C2. Movement of a mechanical system - Revision history

Eantem: /* 3. Find the velocity and the acceleration of point G of the plate relative to the ground. . */

2026-03-03T19:53:20Z

3. Find the velocity and the acceleration of point G of the plate relative to the ground. .

← Older revision		Revision as of 19:53, 3 March 2026
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	:'''Geometric calculation:'''		:'''Geometric calculation:'''
	:<math>\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})</math>		:<math>\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{sin}\theta)+(\odot\dot{\theta})\times(\searrow\Ls)=(\otimes\Ls\psio\text{sin}\theta)+(\nearrow\Ls\dot{\theta})</math>

	[[File:C2-E.Ex1-3-eng.png\|thumb\|right\|300px\|link=]]		[[File:C2-E.Ex1-3-eng.png\|thumb\|right\|300px\|link=]]

Bros: /* ✏️ EXAMPLE C2-3.1: Euler pendulum */

2024-11-09T20:20:11Z

✏️ EXAMPLE C2-3.1: Euler pendulum

← Older revision		Revision as of 20:20, 9 November 2024
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	::Particular cases:		::Particular cases:

	[[File:C2-Ex3-1-4-neut.png\|thumb\|center\|400px\|link=]]		[[File:C2-Ex3-1-5-neut.png\|thumb\|center\|400px\|link=]]
	::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the <math>\psi=0</math> configuration for those two particular cases.		::The dotted circular lines correspond to the approximation of the trajectory in the neighbourhood of the <math>\psi=0</math> configuration for those two particular cases.

Bros: /* ✏️ EXAMPLE C2-1.2: Euler pendulum */

2024-11-09T19:58:42Z

✏️ EXAMPLE C2-1.2: Euler pendulum

← Older revision		Revision as of 19:58, 9 November 2024
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	</center>		</center>

	::Vector <math>\vec{\Or\Cbf}</math> has a constant direction in R but a variable value. Hence, its time derivative is parallel to <math>\vec{\Or\Cbf}</math> with value <math>\~~dotx~~</math>. Vector <math>\vec{\Cbf\Qs}</math>, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to <math>\vec{\Cbf\Qs}</math>, and its value is that of<math>\vec{\Cbf\Qs}</math> times the rate of change of orientation of <math>\vec{\Cbf\Qs}</math> relative to R (<math>\dot\psi</math>):		::Vector <math>\vec{\Or\Cbf}</math> has a constant direction in R but a variable value. Hence, its time derivative is parallel to <math>\vec{\Or\Cbf}</math> with value <math>\dot x</math>. Vector <math>\vec{\Cbf\Qs}</math>, however, has a constant value (L) but variable direction. Consequently, its time derivative is perpendicular to <math>\vec{\Cbf\Qs}</math>, and its value is that of<math>\vec{\Cbf\Qs}</math> times the rate of change of orientation of <math>\vec{\Cbf\Qs}</math> relative to R (<math>\dot\psi</math>):

	[[File:C2-Ex1-2-2-neut.png\|thumb\|center\|300px\|link=]]		[[File:C2-Ex1-2-2-neut-new.png\|thumb\|center\|300px\|link=]]

	::The <math>\vel{Q}{R}</math> 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 <math>\dot x</math> and <math>L\dot\psi</math>, whose directions do correspond to one of those singular directions.		::The <math>\vel{Q}{R}</math> 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 <math>\dot x</math> and <math>L\dot\psi</math>, whose directions do correspond to one of those singular directions.

Bros at 19:52, 9 November 2024

2024-11-09T19:52:05Z

Show changes

Bros at 19:51, 9 November 2024

2024-11-09T19:51:06Z

← Older revision		Revision as of 19:51, 9 November 2024
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	\newcommand{\psio}{\dot{\psi}_0}		\newcommand{\psio}{\dot{\psi}_0}
	\newcommand{\dth}{\dot\theta}		\newcommand{\dth}{\dot\theta}
			\newcommand{\ddth}{\ddot\theta}
	\newcommand{\sth}{\sin{\theta}}		\newcommand{\sth}{\sin{\theta}}
	\newcommand{\cth}{\cos{\theta}}		\newcommand{\cth}{\cos{\theta}}
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	<math>\braq{\accang{disk}{R}}{B} = \vector{\ddot\theta}{\ddot\varphi ~~cos~~\~~theta~~-\dot\varphi\dth ~~sin~~\~~theta~~}{\ddot\psi+\ddot\varphi ~~sin~~\~~theta~~+\dot\varphi\dth ~~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\dth ~~sin~~\~~theta~~+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi ~~sin~~\~~theta~~+\dot\varphi\dth ~~cos~~\~~theta~~}</math>		<math>\braq{\accang{disk}{R}}{B} = \vector{\ddot\theta}{\ddot\varphi \cth-\dot\varphi\dth \sth}{\ddot\psi+\ddot\varphi \sth+\dot\varphi\dth \cth}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi \cth}{\dot\psi+\dot\varphi \sth} = \vector{\ddot\theta-\dot\psi\dot\varphi \cth}{\ddot\varphi \cth-\dot\varphi\dth \sth+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi \sth+\dot\varphi\dth \cth}</math>

	<math>\braq{\velang{disk}{R}}{B'}=\vector{\dot\theta}{\dot\varphi+\dot\psi \sth}{\dot\psi \cth},</math><span>          </span><math>\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'}</math>		<math>\braq{\velang{disk}{R}}{B'}=\vector{\dot\theta}{\dot\varphi+\dot\psi \sth}{\dot\psi \cth},</math><span>          </span><math>\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'}</math>

	<math>\braq{\accang{disk}{R}}{B'} = \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{\dth}{\dot\psi\sth}{\dot\psi\cth}\times\vector{\dot\theta}{\dot\varphi+\dot\psi ~~sin~~\~~theta~~}{\dot\psi ~~cos~~\~~theta~~} = \vector{\ddth-\dot\psi\dot\varphi\cth}{\ddot\varphi+\ddot\psi \sth+\dot\psi\dth\cth}{\ddot\psi \cth-\dot\psi\dth\sth+\dot\varphi\dth}</math>		<math>\braq{\accang{disk}{R}}{B'} = \vector{\ddot\theta}{\ddot\varphi+\ddot\psi \sth+\dot\psi\dot\theta \cth}{\ddot\psi \cth-\dot\psi\dot\theta \sth} + \vector{\dth}{\dot\psi\sth}{\dot\psi\cth}\times\vector{\dot\theta}{\dot\varphi+\dot\psi \sth}{\dot\psi \cth} = \vector{\ddth-\dot\psi\dot\varphi\cth}{\ddot\varphi+\ddot\psi \sth+\dot\psi\dth\cth}{\ddot\psi \cth-\dot\psi\dth\sth+\dot\varphi\dth}</math>
	</center>		</center>
	</small>		</small>

Bros: /* Analytical calculation ➕ */

2024-11-09T19:49:54Z

Analytical calculation ➕

← Older revision		Revision as of 19:49, 9 November 2024
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	<math>\braq{\velang{disk}{R}}{B'}=\vector{\dot\theta}{\dot\varphi+\dot\psi \sth}{\dot\psi \cth},</math><span>          </span><math>\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'}</math>		<math>\braq{\velang{disk}{R}}{B'}=\vector{\dot\theta}{\dot\varphi+\dot\psi \sth}{\dot\psi \cth},</math><span>          </span><math>\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'}</math>

	<math>\braq{\accang{disk}{R}}{B'} = \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{\dth}{\dot\psi\sth}{\dot\psi\cth}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta} = \vector{\ddth-\dot\psi\dot\varphi\cth}{\ddot\varphi+\ddot\psi \sth+\dot\psi\dth\cth}{\ddot\psi \cth-\dot\psi\dth\sth+dot\varphi\dth}</math>		<math>\braq{\accang{disk}{R}}{B'} = \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{\dth}{\dot\psi\sth}{\dot\psi\cth}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta} = \vector{\ddth-\dot\psi\dot\varphi\cth}{\ddot\varphi+\ddot\psi \sth+\dot\psi\dth\cth}{\ddot\psi \cth-\dot\psi\dth\sth+\dot\varphi\dth}</math>
	</center>		</center>
	</small>		</small>

Bros at 19:49, 9 November 2024

2024-11-09T19:49:17Z

← Older revision		Revision as of 19:49, 9 November 2024
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	::Acceleration of <math>\Qs</math> relative to BL:		::Acceleration of <math>\Qs</math> relative to BL:
	<center>		<center>
	<math>\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 \cpsi-\Ls \dot\psi^2 \spsi}{-\Ls \ddot\psi \spsi -\Ls \dot\psi^2 \cpsi}{0}</math>		<math>\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 \cpsi-\Ls \dot\psi^2 \spsi}{\Ls \ddot\psi \spsi +\Ls \dot\psi^2 \cpsi}{0}</math>
	<math>\braq{\acc{Q}{BL}}{B'} = \braq{\dert{\vel{Q}{BL}}{BL}}{B'} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B'}+ \braq{\velang{B'}{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}</math>		<math>\braq{\acc{Q}{BL}}{B'} = \braq{\dert{\vel{Q}{BL}}{BL}}{B'} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B'}+ \braq{\velang{B'}{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}</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	\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 \cpsi-\Ls\dot\psi^2\spsi}{-\Ls\ddot\psi \spsi-\Ls\dot\psi^2\cpsi}{0}		\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 \cpsi-\Ls\dot\psi^2\spsi}{\Ls\ddot\psi \spsi+\Ls\dot\psi^2\cpsi}{0}
	</math>		</math>
	<math>		<math>
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	<center><math>\accang{S}{R}= \dert{\velang{S}{R}}{R}</math></center>		<center><math>\accang{S}{R}= \dert{\velang{S}{R}}{R}</math></center>

	The description of the angular velocity <math>\velang{S}{R}</math> 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 <math>\velang{S}{R}</math> is constant (it is always perpendicular to the plane of motion). ~~Hence~~, the angular acceleration comes exclusively from the change of value of <math>\velang{S}{R}</math>, and it is parallel to <math>\velang{S}{R}</math>. In general motions in space, if <math>\velang{S}{R}</math> is described through Euler rotations, <math>\accang{S}{R}</math> may come from the change of values of (<math>\vecdot\psi</math>, <math>\vecdot\theta</math>,<math>\vecdot\varphi</math>) iand the change of direction of de <math>\vecdot\theta</math> and <math>\vecdot\varphi</math> (<math>\vecdot\psi</math> has always a constant direction in R).		The description of the angular velocity <math>\velang{S}{R}</math> 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 <math>\velang{S}{R}</math> is constant (it is always perpendicular to the plane of motion). In this case, the angular acceleration comes exclusively from the change of value of <math>\velang{S}{R}</math>, and it is parallel to <math>\velang{S}{R}</math>. In general motions in space, if <math>\velang{S}{R}</math> is described through Euler rotations, <math>\accang{S}{R}</math> may come from the change of values of (<math>\vecdot\psi</math>, <math>\vecdot\theta</math>,<math>\vecdot\varphi</math>) iand the change of direction of de <math>\vecdot\theta</math> and <math>\vecdot\varphi</math> (<math>\vecdot\psi</math> has always a constant direction in R).


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	<math>\braq{\accang{disk}{R}}{B}=\vector{\ddot\theta}{\ddot\varphi \~~cth~~-\dot\varphi\~~dot~~\~~psi \sth~~}{\ddot\psi+\ddot\varphi \~~sth~~+\dot\varphi\~~dot~~\~~psi \cth~~}+\vector{0}{0}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi \~~cth~~}{\dot\psi+\dot\varphi \~~sth~~}=\vector{\ddot\theta-\dot\psi\dot\varphi \~~cth~~}{\ddot\varphi \~~cth~~-\dot\varphi\~~dot\psi~~ \~~sth~~+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi \~~sth~~+\dot\varphi\~~dot~~\~~psi \cth~~}</math>		<math>\braq{\accang{disk}{R}}{B} = \vector{\ddot\theta}{\ddot\varphi cos\theta-\dot\varphi\dth sin\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dth 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\dth sin\theta+\dot\psi\dot\theta}{\ddot\psi+\ddot\varphi sin\theta+\dot\varphi\dth cos\theta}</math>

	<math>\braq{\velang{disk}{R}}{B'}=\vector{\dot\theta}{\dot\varphi+\dot\psi \sth}{\dot\psi \cth},</math><span>          </span><math>\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'}</math>		<math>\braq{\velang{disk}{R}}{B'}=\vector{\dot\theta}{\dot\varphi+\dot\psi \sth}{\dot\psi \cth},</math><span>          </span><math>\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'}</math>

	<math>\braq{\accang{disk}{R}}{B'}=\vector{\ddot\theta}{\ddot\varphi+\ddot\psi \~~sth~~+\dot\psi\dot\theta \~~cth~~}{\ddot\psi \~~cth~~-\dot\psi\dot\theta \~~sth~~}+\vector{\dot\~~theta}{0~~}{\dot\psi}\times\vector{\dot\theta}{\dot\varphi+\dot\psi \~~sth~~}{\dot\psi \~~cth~~}=\vector{\~~ddot\theta~~-\dot\psi(\dot\varphi+\~~dot\psi \sth)~~}{\ddot\varphi+\ddot\psi \sth+\dot\psi\~~dot~~\~~theta~~}{\ddot\psi \cth+\dot\~~theta~~\dot\varphi}</math>		<math>\braq{\accang{disk}{R}}{B'} = \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{\dth}{\dot\psi\sth}{\dot\psi\cth}\times\vector{\dot\theta}{\dot\varphi+\dot\psi sin\theta}{\dot\psi cos\theta} = \vector{\ddth-\dot\psi\dot\varphi\cth}{\ddot\varphi+\ddot\psi \sth+\dot\psi\dth\cth}{\ddot\psi \cth-\dot\psi\dth\sth+dot\varphi\dth}</math>
	</center>		</center>
	</small>		</small>

Bros: /* Analytical calculation ➕ */

2024-11-09T19:46:24Z

Analytical calculation ➕

← Older revision		Revision as of 19:46, 9 November 2024
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	=====Analytical calculation ➕=====		=====Analytical calculation ➕=====
	::The vector bases B and B’ are the same as in <span style="text-decoration: underline;">[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum\|'''example C2-1.2''']]</span>.		::The vector bases B and B’ are the same as in <span style="text-decoration: underline;">[[C2. Movement of a mechanical system #✏️ EXAMPLE C2-1.2: Euler pendulum\|'''example C2-1.2''']]</span>.

			[[File:C2-Ex1-2-3-neut.png\|thumb\|right\|200px\|link=]]

	::Acceleration of <math>\Qs</math> relative to BL:		::Acceleration of <math>\Qs</math> relative to BL:

Bros: /* ✏️ EXAMPLE C2-2.2: Euler pendulum */

2024-11-09T19:45:31Z

✏️ EXAMPLE C2-2.2: Euler pendulum

← Older revision		Revision as of 19:45, 9 November 2024
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	::<math>\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}</math>.		::<math>\vel{Q}{R} = \dert{\vec{\Os_\Rs\Cbf}}{R} + \dert{\vecbf{CQ}}{R}</math>.

	:::* <math>\dert{\vec{\Os_\Rs\Cbf}}{R}</math>: constant direction (horizontal), variable value <math>(\dot x)</math>. Thus, its time derivative <math>\ddert{\vec{\Os_\Rs\Cbf}}{R}</math> is horizontal with value <math>\ddot~~\xs~~</math>.		:::* <math>\dert{\vec{\Os_\Rs\Cbf}}{R}</math>: constant direction (horizontal), variable value <math>(\dot x)</math>. Thus, its time derivative <math>\ddert{\vec{\Os_\Rs\Cbf}}{R}</math> is horizontal with value <math>\ddot x</math>.
	:::* <math>\dert{\vecbf{CQ}}{R}</math>: direction perpendicular to the bar, thus variable; variable value <math>\Ls\dot\psi</math>. Thus, its time derivative <math>\ddert{\vecbf{CQ}}{R}</math> has a component perpendicular to <math>\dert{\vecbf{CQ}}{R}</math> (parallel to the bar) with value <math>\Ls\dot\psi\cdot\dot\psi</math> , and another one parallel to <math>\dert{\vecbf{CQ}}{R}</math> (perpendicular to the bar) with value <math>\Ls\ddot\psi</math>.		:::* <math>\dert{\vecbf{CQ}}{R}</math>: direction perpendicular to the bar, thus variable; variable value <math>\Ls\dot\psi</math>. Thus, its time derivative <math>\ddert{\vecbf{CQ}}{R}</math> has a component perpendicular to <math>\dert{\vecbf{CQ}}{R}</math> (parallel to the bar) with value <math>\Ls\dot\psi\cdot\dot\psi</math> , and another one parallel to <math>\dert{\vecbf{CQ}}{R}</math> (perpendicular to the bar) with value <math>\Ls\ddot\psi</math>.

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	::Acceleration of <math>\Qs</math> relative to BL:		::Acceleration of <math>\Qs</math> relative to BL:
	<center>		<center>
	<math>\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 \cpsi-\Ls \dot\psi^2 \spsi}{-\Ls \ddot\psi \spsi -\Ls \dot\psi^2 \cpsi}{0}</math>		<math>\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 \cpsi-\Ls \dot\psi^2 \spsi}{\Ls \ddot\psi \spsi +\Ls \dot\psi^2 \cpsi}{0}</math>

	<math>\braq{\acc{Q}{BL}}{B'} = \braq{\dert{\vel{Q}{BL}}{BL}}{B'} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B'}+ \braq{\velang{B'}{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}</math>		<math>\braq{\acc{Q}{BL}}{B'} = \braq{\dert{\vel{Q}{BL}}{BL}}{B'} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{BL}}{B'}+ \braq{\velang{B'}{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}</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	\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 \cpsi-\Ls\dot\psi^2\spsi}{-\Ls\ddot\psi \spsi-\Ls\dot\psi^2\cpsi}{0}		\braq{\acc{Q}{R}}{B}=\braq{\dert{\vel{Q}{R}}{R}}{B} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B}=\vector{\ddot x+\Ls\ddot\psi \cpsi-\Ls\dot\psi^2\spsi}{\Ls\ddot\psi \spsi+\Ls\dot\psi^2\cpsi}{0}
	</math>		</math>
	<math>		<math>
	\braq{\acc{Q}{R}}{B'}=\braq{\dert{\vel{Q}{R}}{R}}{B'} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B'}+\braq{\velang{B'}{R}\times \OQvec}{B}=\vector{\ddot~~\xs~~ \spsi+\dot x\dot\psi \cpsi}{\ddot~~\xs~~ \cpsi-\dot x\dot\psi \spsi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot x \spsi}{\dot x \cpsi+\Ls\dot\psi}{0}=\vector{\ddot~~\xs~~ \spsi - \Ls \dot\psi^2}{\ddot~~\xs~~ \cpsi+\Ls\ddot\psi}{0}		\braq{\acc{Q}{R}}{B'}=\braq{\dert{\vel{Q}{R}}{R}}{B'} = \frac{\ds}{\ds\ts}\braq{\vel{Q}{R}}{B'}+\braq{\velang{B'}{R}\times \OQvec}{B}=\vector{\ddot x \spsi+\dot x\dot\psi \cpsi}{\ddot x \cpsi-\dot x\dot\psi \spsi +\Ls\ddot\psi}{0}+\vector{0}{0}{\dot\psi}\times\vector{\dot x \spsi}{\dot x \cpsi+\Ls\dot\psi}{0}=\vector{\ddot x \spsi - \Ls \dot\psi^2}{\ddot x \cpsi+\Ls\ddot\psi}{0}
	</math>		</math>
	</center>		</center>

Bros at 19:44, 9 November 2024

2024-11-09T19:44:11Z

Show changes