{RP}}{B}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B}+\braq{\velang{B}{RP}\times \OQvec}{B}=\vector{-r\dot\psi sin\psi}{r\dot\psi cos\psi}{0}+ \vector{0}{0}{-\dot\psi}\times\vector{rcos\psi}{rsin\psi}{0}= \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\vvec_{\Rs\Ps}(\Qs)}{B&#039;} =\braq{\dert{\OQvec}{RP}}{B&#039;}= \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;}+\braq{\velang{B&#039;}{RP}\times \OQvec}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} = \vector{0}{0}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-1.2: Euler pendulum====<br /> ------------<br /> &lt;small&gt;<br /> ::The endpoint &lt;math&gt;\Qs&lt;/math&gt; of &lt;span style=&quot;text-decoration: underline; font-weigth:bold;&quot;&gt;[[C1. Configuration of a mechanical system #✏️ EXAMPLE C1-5.4: Euler pendulum|&#039;&#039;&#039;Euler pendulum&#039;&#039;&#039;]]&lt;/span&gt; describes a circular motion relative to the block. The corresponding velocity &lt;math&gt;\vel{Q}{BL} = \dert{\vecbf{CQ}}{BL}&lt;/math&gt; can be obtained in a similar way as that used in the previous example.<br /> <br /> [[File:C2-Ex1-2-1-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The angle &lt;math&gt;\psi&lt;/math&gt; orientates the bar both relative to the block and the ground, as its origin (vertical line) has a constant orientation in both reference frames<br /> ::The velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground can be obtained as the time derivative of vector &lt;math&gt;\vec{\Or\Qs} (=\vec{\Or\Cbf}+\vecbf{CQ})&lt;/math&gt; relative to the ground:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \vel{Q}{R} = \dert{\vec{\Or\Qs}}{R} = \dert{\vec{\Or\Cbf}}{R}+ \dert{\vec{\Cbf\Qs}}{R}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> ::Vector &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; has a constant direction in R but a variable value. Hence, its time derivative is parallel to &lt;math&gt;\vec{\Or\Cbf}&lt;/math&gt; with value &lt;math&gt;\dot\xs&lt;/math&gt;. Vector &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt;, and its value is that of&lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; times the rate of change of orientation of &lt;math&gt;\vec{\Cbf\Qs}&lt;/math&gt; relative to R (&lt;math&gt;\dot\psi&lt;/math&gt;):<br /> <br /> [[File:C2-Ex1-2-2-neut.png|thumb|center|300px|link=]]<br /> <br /> ::The &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not any of the directions associated with the system (it is not vertical, not horizontal, not parallel to the bar, not perpendicular to the bar). For that reason, it is better to represent it as the addition of the terms &lt;math&gt;\dot\xs&lt;/math&gt; and &lt;math&gt;L\dot\psi&lt;/math&gt;, whose directions do correspond to one of those singular directions.<br /> <br /> ::The first term of the expression &lt;math&gt;\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R}+\dert{\vecbf{CQ}}{R}&lt;/math&gt; corresponds to the velocity of &lt;math&gt;\Cs&lt;/math&gt; relative to the ground &lt;math&gt;\left(\vel{Q}{R} = \dert{\vec{\Or\Cbf}}{R} \right)&lt;/math&gt;, whereas the second one has no physical interpretation: point &lt;math&gt;\Cs&lt;/math&gt; is not fixed in R, thus it is not a position vector in that reference frame.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The two logical vector bases for the calculation are:<br /> <br /> [[File:C2-Ex1-2-3-neut.png|thumb|right|200px|link=]]<br /> <br /> <br /> :::* Basis B (1,2,3) fixed relative to R and BL &lt;math&gt;\Omegavec_\Rs^\Bs=\vec{0},\Omegavec_{\Bs\Ls}^\Bs = \vec{0}&lt;/math&gt;<br /> :::* Basis B&#039; (1&#039;,2&#039;,3&#039;) fixed relative to the bar, thus moving in R and BL: &lt;math&gt;\velang{P}{B&#039;}=\vec{0}&lt;/math&gt;, &lt;math&gt;\velang{RL}{B&#039;} = -\vec{\dot{\psi}}&lt;/math&gt;<br /> <br /> <br /> <br /> ::Projection of the position vector &lt;math&gt;\OQvec&lt;/math&gt; on both bases:<br /> ::&lt;math&gt;\braq{\OQvec}{B} = \vector{\xs+\Ls sin\psi}{\Ls cos\psi}{0}&lt;/math&gt;, &lt;math&gt;\braq{\OQvec}{B&#039;} = \vector{\Ls+\xs sin\psi}{xcos\psi}{0}&lt;/math&gt;<br /> <br /> ::Velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B} = \braq{\dert{\OQvec}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B} = \vector{\dot\xs+\Ls\dot\psi cos\psi}{-\Ls\dot\psi sin\psi}{0}&lt;/math&gt;<br /> &lt;math&gt;\braq{\vel{Q}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\OQvec}{B&#039;} + \braq{\velang{B&#039;}{R} \times \OQvec}{B&#039;} = \vector{\dot\xs sin\psi+\xs\dot\psi cos\psi}{\dot\xs cos\psi - \xs \dot\psi sin \psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{\Ls+\xs sin\psi}{\xs cos\psi}{0}=\vector{\dot\xs sin \psi}{\dot\xs cos\psi + \Ls\dot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> ::If we want to calculate the velocity of &lt;math&gt;\Qs&lt;/math&gt; relative to BL, the position vector to be differentiated is &lt;math&gt;\vecbf{CQ}&lt;/math&gt;:<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\vecbf{CQ}}{B} = \vector{\Ls sin \psi}{\Ls cos \psi}{0}; \braq{\vecbf{CQ}}{B&#039;}=\vector{\Ls}{0}{0}<br /> &lt;/math&gt;&lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.2 Acceleration of a particle==<br /> <br /> The &#039;&#039;&#039;acceleration of a particle &#039;&#039;&#039; &lt;math&gt;\Qs&lt;/math&gt; (or of a point belonging to a rigid body) &#039;&#039;&#039;relative to a reference frame R&#039;&#039;&#039;, &lt;math&gt;\acc{Q}{R}&lt;/math&gt;, is the rate of change of its velocity with time:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R} = \dert{\vel{Q}{R}}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-2.1: rotating platform====<br /> ------<br /> <br /> &lt;small&gt;<br /> ::In the circular motion of point &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;platform relative to the ground &#039;&#039;&#039;]]&lt;/span&gt;, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; comes both from the change of value and the change of orientation of &lt;math&gt;\vel{Q}{R}&lt;/math&gt;. As &lt;math&gt;\vel{Q}{R}&lt;/math&gt; is always perpendicular to &lt;math&gt;\OQvec&lt;/math&gt;, its rate of change of orientation is &lt;math&gt;\dot\psi&lt;/math&gt;, the same as that of &lt;math&gt;\OQvec&lt;/math&gt; :<br /> [[File:C2-Ex2-1-eng.png|thumb|center|450px|link=]]<br /> <br /> ::The &lt;math&gt;\acc{Q}{R}&lt;/math&gt; direction is not any of the directions associated to the system (not the radial direction, not that perpendicular to the radius). For that reason, it is better to represent it as the addition of the two terms &lt;math&gt;\rs\ddot\psi&lt;/math&gt; and &lt;math&gt;r\dot\psi^2&lt;/math&gt; , whose directions do correspond to one of those singular directions.<br /> <br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.1: rotating platform|&#039;&#039;&#039;example C2-1.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> &lt;center&gt;&lt;math&gt;<br /> \braq{\acc{Q}{R}}{B} = \braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B} = \vector{-\rs \ddot\psi sin\psi - \rs \dot\psi^2cos\psi}{\rs\ddot\psi cos\psi - \rs \dot\psi^2sin\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;} = \braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;} + \braq{\velang{B&#039;}{R} \times \vel{Q}{R}}{B} = \vector{0}{\rs\ddot\psi}{0} + \vector{0}{0}{\dot\psi} \times \vector{0}{\rs\dot\psi}{0} = \vector{-\rs\dot\psi^2}{\rs\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-2.2: Euler pendulum====<br /> ------<br /> &lt;small&gt;<br /> ::The calculation of the acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to the ground (R) is laborious because the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; comes from the addition of two terms:<br /> ::&lt;math&gt;\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}&lt;/math&gt;.<br /> <br /> :::* &lt;math&gt;\dert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt;: constant direction (horizontal), variable value &lt;math&gt;(\dot\xs)&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vec{\Os_\Rs\Cbf}}{R}&lt;/math&gt; is horizontal with value &lt;math&gt;\ddot\xs&lt;/math&gt;.<br /> :::* &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt;: direction perpendicular to the bar, thus variable; variable value &lt;math&gt;\Ls\dot\psi&lt;/math&gt;. Thus, its time derivative &lt;math&gt;\ddert{\vecbf{CQ}}{R}&lt;/math&gt; has a component perpendicular to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (parallel to the bar) with value &lt;math&gt;\Ls\dot\psi\cdot\dot\psi&lt;/math&gt; , and another one parallel to &lt;math&gt;\dert{\vecbf{CQ}}{R}&lt;/math&gt; (perpendicular to the bar) with value &lt;math&gt;\Ls\ddot\psi&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex2-2-neut.png|thumb|center|300px|link=]]<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> -----------<br /> -----------<br /> <br /> ==C2.3 Intrinsic directions. Intrinsic components of the acceleration==<br /> A simple drawing shows that the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; relative to a reference frame R is always tangent to the trajectory it describes in R (&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039;). Its direction is the &#039;&#039;&#039;tangential direction&#039;&#039;&#039;.<br /> [[File:C2-1-eng.png|thumb|center|375px|link=|]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.1&#039;&#039;&#039; The velocity vector is always tangent to the trajectory&lt;/center&gt;&lt;/small&gt;<br /> <br /> In a general case, the velocity &lt;math&gt;\vel{Q}{R}&lt;/math&gt; changes both its value and its direction. Hence, the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has two components, one associated with the change of value (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and another one associated with the change of direction (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;). Those components are the &#039;&#039;&#039;intrinsic components of the acceleration&#039;&#039;&#039;, and they are called &#039;&#039;&#039;tangential component&#039;&#039;&#039; &lt;math&gt;\accs{Q}{R}&lt;/math&gt; and &#039;&#039;&#039;normal component&#039;&#039;&#039; &lt;math&gt;\accn{Q}{R}&lt;/math&gt;, respectively:<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\accs{Q}{R}+\accn{Q}{R}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;, the tangential component is perpendicular to the radius, and the normal one is parallel to the radius and pointing to the center of the trajectory (&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039;):<br /> [[File:C2-2-neut.png|thumb|center|275px|link=]]<br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.2&#039;&#039;&#039; Intrinsic components of the acceleration in a circular motion&lt;/center&gt;&lt;/small&gt;<br /> That result may be used locally for any other movement. Indeed, as the calculation of the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; with respect to a reference frame R (&lt;math&gt;\vel{Q}{R}&lt;/math&gt;) calls for two consecutive position vectors (or, what is the same, two consecutive points of the trajectory), that of the acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; calls for three:<br /> &lt;center&gt;<br /> &lt;math&gt;\acc{Q}{R}=\dert{\vel{Q}{R}}{R}\simeq\frac{\vvec_\Rs(\textbf{Q},\textrm{t+dt})-\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}\equiv\frac{\Delta\vvec_\Rs(\textbf{Q},\textrm t)}{\Delta t(\rightarrow0)}&lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> The calculation of vector &lt;math&gt;\Delta\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; calls for three consecutive points of the trajectory (two for each velocity, where the last point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm t)&lt;/math&gt; and the first point to calculate &lt;math&gt;\vvec_\Rs(\textbf{Q},\textrm{t+dt})&lt;/math&gt; are the same). These three points define a plane (&#039;&#039;&#039;osculating plane&#039;&#039;&#039;), and there is just one circle containing the three of them. That is: any trajectory may be approximated &lt;span style=&quot;text-decoration: underline;&quot;&gt;locally&lt;/span&gt; by a circle (&#039;&#039;&#039;osculating circle&#039;&#039;&#039;). The center and the radius of that circle are the &#039;&#039;&#039;center of curvature&#039;&#039;&#039; and the &#039;&#039;&#039;radius of curvature&#039;&#039;&#039; of the trajectory of Q relative R (&lt;math&gt;\textrm{CC}_\textrm{R}(\textbf{Q})&lt;/math&gt; and &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; respectively). The results obtained for the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt; may be used locally to calculate &lt;math&gt;\Re_\textrm{R}(\textbf Q)&lt;/math&gt; (&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039;).<br /> <br /> [[File:C2-3-eng.png|thumb|center|500px|link=]]<br /> <br /> &lt;small&gt;&lt;center&gt;&#039;&#039;&#039;Figure C2.3&#039;&#039;&#039; local geometry of the trajectory of a particle Q relative to a reference frame R&lt;/center&gt;&lt;/small&gt;<br /> Both the radius of curvature and the position of the center of curvature change along the trajectory in general. In rectilinear spans, as there is no change in the velocity direction, the normal component of the acceleration is zero, and the radius of curvature becomes infinite. <br /> <br /> The &#039;&#039;&#039;tangential unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{s}&lt;/math&gt; (&lt;math&gt;\vecbf{s}=\velo{R}/|\velo{R}|=\accso{R}/|\accso{R}|&lt;/math&gt;) and the &#039;&#039;&#039;normal unit vector &#039;&#039;&#039; &lt;math&gt;\vecbf{n}&lt;/math&gt; (&lt;math&gt;\vecbf{n}=\accno{R}/|\accno{R}|&lt;/math&gt;) may be completed with a third unit vector &lt;math&gt;\vecbf{b}&lt;/math&gt; orthogonal to the other two (&#039;&#039;&#039;binormal unit vector&#039;&#039;&#039;, &lt;math&gt;\vecbf{b}\equiv\vecbf{s}\times\vecbf{n}&lt;/math&gt;), and constitute the &#039;&#039;&#039;intrinsic basis&#039;&#039;&#039; or &#039;&#039;&#039;Frenet basis&#039;&#039;&#039; for the motion of &lt;math&gt;\Qs&lt;/math&gt; in the reference frame R.<br /> <br /> <br /> ====✏️ EXAMPLE C2-3.1: Euler pendulum====<br /> ---------<br /> &lt;small&gt;<br /> ::In the circular motion of the endpoint &lt;math&gt;\Qs&lt;/math&gt; of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;bar relative to the block&#039;&#039;&#039;]]&lt;/span&gt;, the two intrinsic components of the acceleration &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; are nonzero. Their values and directions are those of the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;circular motion&#039;&#039;&#039;]]&lt;/span&gt;:<br /> :::* tangential acceleration &lt;math&gt; \accs{Q}{BL}&lt;/math&gt;: parallel to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\ddot\psi&lt;/math&gt;.<br /> :::* normal acceleration &lt;math&gt;\accn{Q}{BL}&lt;/math&gt; : perpendicular to &lt;math&gt;\vel{Q}{BL}&lt;/math&gt; with value L&lt;math&gt;\dot\psi^2&lt;/math&gt;.<br /> [[File:C2-Ex3-1-1-neut.png|thumb|center|300px|link=]]<br /> ::Though it is evident that the radius of curvature of the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to BL is L (it is a circular motion), it can also be obtained as &lt;math&gt;\frac{\vecbf{v}_{\textrm{BL}}^2(\Qs)}{|\accn{Q}{BL}|}=\frac{(\Ls\dot\psi)^2}{\Ls\dot\psi^2}=\Ls&lt;/math&gt;.<br /> <br /> ::The acceleration &lt;math&gt;\acc{Q}{R}&lt;/math&gt; has been described in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.2: Euler pendulum|&#039;&#039;&#039;example C2-2.2&#039;&#039;&#039;]]&lt;/span&gt; as the addition of three terms (the two horizontal ones corresponding to &lt;math&gt;\acc{Q}{BL}&lt;/math&gt; plus a permanently horizontal one with value &lt;math&gt;\ddot\xs&lt;/math&gt;). Identifying in that case which is the tangential component (parallel to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) and which is the normal one (orthogonal to &lt;math&gt;\vel{Q}{R}&lt;/math&gt;) is not straightforward, as the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction is not that of a singular direction of the problem &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-1.2: Euler pendulum|&#039;&#039;&#039;example C2-1.2&#039;&#039;&#039;]]&lt;/span&gt;.<br /> ::That identification is straightforward in two particular configurations where the &lt;math&gt;\vel{Q}{R}&lt;/math&gt; direction (which is the tangential direction) is horizontal:<br /> [[File:C2-Ex3-1-2-eng.png|thumb|center|400px|link=]]<br /> ::The radius of curvature of the pendulum endpoint relative to the ground for the &lt;math&gt;\psi=0&lt;/math&gt; configuration is:<br /> [[File:C2-Ex3-1-3-eng.png|thumb|center|400px|link=]]<br /> ::The center of curvature is always above &lt;math&gt;\Qs&lt;/math&gt; because the normal acceleration points in that direction.<br /> ::Particular cases:<br /> <br /> [[File:C2-Ex3-1-4-neut.png|thumb|center|400px|link=]]<br /> ::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the &lt;math&gt;\psi=0&lt;/math&gt; configuration for those two particular cases. <br /> <br /> ::Though it is a laborious, it is possible to calculate &lt;math&gt;\re{Q}{R}&lt;/math&gt; for a general configuration if we remember that only the parallel components participate in the scalar product &lt;math&gt;\vel{Q}{R}\cdot\acc{Q}{R}&lt;/math&gt; (and so &lt;math&gt;\accs{Q}{R}&lt;/math&gt;), and that only the orthogonal components participate in the cross product &lt;math&gt;\vel{Q}{R}\times\acc{Q}{R}&lt;/math&gt;, (and so &lt;math&gt;\accn{Q}{R}&lt;/math&gt;) &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-3.1: Euler pendulum|(&#039;&#039;&#039;example C2-3.1&#039;&#039;&#039;]]&lt;/span&gt; analytical). The result is:<br /> <br /> &lt;center&gt;&lt;math&gt;\re{Q}{R}=\frac{\textbf{v}_{\Rs}^2(\Qs)}{|\accn{Q}{R}|}=\frac{\left[\dot\xs^2+\left(\Ls\dot\psi\right)^2+2\Ls\dot\xs\dot\psi cos\psi\right]^{3/2}}{\left|\Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)\right|}&lt;/math&gt;&lt;/center&gt;<br /> <br /> ::When the calculated expressions are complicated (as the previous one), it is advisable to check that it works in simple situations to avoid easily detectable errors. For example:<br /> <br /> :::*If &lt;math&gt;\dot\xs=0&lt;/math&gt; permanently (that is, &lt;math&gt;\ddot\xs=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is circular with radius L:<br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\xs=0, \ddot\xs=0}=\frac{\left(\Ls^2\dot\psi^2\right)^{3/2}}{\Ls\dot\psi^2\Ls\dot\psi}=\Ls&lt;/math&gt;<br /> <br /> :::*If &lt;math&gt;\dot\psi=0&lt;/math&gt; permanently (&lt;math&gt;\ddot\psi=0&lt;/math&gt;), the trajectory of &lt;math&gt;\Qs&lt;/math&gt; relative to R is rectilinear, and the radius of curvature has to be infinite: <br /> <br /> :::&lt;math&gt;\re{Q}{R}\big]_{\dot\psi=0, \ddot\psi=0}=\frac{(\dot\xs^2)^{3/2}}{0}\rightarrow\infty&lt;/math&gt;<br /> &lt;div&gt;<br /> =====Analytical calculation ➕=====<br /> ::The vector bases B and B’ are the same as in &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-2.1: rotating platform|&#039;&#039;&#039;example C2-2.1&#039;&#039;&#039;]]&lt;/span&gt;.<br /> <br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to BL:<br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B} = \braq{\dert{\vel{Q}{BL}}{BL}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B} = \vector{\Ls\ddot\psi cos\psi-\Ls \dot\psi^2 sin\psi}{-\Ls \ddot\psi sin\psi -\Ls \dot\psi^2 cos\psi}{0}&lt;/math&gt; <br /> &lt;math&gt;\braq{\acc{Q}{BL}}{B&#039;} = \braq{\dert{\vel{Q}{BL}}{BL}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B&#039;}+ \braq{\velang{B&#039;}{BL}\times \OQvec}{B} = \vector{0}{\Ls\ddot\psi}{0} + \vector{0}{0}{\dot\psi}\times \vector{0}{\Ls\dot\psi}{0} = \vector{-\Ls\dot\psi^2}{\Ls\ddot\psi}{0}&lt;/math&gt;<br /> &lt;/center&gt;<br /> ::Acceleration of &lt;math&gt;\Qs&lt;/math&gt; relative to R:<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot\xs+\Ls\ddot\psi cos\psi-\Ls\dot\psi^2sin\psi}{-\Ls\ddot\psi sin\psi-\Ls\dot\psi^2cos\psi}{0}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \braq{\acc{Q}{R}}{B&#039;}=\braq{\dert{\vel{Q}{R}}{R}}{B&#039;} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times \OQvec}{B}=\vector{\ddot\xs sin\psi+\dot\xs\dot\psi cos\psi}{\ddot\xs cos\psi-\dot\xs\dot\psi sin\psi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot\xs sin\psi}{\dot\xs cos\psi+\Ls\dot\psi}{0}=\vector{\ddot\xs sin\psi - \Ls \dot\psi^2}{\ddot\xs cos\psi+\Ls\ddot\psi}{0}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> [[File:C2-Ex3-1-6-neut.png|thumb|center|500px|link=]]<br /> <br /> ::The calculation of the radius of curvature in the general configuration is cumbersome. As it is a planar motion, and the velocity and the acceleration only have two components, the third component will not be shown. The vector basis is B (but the same result would be obtained through the vector basis B’).<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \braq{\vel{Q}{R}}{B} = \vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls \dot\psi sin\psi}, \braq{\acc{Q}{R}}{B} = \vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\acc{Q}{R}\times\frac{\vel{Q}{R}}{\abs{\vel{Q}{R}}}}<br /> &lt;/math&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{\frac{1}{\sqrt{({\dot\xs + \Ls\dot\psi cos\psi)^2+(\Ls \dot\psi sin\psi)^2}}}\vecdosd{\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi}{\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi}\times\vecdosd{\dot\xs + \Ls\dot\psi cos\psi}{\Ls\dot\psi sin\psi}}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=\abs{<br /> \frac<br /> {<br /> (\ddot\xs + \Ls \ddot\psi cos \psi -\Ls \dot\psi^2sin\psi)\Ls \dot\psi sin\psi-(\Ls\ddot\psi sin\psi + \Ls \dot\psi^2 cos\psi)(\dot\xs + \Ls\dot\psi cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \abs{\accn{Q}{R}}=<br /> \abs{<br /> \frac<br /> {<br /> \Ls\ddot\xs\dot\psi sin\psi-\Ls\dot\xs\ddot\psi sin\psi-L^2\dot\psi^3-\Ls\dot\xs\dot\psi^2cos\psi<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }=<br /> \abs{<br /> \frac<br /> {<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> {<br /> \sqrt{\dot\xs^2+(\Ls\dot\psi)^2+2\Ls \dot\xs\dot\psi cos\psi}<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \Re_\Rs(\Qs)=\frac{\textrm{v}^2_\Rs(\Qs)}{\abs{\accn{Q}{R}}}=<br /> \frac<br /> {<br /> \left( \dot\xs^2+(\Ls\dot\psi)^2+2\Ls\dot\xs\dot\psi cos\psi\right)^{3/2}<br /> }<br /> {<br /> \abs{<br /> \Ls(\ddot\xs\dot\psi-\dot\xs\ddot\psi)sin\psi-\Ls\dot\psi^2(\Ls\dot\psi+\dot\xs cos\psi)<br /> }<br /> }<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/div&gt;&lt;/small&gt;<br /> <br /> <br /> ---------<br /> ---------<br /> <br /> ==C2.4 Angular velocity of a rigid body==<br /> The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|&#039;&#039;&#039;configuration of a rigid body&#039;&#039;&#039;]]&lt;/span&gt; S relative to a reference frame R is totally defined through the position of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body and the orientation of S relative to R (described, for instance, by means of &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt;). Similarly, the evolution of the configuration relative to R can be described through the velocity of a point &lt;math&gt;\Qs&lt;/math&gt; of the rigid body &lt;math&gt;\vel{Q}{R}&lt;/math&gt;, and the &#039;&#039;&#039;angular velocity&#039;&#039;&#039; of the rigid body &lt;math&gt;\velang{S}{R}&lt;/math&gt; (rate of change of orientation with time). When the orientation relative to R is constant with time, we say that the rigid body has a &#039;&#039;&#039;translational motion&#039;&#039;&#039; &lt;math&gt;\left(\velang{S}{R}=0\right)&lt;/math&gt;.<br /> <br /> <br /> ===Simple rotation===<br /> <br /> The orientation of a rigid body with planar motion relative to a reference frame R &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;is totally defined by an angle &lt;math&gt;\psi&lt;/math&gt;&#039;&#039;&#039;]]&lt;/span&gt;.If that orientation changes, &lt;math&gt;\dot\psi\neq0&lt;/math&gt; . <br /> <br /> Giving the value of &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt; is not enough to define how the orientation of a rigid body changes when its motion is a planar one.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.1: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> :{|<br /> |-<br /> | <br /> [[File:C2-Ex4-1-1-eng.png|250px|thumb|link=]]<br /> || The wheel has a planar motion relative to R. Its center &lt;math&gt;\Cs&lt;/math&gt; is fix fixed in R, and its orientation changes with a rate &lt;math&gt;\dot\psi&lt;/math&gt; &lt;math&gt;[rad/s]&lt;/math&gt;. With just that information, we cannot infer the motion it describes. For instance, that information might correspond to any of the following cases:<br /> |}<br /> [[File:C2-Ex4-1-2-neut.png|400px|thumb|center|link=]]<br /> <br /> :::* Case (a): angle &lt;math&gt;\psi&lt;/math&gt; defined on the horizontal plane; the plane of motion is horizontal.<br /> :::* Case (b): angle &lt;math&gt;\psi&lt;/math&gt; defined on a vertical plane; the plane of motion is vertical.<br /> <br /> ::If nothing is said about the plane where the angle has been defined (and that is equivalent to giving a direction: the direction perpendicular to the plane), the motion is not defined univocally.<br /> &lt;/small&gt;<br /> ------------<br /> <br /> <br /> Thus, the movement associated with a change in orientation is defined by the rate of change of the angle plus a direction. The mathematical object including those two features is a vector. Hence, the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is a vector. The convention to associate a direction to that vector is the screw rule (or the &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[Vector calculus#V.2 Operations between vectors with geometric representation|&#039;&#039;&#039;right hand rule&#039;&#039;&#039;]]&lt;/span&gt;, or the corkscrew rule,).<br /> <br /> ====✏️ EXAMPLE C2-4.2: wheel with planar motion====<br /> ---------<br /> &lt;small&gt;<br /> ::The angular velocity associated with movements (a) and (b) in the previous example is:<br /> [[File:C2-Ex4-2-1-eng.png|450px|thumb|center|link=]]<br /> &lt;/small&gt;<br /> -------------<br /> <br /> ===Rotation in space===<br /> The orientation of a rigid body moving in space relative to a reference frame R may be given through three &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|&#039;&#039;&#039;Euler angles&#039;&#039;&#039;]]&lt;/span&gt; &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;. We may associate an angular velocity to the change of each of those angles.<br /> <br /> <br /> ====✏️ EXAMPLE C2-4.3: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|&#039;&#039;&#039;orientation of a gyroscope&#039;&#039;&#039;]]&lt;/span&gt; relative to the ground (R) may be given through three Euler angles. The angular velocities associated with &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt; have the following interpretation: <br /> <br /> ::&lt;math&gt;\vecdot\psi=\velang{fork}{R}&lt;/math&gt;, &lt;math&gt;\vecdot\theta=\velang{arm}{fork}&lt;/math&gt;, &lt;math&gt;\vecdot\varphi=\velang{disk}{arm}&lt;/math&gt;.<br /> <br /> <br /> [[File:C2-Ex4-3-1-eng-jpg.jpg|thumb|400px|center|link=]]<br /> ::Those angular velocities can be projected on any vector basis suggested by the problem:<br /> :::* Vector basis &lt;math&gt;\Bs_\Rs&lt;/math&gt; fixed to the reference frame <br /> :::* Vector basis &lt;math&gt;\Bs&lt;/math&gt; fixed to the fork (it can be generated from &lt;math&gt;\Bs_\Rs&lt;/math&gt; through the &lt;math&gt;\dot\psi&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs&#039;&lt;/math&gt; fixed to the arm (it can be generated from &lt;math&gt;\Bs&lt;/math&gt; through the &lt;math&gt;\dot\theta&lt;/math&gt; rotation)<br /> :::* Vector basis &lt;math&gt;\Bs_\textrm{V}&lt;/math&gt; fixed to the disk<br /> <br /> [[File:C2-Ex4-3-2-eng-jpg.jpg|thumb|400px|center|link=]]<br /> <br /> ::Nevertheless, it is advisable to choose a vector basis where the maximum number of rotations have the direction of one of the axes in the basis, in order to minimize the projections. As the axes of the three rotations do not correspond to an orthogonal trihedral, it will always be necessary to project at least one of the angular velocities (&lt;math&gt;\vec{\dot{\psi}}, \vec{\dot{\theta}}, \vec{\dot{\varphi}}&lt;/math&gt;). With a proper choice of the vector basis, the angular velocities to be projected will be contained on a plane defined by two axes of the vector basis, and that simplifies the operation. Hence, the best choices are B or B’. The angular velocities that will have two components will be &lt;math&gt;\vec{\dot{\varphi}}&lt;/math&gt;, when we choose B, and &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; when we choose B’:<br /> <br /> ::{|<br /> |&lt;math&gt;\braq{\velang{fork}{R}}{B}=\vector{0}{0}{\dot\psi}, \braq{\velang{arm}{fork}}{B}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B}=\vector{0}{\dot{\varphi}cos\theta}{\dot{\varphi}sin\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{fork}{R}}{B&#039;}=\vector{0}{\dot{\psi}sin\theta}{\dot{\psi}cos\theta}, \braq{\velang{arm}{fork}}{B&#039;}=\vector{\dot{\theta}}{0}{0}, \braq{\velang{disk}{arm}}{B&#039;}=\vector{0}{\dot{\varphi}}{0}&lt;/math&gt;<br /> |[[File:C2-Ex4-3-3-neut.png|thumb|right|200px|link=]]<br /> |}<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> ---------<br /> ---------<br /> <br /> ==C2.5 Angular acceleration of a rigid body==<br /> <br /> The &#039;&#039;&#039;angular acceleration&#039;&#039;&#039; of a rigid body S relative to a reference frame R (&lt;math&gt;\accang{S}{R}&lt;/math&gt;) is the time derivative of its angular velocity relative to R:<br /> <br /> &lt;center&gt;&lt;math&gt;\accang{S}{R}= \dert{\velang{S}{R}}{R}&lt;/math&gt;&lt;/center&gt;<br /> <br /> The description of the angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; may be any (rotations about fixed axes, Euler rotations...). When the rigid body has a planar motion relative to R, the direction of its angular velocity &lt;math&gt;\velang{S}{R}&lt;/math&gt; is constant (it is always perpendicular to the plane of motion). Hence, the angular acceleration comes exclusively from the change of value of &lt;math&gt;\velang{S}{R}&lt;/math&gt;, and it is parallel to &lt;math&gt;\velang{S}{R}&lt;/math&gt;. In general motions in space, if &lt;math&gt;\velang{S}{R}&lt;/math&gt; is described through Euler rotations, &lt;math&gt;\accang{S}{R}&lt;/math&gt; may come from the change of values of (&lt;math&gt;\vecdot\psi&lt;/math&gt;, &lt;math&gt;\vecdot\theta&lt;/math&gt;,&lt;math&gt;\vecdot\varphi&lt;/math&gt;) iand the change of direction of de &lt;math&gt;\vecdot\theta&lt;/math&gt; and &lt;math&gt;\vecdot\varphi&lt;/math&gt; (&lt;math&gt;\vecdot\psi&lt;/math&gt; has always a constant direction in R).<br /> <br /> <br /> ==== ✏️ EXAMPLE C2-5.1: gyroscope====<br /> ---------<br /> &lt;div&gt;<br /> &lt;small&gt;<br /> ::The fork of a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt; has a planar motion relative to the ground (R), and its angular velocity is vertical: &lt;math&gt;\velang{fork}{R}=\vecdot\psi&lt;/math&gt; Its angular acceleration is also vertical, with value &lt;math&gt;\ddot{\psi}: \accang{S}{R}=\vec{\ddot{\psi}}&lt;/math&gt;.<br /> <br /> ::The angular acceleration of the disk is more complicated. It can be obtained through the geometric time derivative of &lt;math&gt;\velang{disk}{R}=\vecdot\psi+\vecdot\theta+\vecdot\varphi&lt;/math&gt;. The rotation &lt;math&gt;\vecdot\varphi&lt;/math&gt; can be decomposed in a vertical component with value &lt;math&gt;\dot\varphi\textrm{sin}\theta&lt;/math&gt;, and a horizontal one with value &lt;math&gt;\dot\varphi\textrm{cos}\theta&lt;/math&gt;. The vertical component can only change its value, whereas the horizontal its value and its direction (because of &lt;math&gt;\vecdot\psi&lt;/math&gt;).<br /> <br /> <br /> [[File:C2-Ex5-1-neut.png|thumb|center|400px|link=]]<br /> <br /> <br /> &lt;center&gt;<br /> {|<br /> |<br /> Time derivative of the vertical components <br /> <br /> [[File:C2-Ex5-3-neut.png|thumb|center|350px|link=]]<br /> <br /> |<br /> :::Time derivative of the horizontal components <br /> <br /> :::[[File:C2-Ex5-4-neut.png|thumb|center|350px|link=]]<br /> |}&lt;/center&gt;<br /> <br /> <br /> =====Analytical calculation ➕=====<br /> ::The same result is obtained if the time derivative is performed analytically through the vector basis rotating with &lt;math&gt;\vecdot\psi&lt;/math&gt; relative to R or that rotating with &lt;math&gt;\vecdot\psi+\vecdot\theta&lt;/math&gt; (also relative to R): <br /> <br /> &lt;center&gt;<br /> &lt;math&gt;\braq{\velang{}{R}}{B}=\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B}=\braq{\dert{\velang{disk}{R}}{R}}{B}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B}+\braq{\velang{B}{R}\times\velang{disk}{R}}{B}&lt;/math&gt;<br /> <br /> <br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi cos\theta}{\dot\psi+\dot\varphi sin\theta}=\vector{\ddot\theta-\dot\psi\dot\varphi cos\theta}{\ddot\varphi cos\theta-\dot\varphi\dot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dot\psi cos\theta}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\velang{disk}{R}}{B&#039;}=\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta},&lt;/math&gt;&lt;span&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;/span&gt;&lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\braq{\dert{\velang{disk}{R}}{R}}{B&#039;}=\frac{\textrm{d}}{\textrm{dt}}\braq{\velang{disk}{R}}{B&#039;}+\braq{\velang{B&#039;}{R}\times\velang{disk}{R}}{B&#039;}&lt;/math&gt;<br /> <br /> &lt;math&gt;\braq{\accang{disk}{R}}{B&#039;}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta cos\theta}{\ddot\psi cos\theta-\dot\psi\dot\theta sin\theta}+\vector{\dot\theta}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta}=\vector{\ddot\theta-\dot\psi(\dot\varphi+\dot\psi sin\theta)}{\ddot\varphi+\ddot\psi sin\theta+\dot\psi\dot\theta}{\ddot\psi cos\theta+\dot\theta\dot\varphi}&lt;/math&gt;<br /> &lt;/center&gt;<br /> &lt;/small&gt;<br /> &lt;/div&gt;<br /> <br /> <br /> ----------<br /> ----------<br /> <br /> ==C2.6 Particle kinematics VS rigid body kinematics==<br /> Particle (point) and rigid body are two very different models. From a kinematic point of view, the second one is richer because it includes the concept of rotation (not applicable to particles, as they cannot be orientated because they have no dimensions). Because of rotations, points of a same rigid boy may describe different trajectories.<br /> <br /> One has to bear that in mind in order not to use erroneously concepts that only apply to one of the models when talking about the other. The following examples illustrate some wrong statements and some correct ones.<br /> <br /> <br /> <br /> ====✏️ EXAMPLE C2-6.1: particle inside a circular guide====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-1-neut_REV01.png|thumb|left|180px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Vector &lt;math&gt;\vec{\textbf{OP}}&lt;/math&gt; rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a circular trajectory relative to R (or has a circular motion relative to): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.2: particle on an incline====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-2-neut.png|thumb|left|200px|link=]]<br /> |Particle &lt;math&gt;\Ps&lt;/math&gt; has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> Particle &lt;math&gt;\Ps&lt;/math&gt; describes a rectilinear trajectory relative to R (or has a rectilinear motion relative to R): &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.3: wheel with a nonsliding contact with the ground and with planar motion====<br /> ------------<br /> &lt;small&gt;<br /> [[File:C2-Ex6-3-neut.png|thumb|center|540px|link=]]<br /> ::Points on the wheel rotate relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The wheel rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a translational motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> ::The center of the wheel has a rectilinear motion relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&#039;&#039;&#039;Some points on a rotating rigid body may have rectilinear motion.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/ED3LXV6JWCA?start=11&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.1&#039;&#039;&#039; Visualització de les trajectòries de punts&lt;/small&gt;&lt;/center&gt;<br /> <br /> ====✏️ EXAMPLE C2-6.4: motion of a ferris wheel====<br /> ------------<br /> &lt;small&gt;<br /> ::{|<br /> |[[File:C2-Ex6-4-1-eng.png|thumb|left|200px|link=]]<br /> |The ring rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> <br /> <br /> <br /> The cabin rotates relative to R: &#039;&#039;&#039;&lt;span style=&quot;color:red;&quot;&gt;WRONG&lt;/span&gt;&#039;&#039;&#039; if we neglect the pendulum motion, the ground and the ceiling of the cabin are always parallel to te ground, so it does not rotate).<br /> <br /> <br /> The cabin has a translational motion relative to R &#039;&#039;&#039;&lt;span style=&quot;color:rgb(24,182,96);&quot;&gt;CORRECT&lt;/span&gt;&#039;&#039;&#039;<br /> |-<br /> |[[File:C2-Ex6-4-2-neut.png|thumb|left|200px|link=]]<br /> |In this case, all points in the cabin have circular motions with the same radius relative to R, but different center of curvature.<br /> <br /> <br /> In such a case, we may combine a concept from rigid body kinematics (translational motion) with a concept from particle kinematics (circular motion) to describe the motion of the cabin:<br /> <br /> <br /> The cabin has a &#039;&#039;&#039;translational circular motion&#039;&#039;&#039; relative to R.<br /> |}<br /> <br /> &lt;center&gt;&#039;&#039;&#039;Points in a rigid body with a translational motion may describe curvilinear trajectories.&#039;&#039;&#039;&lt;/center&gt;<br /> &lt;/small&gt;<br /> <br /> <br /> --------<br /> --------<br /> <br /> ==C2.7 Degrees of freedom==<br /> As we have seen through various examples in this unit, the velocities of the points in a mechanical system depend on a set of scalar variables with dimensions or . The minimum set of scalar variables of this sort needed to describe the system motion is the set of the &#039;&#039;&#039;degrees of freedom&#039;&#039;&#039; (DOF) of the system.<br /> <br /> When the system is just a free rigid body moving in space (without any contact with material objects), &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C4. Rigid body kinematics#C4.1 Velocity distribution|&#039;&#039;&#039;the number of DOF is 6&#039;&#039;&#039;]]&lt;/span&gt;: three associated with the motion of one point (for instance, &lt;math&gt;(\dot{\textrm{x}}, \dot{\textrm{y}}, \dot{\textrm{z}})&lt;/math&gt;) and three associated with the change of orientation of the rigid body (for instance, &lt;math&gt;(\dot{\psi}, \dot{\theta}, \dot{\varphi})&lt;/math&gt;).<br /> <br /> In mechanical engineering, the usual mechanical systems are multibody systems: sets of rigid bodies &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|&#039;&#039;&#039;linked&#039;&#039;&#039;]]&lt;/span&gt; through revolute joints, spherical joints... Because of these links (or constraints), the mechanical state of each rigid body (that is, its configuration in space and its motion) is related to that of the other rigid bodies: in a multibody System with N rigid bodies, the number of DOF is lower than 6N.<br /> <br /> <br /> ------------<br /> ------------<br /> <br /> ==C2.8 Usual constraints in mechanical systems==<br /> <br /> &#039;&#039;&#039;Falta paragraf versio catala&#039;&#039;&#039;<br /> <br /> <br /> &lt;center&gt;&lt;small&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;background-color:white; text-align:left&quot;<br /> |-<br /> |&lt;center&gt;[[File:C2-8-eng.png|thumb|center|175px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;With sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions), and two independent translational motions (along the two tangential directions)&lt;br&gt;&lt;br&gt;<br /> &#039;&#039;&#039;Without sliding:&#039;&#039;&#039; &lt;br&gt;<br /> It allows three independent rotations between the rigid bodies (about the normal direction and the two tangential directions)<br /> <br /> |-<br /> |&lt;center&gt;[[File:C2-8-revolucio-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;revolute joint&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1<br /> |-<br /> |&lt;center&gt;[[File:C2-8-cilindric-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;cylindrical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 1, and a translational motion (displacement without rotation) along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-prismatic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;prismatic joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows a translational motion between the two rigid bodies along axis 1.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-esferic-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;spherical joint &#039;&#039;&#039;&lt;br&gt;<br /> Allows three independent rotations between the two rigid bodies about axes 1, 2, 3.<br /> |-<br /> |&lt;center&gt;[[File:C2-8-helicoidal-neut.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;helical joint (screw)&#039;&#039;&#039;&lt;br&gt;<br /> Allows a rotation between the two rigid bodies about axis 3; this rotation provokes a displacement along axis 3. The relationship between the rotation and the displacement is given by the screw pitch e [mm/volta].<br /> |-<br /> |&lt;center&gt;[[File:C2-8-Cardan-rev.png|thumb|center|220px|link=]]&lt;/center&gt;||<br /> &#039;&#039;&#039;Cardan joint (universal joint)&#039;&#039;&#039;&lt;br&gt;<br /> Allows two independent rotations between the two rigid bodies about axes 1, 3.<br /> |}<br /> &lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/067F1MQVICs&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.2&#039;&#039;&#039; Junta Cardan (junta universal o de creueta)&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> &lt;center&gt;&lt;html&gt;&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube-nocookie.com/embed/K-xIHJErByk&quot; title=&quot;YouTube video player&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share&quot; allowfullscreen&gt;&lt;/iframe&gt;&lt;/html&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;small&gt;&#039;&#039;&#039;Video C2.3&#039;&#039;&#039; Graus de Llibertat d&#039;una roda emb moviment pla i contacte amb el terra&lt;/small&gt;&lt;/center&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.1: gyroscope====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::In a &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Movement of a mechanical system#✏️ EXAMPLE C2-4.3: gyroscope|&#039;&#039;&#039;gyroscope&#039;&#039;&#039;]]&lt;/span&gt;, the support does not move relative to the ground (R). There are revolute joints between the fork and the support, between the arm and the fork, and between the disk and the arm. All that can be represented through a simplified diagram:<br /> [[File:C2-Ex8-1-eng.png|thumb|center|600px|link=]]<br /> <br /> ::The position of point &lt;math&gt;\Os&lt;/math&gt; relative to the ground is constant. Hence, the gyroscope configuration is totally defined by the three angles &lt;math&gt;(\psi,\theta,\varphi)&lt;/math&gt;: the gyroscope has 3 IC relative to the ground.<br /> <br /> ::Regarding its motion, as the variation of any of those angles does not imply that of the other two, their time evolutions are independent: the gyroscope has 3 DOF relative to the ground, and they may be described through &lt;math&gt;(\dot\psi,\dot\theta,\dot\varphi)&lt;/math&gt;.<br /> |}<br /> &lt;/small&gt;<br /> <br /> ====✏️ EXEMPLE C2-8.2: tricycle====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The tricycle is a system with 5 rigid bodies: the chassis, the handlebar and the three wheels. There is no element fixed to the ground. There are revolute joints between the rear wheels and the chassis, between the handlebar and the chassis, and between the front wheel and the handlebar. Moreover, the wheels are in contact with the ground: that too is a restriction (or a constraint). If it moves on horizontal ground without sliding, that contact may be idealized as a single-point contact without sliding (whether a contact is a sliding or a nonsliding one depends on the system dynamics; in kinematics, sliding or nonsliding is a hypothesis).<br /> [[File:C2-Ex8-2-1-eng.png|thumb|650px|center|link=]]<br /> [[File:C2-Ex8-2-2-neut.png|thumb|450px|center|link=]]<br /> ::A good way to determine the number of DOF of a system relative to a reference frame is to count up how many motions have to be blocked to reach a complete rest. In a tricycle, if we block the motion of point &lt;math&gt;\Os&lt;/math&gt; (that can only be in the longitudinal direction of the wheels do not skid), the chassis would still be able to rotate about a vertical axis through &lt;math&gt;\Os&lt;/math&gt;. If we block that rotation &lt;math&gt;(\dot\psi=0)&lt;/math&gt;, the rear wheels are blocked, but the handlebar and the front wheel may still rotate about the vertical axis through the wheel center &lt;math&gt;(\dot\psi&#039;\neq 0)&lt;/math&gt;. If we blocked that last motion, the tricycle is at rets. We have blocked three motions, hence the tricycle has 3 DOF.<br /> |}<br /> &lt;/small&gt;<br /> <br /> <br /> ====✏️ EXAMPLE C2-8.3: spherical shell on a platform====<br /> ------------<br /> {|:<br /> &lt;small&gt;<br /> ::The system contains 4 rigid bodies: the platform, the shell, the arm and the fork. There are revolute joints between the platform and the ground, between the shell and the arm, between the arm and the fork, and between the fork and the ceiling (or the ground). Moreover, between shell and platform there is a single-point contact without sliding. <br /> <br /> [[File:C2-Ex8-3-eng.png|thumb|500px|center|link=]]<br /> ::The DOF of the System relative to the ground (R) can be discovered by blocking different motions until reaching a total rest:<br /> :::* block the rotation of the platform relative to the ground<br /> :::* block the rotation of the fork relative to the ground<br /> ::Under those conditions, though the revolute joint between shell and arm allows a rotation, that rotation would provoke a sliding motion between shell and platform, and that is not consistent with the hypothesis of nonsliding contact. Hence, the system is at rest: it has 2 DOF relative to the ground.<br /> |}&lt;/small&gt;<br /> <br /> <br /> -------------<br /> -------------<br /> <br /> ==C2.E General examples==<br /> ====🔎 Example C2-E.1: rotating pendulum====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> The plate is articulated at point &lt;math&gt;\Os&lt;/math&gt; O to a fork, which rotates with constant angular velocity &lt;math&gt;\psio&lt;/math&gt; relative to the ground (T). Between fork and ground (ceiling), and between plate and fork there are revolute joints.<br /> <br /> <br /> [[File:C2-E.Ex1-1-eng.png|thumb|400px|center|link=]]<br /> <br /> =====1. How many degrees of freedom (DoF) has the system? Describe them.=====<br /> &lt;div&gt; <br /> :The fork has a simple rotation relative to the ground about a vertical axis. <br /> <br /> :Independently from that rotation, the plate may rotate about the horizontal axis of the fork.<br /> <br /> :Those two motions are independent because, if we block one of them, the other one may still take place.<br /> <br /> :Hence, the system has two degrees of freedom.<br /> &lt;/div&gt;<br /> <br /> =====2. Find the angular velocity and the angular acceleration of the plate relative to the ground. =====<br /> &lt;div&gt;<br /> :The angular velocity of the plate is the superposition of &lt;math&gt;\vecdot\psi_0&lt;/math&gt; (1st Euler rotation, axis fixed to the ground) and &lt;math&gt;\vecdot\theta&lt;/math&gt; (2nd Euler rotation, axis rotating with &lt;math&gt;\vecdot\psi_0&lt;/math&gt;relative to the ground): &lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> <br /> <br /> :&lt;math&gt;\velang{plate}{E}=\vecdot\psi_0+\vecdot\theta=(\Uparrow \psio)+(\odot \dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-2-neut.png|thumb|right|200px|link=]]<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{\velang{plate}{E}}{E}=\dert{\vecdot\psi_0}{E}+\dert{\vecdot\theta}{E}=\dert{(\Uparrow \psio)}{E}+\dert{(\odot \dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :As &lt;math&gt;\vecdot\psi_0&lt;/math&gt; has constant value and direction, the angular acceleration will be associated only to the change of value and direction of &lt;math&gt;\vecdot\theta&lt;/math&gt;.It is a vector with variable value which rotates about a vertical axis because of the 1st Euler rotation &lt;math&gt;(\Omegavec^{\vecdot\theta}_\textrm{T} =\vecdot\psi_0)&lt;/math&gt;.<br /> <br /> :&lt;math&gt;\accang{plate}{E}=\dert{(\odot\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vecdot{\theta}}{|\vecdot{\theta}|}\right]+[\velang{$\vecdot{\theta}$}{$\Ts$}\times\vecdot{\theta}]=[\odot\ddot{\theta}]+[(\Uparrow\psio)\times(\odot\dot{\theta})]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The time derivative of the angular velocity can also be done analytically. The vector basis where the &lt;math&gt;\velang{plate}{E}&lt;/math&gt; projection is straightforward is the vector basis fixed to the fork &lt;math&gt;(\velang{B}{E}=\vecdot{\psi}_0)&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{plate}{E}}{B}=\vector{\dot{\theta}}{0}{\psio}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{plate}{E}}{B}=\braq{\dert{\velang{plate}{E}}{E}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{plate}{E}}{B}+\braq{\velang{B}{E}}{B}\times\braq{\velang{plate}{E}}{B}=\vector{\ddot{\theta}}{0}{0}+\vector{0}{0}{\psio}\times\vector{\dot{\theta}}{0}{\psio}=\vector{\ddot{\theta}}{\psio\dot{\theta}}{0}&lt;/math&gt;<br /> &lt;/div&gt;<br /> <br /> =====3. Find the velocity and the acceleration of point G of the plate relative to the ground. . =====<br /> &lt;div&gt;<br /> :As point &lt;math&gt;\Os&lt;/math&gt; is fixed to the ground, &lt;math&gt;\OGvec&lt;/math&gt; vector can be taken as position vector in the ground frame. Its value L is constant, but its direction is not because of &lt;math&gt;\psio&lt;/math&gt; and &lt;math&gt;\vecdot{\theta}&lt;/math&gt;: &lt;math&gt;\OGvec=(\searrow\Ls)^{*}&lt;/math&gt;.<br /> <br /> :&#039;&#039;&#039;Geometric calculation:&#039;&#039;&#039;<br /> :&lt;math&gt;\vel{G}{E}=\dert{\OGvec}{E}=[\text{change of direction}]_\Es=\velang{$\OGvec$}{E}\times\OGvec=(\vec{\dot{\psi}}_0+\vec{\dot{\theta}})\times\OGvec=\left[(\Uparrow\psio)+(\odot\dot{\theta})\right]\times(\searrow\Ls)=(\Uparrow\psio)\times(\rightarrow\Ls\text{cos}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> [[File:C2-E.Ex1-3-eng.png|thumb|right|300px|link=]]<br /> <br /> That velocity has variable value and direction, thus the acceleration has both parallel component and orthogonal component to the velocity. <br /> <br /> :&lt;math&gt;\acc{G}{E}=\dert{\vel{G}{E}}{E}=\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}+\dert{(\nearrow\Ls\dot{\theta})}{E}&lt;/math&gt;<br /> <br /> :The &lt;math&gt;(\otimes\Ls\psio\text{sin}\theta)&lt;/math&gt; vector rotates relative to the ground just because of &lt;math&gt;\psio&lt;/math&gt;, whereas the &lt;math&gt;(\nearrow\Ls\dot{\theta})&lt;/math&gt; vector rotates because of &lt;math&gt;\vec{\psio}&lt;/math&gt; and &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> :&lt;math&gt;\hspace{3.1cm}=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)\right]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+\left[\leftarrow\Ls\psio^2\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{E}=[\text{change of value}]+[\text{change of direction}]_\Es=&lt;/math&gt;<br /> &lt;math&gt;\hspace{2.9cm}=\left[\nearrow\Ls\ddot{\theta}\right]+\left[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})\right]=\left[\nearrow\Ls\ddot{\theta}\right]+\left[(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)+(\nwarrow\Ls\dot{\theta}^2)\right]<br /> &lt;/math&gt;<br /> :Finally: &lt;math&gt;\acc{P}{E}=(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nwarrow\Ls\dot{\theta}^2)+(\nearrow\Ls\ddot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Analytical calculation:&#039;&#039;&#039;<br /> :The whole calculation can be done analytically. The vector basis where the projection of &lt;math&gt;\OGvec&lt;/math&gt;is straightforward is fixed to the plate (base B’). That vector basis changes its orientation whenever the values of &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\theta&lt;/math&gt; change. Hence, the angular velocity of the vector basis is &lt;math&gt;\velang{B&#039;}{E}=\vec{\psio}+\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B&#039;}=\vector{0}{0}{-L}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{E}}{B&#039;}=\braq{\dert{\OGvec}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{0}{0}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{0}{0}{-\Ls}=\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{E}}{B&#039;}=\braq{\dert{\vel{G}{E}}{E}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{E}}{B&#039;}+\braq{\velang{B&#039;}{E}}{B&#039;}\times\braq{\vel{G}{E}}{B&#039;}=\vector{-\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}}{0}+\vector{\dot{\theta}}{\psio\text{sin}\theta}{\psio\text{cos}\theta}\times\vector{-\Ls\psio\text{sin}\theta}{\Ls\dot{\theta}}{0}=\vector{-2\Ls\psio\dot{\theta}\text{cos}\theta}{\Ls\ddot{\theta}-\Ls\psio^2\text{sin}\theta\text{cos}\theta}{\Ls\dot{\theta}^2+\Ls\psio^2\text{sin}^2\theta}&lt;/math&gt;<br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.2: placa articulada giratòria====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> La placa rectangular està unida a un suport giratori a través de dues barres paral·leles amb articulacions als extrems. Una tercera barra està enllaçada a la placa a través d’una &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;ròtula esfèrica&#039;&#039;&#039;]]&lt;/span&gt; a P i al suport a través d’un &lt;span style=&quot;text-decoration: underline;&quot;&gt;[[C2. Moviment d&#039;un sistema mecànic#C2.8 Enllaços habituals en els sistemes mecànics|&#039;&#039;&#039;enllaç cilíndric&#039;&#039;&#039;]]&lt;/span&gt;. El suport gira amb velocitat angular &lt;math&gt;\vecdot{\psi}&lt;/math&gt; variable respecte del terra (T).<br /> <br /> [[Fitxer:C3-E.Ex2-1-cat-esp.png|thumb|center|450px|link=]]<br /> <br /> [[Fitxer:C2-E.Ex2-2-cat.png|thumb|center|450px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :El suport pot girar lliurement al voltant de l’eix vertical fix a terra (rotació simple). Si el suport s’atura respecte del terra, el sistema encara es pot moure. <br /> :Respecte del suport, les barres &lt;math&gt;\OCvec&lt;/math&gt; poden fer una rotació simple al voltant de l’eix horitzontal perpendicular a les barres i que passa per &lt;math&gt;\Os&lt;/math&gt;. Si una d’aquestes barres s’atura respecte del suport, ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt;, ni la barra en colze es poden moure. Una anàlisi alternativa d’aquest segon grau de llibertat és veure que si la barra en colze s’atura (si s’atura la seva translació vertical respecte del suport), ni la placa, ni les barres &lt;math&gt;\OCvec&lt;/math&gt; es poden moure respecte del suport.<br /> :Per tant, el sistema té dos graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de la placa respecte del terra.=====<br /> &lt;div&gt;<br /> :La velocitat angular de la placa és la superposició de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :&lt;math&gt;\velang{placa}{T}=\vec{\dot{\psi}}+\vec{\dot{\theta}}=(\Uparrow\dot{\psi})+(\otimes\dot{\theta})&lt;/math&gt;<br /> <br /> :L’acceleració angular prové del canvi de valor de &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;, i del de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\accang{placa}{T}=\dert{\velang{placa}{T}}{T}=\dert{(\Uparrow\dot{\psi})}{T}=\dert{(\otimes\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\Uparrow\dot{\psi})}{T}=[\text{canvi de valor}]=\ddot{\psi}\frac{\vec{\dot{\psi}}}{|\vec{\dot{\psi}}|}=(\Uparrow\ddot{\psi})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\otimes\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\otimes\dot{\theta})\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\accang{placa}{T}=(\Uparrow\ddot{\psi})+(\otimes\ddot{\theta})+(\Leftarrow\dot{\psi}\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> <br /> :La derivada de la velocitat angular es pot fer també de manera analítica. La base vectorial B en la qual la projecció de &lt;math&gt;\velang{placa}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{placa}{T}}{B}=\vector{0}{\dot{\theta}}{\dot{\psi}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{placa}{T}}{B}=\braq{\dert{\velang{placa}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{placa}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{placa}{T}}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{0}{\ddot{\theta}}{\ddot{\psi}}+\vector{0}{0}{\dot{\psi}}\times\vector{0}{\dot{\theta}}{\dot{\psi}}=\vector{-\dot{\psi}\dot{\theta}}{\ddot{\theta}}{\ddot{\psi}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt Q de la placa respecte del terra. =====<br /> &lt;div&gt;<br /> :Ja que el punt &lt;math&gt;\Os&lt;/math&gt; és fix al terra, el vector &lt;math&gt;\OQvec&lt;/math&gt; és un vector de posició per a la referencia del terra. El seu valor és &lt;math&gt;2\Ls\text{cos}\theta&lt;/math&gt;, i la seva direcció és sempre horitzontal. La velocitat de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor (doncs &lt;math&gt;\theta&lt;/math&gt; és variable) com del canvi de direcció respecte del terra (ocasionat per la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt; del suport).<br /> <br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> [[Fitxer:C2-E.Ex2-4-neut.png|thumb|right|230px|link=]]<br /> <br /> :&lt;math&gt;\OQvec=(\rightarrow 2\Ls\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{Q}{T}=\dert{\OQvec}{T}=\dert{(\rightarrow 2\Ls\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\rightarrow -2\Ls\dot{\theta}\text{sin}\theta\right]+\left[(\Uparrow\dot{\psi})\times(\rightarrow 2\Ls\text{cos}\theta)\right]=\left[\leftarrow 2\Ls\dot{\theta}\text{sin}\theta\right]+\left[\otimes 2\Ls\dot{\psi}\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :L’acceleració de &lt;math&gt;\Qs&lt;/math&gt; prové tant del canvi de valor com del canvi de direcció (associat a &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;) dels dos termes de la velocitat:<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{Q}{T}}{T}=\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}+\dert{\otimes 2\Ls\dot{\psi}\text{cos}\theta}{T}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\leftarrow 2\Ls\dot{\theta}\text{sin}\theta)\right]=\left[\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)\right]+\left[\odot 2\Ls\dot{\psi}\dot{\theta}\text{sin}\theta\right]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes 2\Ls\dot{\psi}\text{cos}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[(\Uparrow\dot{\psi})\times(\otimes 2\Ls\dot{\psi}\text{cos}\theta)\right]=\left[\otimes 2\Ls(\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta)\right]+\left[\leftarrow 2\Ls\dot{\psi}^2\text{cos}\theta\right]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{Q}{T}=(\leftarrow 2\Ls(\ddot{\theta}\text{sin}\theta+(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta))+(\odot 4\Ls\dot{\psi}\dot{\theta}\text{sin}\theta)+(\otimes 2\Ls\ddot{\psi}\text{cos}\theta)&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> [[Fitxer:C3-E.Ex2-3-neut.png|thumb|right|150px|link=]]<br /> :La derivada també es pot fer de manera analítica. La base vectorial B en la qual la projecció del vector &lt;math&gt;\OPvec&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\OQvec}{B}=\vector{2\Ls\text{cos}\theta}{0}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{Q}{T}}{B}=\braq{\dert{\OQvec}{T}}{B}=\frac{d}{dt}\braq{\OQvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OQvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{0}{0}+\vector{0}{0}{\dot{\psi}}\times\vector{2\Ls\text{cos}\theta}{0}{0}=\vector{-2\Ls\dot{\theta}\text{sin}\theta}{2\Ls\dot{\psi}\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{Q}{T}}{B}=\braq{\dert{\vel{Q}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{Q}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{Q}{T}}{B}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-\dot{\psi}\dot{\theta}\text{sin}\theta}{0}+\vector{0}{0}{\dot{\psi}}\times 2\Ls\vector{-\dot{\theta}\text{sin}\theta}{\dot{\psi}\text{cos}\theta}{0}=2\Ls\vector{-\ddot{\theta}\text{sin}\theta-(\dot{\psi}^2+\dot{\theta}^2)\text{cos}\theta}{\ddot{\psi}\text{cos}\theta-2\dot{\psi}\dot{\theta}\text{sin}\theta}{0}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> |}&lt;/small&gt;<br /> <br /> ====🔎 Exercici C2-E.3: pèndol giratori amb punt de suspensió mòbil====<br /> ---------<br /> ::{|:<br /> &lt;small&gt;<br /> El pèndol en forma d’anella està articulat al suport, el qual té un enllaç prismàtic amb la guia. La guia està articulada al sostre, i la seva velocitat angular respecte d’aquest &lt;math&gt;(\vec{\psio})&lt;/math&gt; es manté constant. La molla entre suport i guia garanteix que el primer no caigui a terra quan el sistema està aturat.<br /> <br /> [[Fitxer:C2-E.Ex3-1-cat.png|thumb|center|600px|link=]]<br /> <br /> =====1. Quants graus de llibertat (GL) té el sistema? Descriu-los.=====<br /> &lt;div&gt;<br /> :La guia pot girar al voltant de l’eix vertical que passa per &lt;math&gt;\Os &#039;&lt;/math&gt; (rotació simple).<br /> <br /> :Independentment, el suport es pot traslladar al llarg de la guia (translació rectilínia).<br /> <br /> :Finalment, si els dos moviments anteriors s’aturen, l’anella encara pot fer una rotació simple al voltant de l’eix horitzontal que passa per &lt;math&gt;\Os &#039;&lt;/math&gt;, és perpendicular al pla de l’anella i és fix al suport.<br /> <br /> :Per tant, el sistema té 3 graus de llibertat.<br /> <br /> &lt;/div&gt;<br /> <br /> =====2. Determina la velocitat angular i l’acceleració angular de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :La velocitat angular de l’anella és la superposició de &lt;math&gt;(\vec{\psio})&lt;/math&gt; (1a rotació d’Euler, eix fix a terra) i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt; (2a rotació d’Euler, eix afectat de la rotació &lt;math&gt;\vec{\dot{\psi}}&lt;/math&gt;):<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&lt;math&gt;\velang{anella}{T}=\vec{\psio}+\vec{\dot{\theta}}=(\Uparrow\psio)+(\odot\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\accang{anella}{T}=\dert{\velang{anella}{T}}{T}=\dert{(\vec{\psio}+\vec{\dot{\theta}})}{T}=\dert{\vec{\psio}}{T}+\dert{\vec{\dot{\theta}}}{T}&lt;/math&gt;<br /> :&lt;math&gt;=\dert{(\Uparrow\psio)}{T}+\dert{(\odot\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :L’acceleració angular prové exclusivament del canvi de valor i direcció de &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;, ja que &lt;math&gt;\vec{\psio}&lt;/math&gt; és de valor i direcció constant.<br /> <br /> :&lt;math&gt;\accang{anella}{T} = \dert{\velang{anella}{T}}{T} = \dert{(\odot\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=&lt;/math&gt;<br /> :&lt;math&gt;=\left[\ddot{\theta}\frac{\vec{\dot{\theta}}}{|\vec{\dot{\theta}}|}\right]+\left[\velang{$\vec{\dot{\theta}}$}{T}\times\vec{\dot{\theta}}\right]=[\odot\ddot{\theta}]+\left[(\Uparrow\dot{\psi})\times(\odot\dot{\theta})\right]=(\odot\ddot{\theta})+(\Rightarrow\psio\dot{\theta})&lt;/math&gt;<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039;<br /> <br /> :La derivada de la velocitat angular de l’anella es pot fer també de manera analítica. La base vectorial en la qual la projecció de &lt;math&gt;\velang{anella}{T}&lt;/math&gt; és immediata és la fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\dot{\psi}})&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\braq{\velang{anella}{T}}{B}=\vector{0}{\psio}{\dot{\theta}}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\accang{anella}{T}}{B}=\braq{\dert{\velang{anella}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\velang{anella}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\velang{anella}{T}}{B}=\vector{0}{0}{\ddot{\theta}}+\vector{0}{\psio}{0}\times\vector{0}{\psio}{\dot{\theta}}=\vector{\psio\dot{\theta}}{0}{\ddot{\theta}}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> =====3. Determina la velocitat i l’acceleració del punt G de l’anella respecte del terra.=====<br /> &lt;div&gt;<br /> :&#039;&#039;&#039;Càlcul geomètric:&#039;&#039;&#039;<br /> <br /> :El vector &lt;math&gt;\vec{\Os&#039;\Gs}&lt;/math&gt; és un vector de posició de &lt;math&gt;\Gs&lt;/math&gt; a la referencia del terra, ja que &lt;math&gt;\Os&#039;&lt;/math&gt; és un punt fix a terra.<br /> <br /> :&lt;math&gt;\vec{\Os&#039;\Gs}=\vec{\Os&#039;\Os}+\vec{\Os\Gs}=(\downarrow \textrm{x})+(\searrow \Ls)^*&lt;/math&gt;<br /> <br /> :&lt;math&gt;\vel{G}{T}=\dert{\vec{\Os&#039;\Gs}}{T}=\dert{\vec{\Os&#039;\Os}}{T}+\dert{\vec{\Os\Gs}}{T}=\dert{(\downarrow \textrm{x})}{T}+\dert{(\searrow \Ls)}{T}&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-3-cat.png|thumb|right|250px|link=]]<br /> <br /> :El terme &lt;math&gt;(\downarrow \text{x})&lt;/math&gt; té valor variable però orientació constant, en tant que el terme &lt;math&gt;(\searrow \Ls)&lt;/math&gt;, de valor constant, canvia d’orientació respecte del terra per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;:<br /> <br /> :&lt;math&gt;\dert{\vec{\Os&#039;\Os}}{T}=\dert{(\downarrow \textrm{x})}{T}=[\text{canvi de valor}]=(\downarrow\dot{\text{x}})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{\vec{\Os\Gs}}{T}=\dert{(\searrow \Ls)}{T}=[\text{canvi de direcció}]_\Ts=\velang{$\OGvec$}{T}\times\OGvec=((\Uparrow\psio)+(\odot\dot{\theta}))\times(\searrow \Ls)=&lt;/math&gt;<br /> :&lt;math&gt;=(\Uparrow\psio)\times(\rightarrow\Ls\text{sin}\theta)+(\odot\dot{\theta})\times(\searrow \Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\vel{G}{T}=(\downarrow\dot{\text{x}})+(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\acc{Q}{T}=\dert{\vel{G}{T}}{T}=\dert{(\downarrow\dot{\text{x}})}{T}+\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}+\dert{(\nearrow\Ls\dot{\theta})}{T}&lt;/math&gt;<br /> <br /> :Tots tres termes de la velocitat són de valor variable, i només els dos últims giren (canvien de direcció) respecte del terra. El segon, que és perpendicular al pla de l’anella, només gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt;, en tant que el tercer gira per causa de &lt;math&gt;\vec{\psio}&lt;/math&gt; i &lt;math&gt;\vec{\dot{\theta}}&lt;/math&gt;. La derivada de cadascun d’aquests termes és:<br /> <br /> :&lt;math&gt;\dert{(\downarrow\dot{\text{x}})}{T}=[\text{canvi de valor}]=(\downarrow\ddot{x})&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\otimes\Ls\psio\text{sin}\theta)}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[(\Uparrow\psio)\times(\otimes\Ls\psio\text{sin}\theta)]=[\otimes\Ls\psio\dot{\theta}\text{cos}\theta]+[\leftarrow\Ls\psio^2\text{sin}\theta]&lt;/math&gt;<br /> <br /> :&lt;math&gt;\dert{(\nearrow\Ls\dot{\theta})}{T}=[\text{canvi de valor}]+[\text{canvi de direcció}]_\Ts=[\nearrow\Ls\ddot{\theta}]+[((\Uparrow\psio)+(\odot\dot{\theta}))\times(\nearrow\Ls\dot{\theta})]=[\nearrow\Ls\ddot{\theta}]+[(\nwarrow\Ls\dot{\theta}^2)+(\otimes\Ls\psio\dot{\theta}\text{cos}\theta)]&lt;/math&gt;<br /> <br /> :Per tant, &lt;math&gt;\acc{G}{T}=(\downarrow\ddot{x})+(\leftarrow\Ls\psio^2\text{sin}\theta)+(\nearrow\Ls\ddot{\theta})+(\nwarrow\Ls\dot{\theta}^2)+(\otimes 2\Ls\psio\dot{\theta}\text{cos}\theta)&lt;/math&gt;<br /> <br /> [[Fitxer:C2-E.Ex3-2-neut.png|thumb|right|300px|link=]]<br /> <br /> :&#039;&#039;&#039;Càlcul analític:&#039;&#039;&#039; <br /> <br /> :Tot el càlcul es pot fer de manera analítica. El vector &lt;math&gt;\OGvec=\vec{\Os\Os&#039;}+\vec{\Os&#039;\Gs}&lt;/math&gt; té el primer terme vertical, i per tant la seva projecció és immediata a la base B fixa al suport &lt;math&gt;(\velang{B}{T}=\vec{\psio})&lt;/math&gt;; el segon terme, en canvi, es projecta immediatament a la base B’ fixa a l’anella &lt;math&gt;(\velang{B&#039;}{T}=\vec{\psio}+\vec{\dot{\theta}})&lt;/math&gt;. Qualsevol de les dues pot ser adequada.<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\Ls\text{sin}\theta}{-\text{x}-\Ls\text{cos}\theta}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B}=\braq{\dert{\OGvec}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B}+\braq{\velang{B}{T}}{B}\times\braq{\OGvec}{B}=&lt;/math&gt;<br /> :&lt;math&gt;=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}+\vector{0}{\psio}{0}\times\vector{\Ls\text{sin}\theta}{-x-\Ls\text{cos}\theta}{0}=\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B}=\braq{\dert{\vel{G}{T}}{T}}{B}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B}+\braq{\velang{B}{T}}{B}\times\braq{\vel{G}{T}}{B}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta-\dot{\theta}^2\text{cos}\theta)}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{0}{\psio}{0}\times\vector{\Ls\dot{\theta}\text{cos}\theta}{-\dot{x}+\Ls\dot{\theta}\text{sin}\theta}{0}=\vector{\Ls(\ddot{\theta}\text{cos}\theta-\dot{\theta}^2\text{sin}\theta)}{-\ddot{x}+\Ls(\ddot{\theta}\text{sin}\theta+\dot{\theta}^2\text{cos}\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> :&lt;span style=&quot;text-decoration: underline;&quot;&gt;Càlcul a la base B&#039;&lt;/span&gt;<br /> <br /> :&lt;math&gt;\braq{\OGvec}{B}=\vector{\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\vel{G}{T}}{B&#039;}=\braq{\dert{\OGvec}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\OGvec}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\OGvec}{B&#039;}=\vector{-\dot{x}\text{sin}\theta-\xs\dot{\theta}\text{cos}\theta}{-\dot{x}\text{cos}\theta+\xs\dot{\theta}\text{sin}\theta}{0}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\xs\text{sin}\theta}{-\xs\text{cos}\theta-\Ls}{0}=\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}&lt;/math&gt;<br /> <br /> :&lt;math&gt;\braq{\acc{G}{T}}{B&#039;}=\braq{\dert{\vel{G}{T}}{T}}{B&#039;}=\frac{\ds}{\ds\ts}\braq{\vel{G}{T}}{B&#039;}+\braq{\velang{B&#039;}{T}}{B&#039;}\times\braq{\vel{G}{T}}{B&#039;}=\vector{-\ddot{x}\text{sin}\theta-\dot{x}\dot{\theta}\text{cos}\theta+\Ls\ddot{\theta}}{-\ddot{x}\text{cos}\theta+\dot{x}\dot{\theta}\text{sin}\theta}{-\Ls\psio\dot{\theta}\text{cos}\theta}+\vector{\psio\text{sin}\theta}{\psio\text{cos}\theta}{\dot{\theta}}\times\vector{-\dot{x}\text{sin}\theta+\Ls\dot{\theta}}{-\dot{x}\text{cos}\theta}{-\Ls\psio\text{sin}\theta}=\vector{-\ddot{x}\text{sin}\theta+\Ls(\ddot{\theta}-\psio^2\text{sin}\theta\text{cos}\theta)}{-\ddot{x}\text{cos}\theta+\Ls(\dot{\theta}^2+\psio^2\text{sin}^2\theta)}{-2\Ls\psio\dot{\theta}\text{cos}\theta}&lt;/math&gt;<br /> <br /> &lt;/div&gt;<br /> <br /> |}&lt;/small&gt;<br /> <br /> -------------<br /> <br /> <br /> &#039;&#039;&#039;*NOTA:&#039;&#039;&#039; En aquest web (i per manca de símbols de fletxa més precisos), tot i que les fletxes &lt;math&gt;\nearrow&lt;/math&gt;, &lt;math&gt;\swarrow&lt;/math&gt;, &lt;math&gt;\nwarrow&lt;/math&gt; i &lt;math&gt;\searrow&lt;/math&gt; semblen indicar que els vectors formen un angle de 45° amb la direcció vertical, no té per què ser així. Cal interpretar les fletxes de manera qualitativa, observant el dibuix que sempre acompanya aquest tipus de notació. Per exemple, l’apartat 3 de l’exercici C2-E.1, el vector &lt;math&gt;\OPvec&lt;/math&gt; forma un angle &lt;math&gt;\theta&lt;/math&gt; genèric amb la direcció vertical. Si el valor de l’angle &lt;math&gt;\theta&lt;/math&gt; és menor de 90° (com a la figura), el vector &lt;math&gt;\OPvec&lt;/math&gt; té component cap a baix i cap a la dreta.<br /> <br /> <br /> &lt;p align=&quot;right&quot;&gt;&lt;small&gt;© Universitat Politècnica de Catalunya. [[Mecànica:Drets d&#039;autor |All rights reserved]]&lt;/small&gt;&lt;/p&gt;<br /> -------------<br /> -------------<br /> <br /> <br /> <br /> <br /> &lt;center&gt;<br /> [[C1. Configuration of a mechanical system|&lt;&lt;&lt; C1. Configuration of a mechanical system]]<br /> <br /> [[C3. Composition of movements|C3. Composition of movements &gt;&gt;&gt;]]<br /> &lt;/center&gt;</div>

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