D7. Examples of 3D dynamics - Revision history

Eantem: /* ✏️ EXAMPLE D7.5: rotating ring */

2026-07-15T11:17:40Z

✏️ EXAMPLE D7.5: rotating ring

← Older revision		Revision as of 11:17, 15 July 2026
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	<br>		<br>
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	:<big><span style="text-decoration: underline;">'''OPTION 2'''</span></big>		:<big><span style="text-decoration: underline;">'''OPTION 2'''</span></big>

2026-07-15T11:08:45Z

✏️ EXAMPLE D7.5: rotating ring

← Older revision		Revision as of 11:08, 15 July 2026
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	:Since the motor torque acts between P1 and the ring, the two system options for applying the vector theorems are: ring, P1. Since the equation of motion for the <math>\psi</math> coordinate and the normal force have already been determined, the number of unknowns in these two cases is:		:Since the motor torque acts between P1 and the ring, the two system options for applying the vector theorems are: ring, P1. Since the equation of motion for the <math>\psi</math> coordinate and the normal force have already been determined, the number of unknowns in these two cases is:

	[[File:D7-Ex5-7-eng.png\|thumb\|center\|~~400px~~\|link=]]		[[File:D7-Ex5-7-eng.png\|thumb\|center\|450px\|link=]]
	<center> 5 constraint unk. + <math>\Gamma = </math> 6 unk.<math>\:\:\:\:\:\:\:\:\:\:</math> 9 constraint unk. + <math>\Gamma = </math> 10 unk.</center>		<center> 5 constraint unk. + <math>\Gamma = </math> 6 unk.<math>\:\:\:\:\:\:\:\:\:\:</math> 9 constraint unk. + <math>\Gamma = </math> 10 unk.</center>

2026-07-15T11:07:26Z

✏️ EXAMPLE D7.5: rotating ring

← Older revision		Revision as of 11:07, 15 July 2026
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	:The best option, then, is to apply the vector theorems to the ring. The external interactions on that rigid body are:		:The best option, then, is to apply the vector theorems to the ring. The external interactions on that rigid body are:
	:[[File:D7-Ex5-8-neut.png\|thumb\|left\|~~200px~~\|link=]]		:[[File:D7-Ex5-8-neut.png\|thumb\|left\|250px\|link=]]
	:The first component of the AMT at <math>\Os</math> is free of constraint unknowns (it contains the moment of the friction force, but N is no longer an unknown). Therefore:		:The first component of the AMT at <math>\Os</math> is free of constraint unknowns (it contains the moment of the friction force, but N is no longer an unknown). Therefore:

2026-07-15T10:43:46Z

D7.1 Analysis of the equations of motion

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	==D7.1 Analysis of the equations of motion==		==D7.1 Analysis of the equations of motion==

	Obtaining the equations of motion of the free DoF of multibody systems is not, in general, the objective of dynamics problems, but rather a previous step to their integration in order to know how the coordinates that describe the configuration of the system evolve over time:		Obtaining the equations of motion of the free DoF (<math>\dot{\qs}_{i}</math>) of multibody systems is not, in general, the objective of dynamics problems, but rather a previous step to their integration in order to know how the coordinates that describe the configuration of the system evolve over time:
	<center>		<center>
	<math>\ddot{\qs}_{i}\xrightarrow[]{\int{\ds\ts}}\dot{\qs}_\is\xrightarrow[]{\int{\ds\ts}}{\qs_\is}</math>		<math>\ddot{\qs}_{i}\xrightarrow[]{\int{\ds\ts}}\dot{\qs}_\is\xrightarrow[]{\int{\ds\ts}}{\qs_\is}</math>

2026-03-18T18:26:05Z

✏️ EXAMPLE D7.4: rotating ball

← Older revision		Revision as of 18:26, 18 March 2026
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	:<math>\begin{aligned}\left\{\vec{\Hs}_{\text{RTO}}(\Os)\right\} &= \left(\frac{2}{5}\ms\rs^2\mat{1}{0}{0}{0}{1}{0}{0}{0}{1} + \ms\Ls^2\mat{0}{0}{0}{0}{1}{0}{0}{0}{1}\right)\vector{(\Ls/\rs)\Omega_0}{\Omega_0}{0}=\ms\Omega_0\vector{(2/5)\rs\Ls}{(2/5)\rs^2+\Ls^2}{0}		:<math>\begin{aligned}\left\{\vec{\Hs}_{\text{RTO}}(\Os)\right\} &= \left(\frac{2}{5}\ms\rs^2\mat{1}{0}{0}{0}{1}{0}{0}{0}{1} + \ms\Ls^2\mat{0}{0}{0}{0}{1}{0}{0}{0}{1}\right)\vector{(\Ls/\rs)\Omega_0}{\Omega_0}{0}=\ms\Omega_0\vector{(2/5)\rs\Ls}{(2/5)\rs^2+\Ls^2}{0}
	\end{aligned}</math>		\end{aligned}</math>



	:<math>\left.\begin{aligned}		:<math>\left.\begin{aligned}

2026-03-18T18:25:45Z

✏️ EXAMPLE D7.4: rotating ball

← Older revision		Revision as of 18:25, 18 March 2026
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	:<math>\vec{\Hs}_{\text{RTO}}(\Os) = ([\Is\Is(\Gs)]+[\Is\Is^\oplus(\Os)])\velang{ball}{RTO=E}</math>		:<math>\vec{\Hs}_{\text{RTO}}(\Os) = ([\Is\Is(\Gs)]+[\Is\Is^\oplus(\Os)])\velang{ball}{RTO=E}</math>

	:<math>\begin{aligned}\left\{\vec{\Hs}_{\text{RTO}}(\Os)\right\} &= \left(\frac{2}{5}\ms\rs^2\mat{1}{0}{0}{0}{1}{0}{0}{0}{1} + \ms\Ls^2\mat{0}{0}{0}{0}{1}{0}{0}{0}{1}\right)\vector{(\Ls/\rs)\Omega_0}{\Omega_0}{0}~~=\\~~		:<math>\begin{aligned}\left\{\vec{\Hs}_{\text{RTO}}(\Os)\right\} &= \left(\frac{2}{5}\ms\rs^2\mat{1}{0}{0}{0}{1}{0}{0}{0}{1} + \ms\Ls^2\mat{0}{0}{0}{0}{1}{0}{0}{0}{1}\right)\vector{(\Ls/\rs)\Omega_0}{\Omega_0}{0}=\ms\Omega_0\vector{(2/5)\rs\Ls}{(2/5)\rs^2+\Ls^2}{0}
	&=\ms\Omega_0\vector{(2/5)\rs\Ls}{(2/5)\rs^2+\Ls^2}{0}
	\end{aligned}</math>		\end{aligned}</math>

	~~<math></math>~~





	:<math>\left.\begin{aligned}		:<math>\left.\begin{aligned}

2026-03-18T18:24:23Z

✏️ EXAMPLE D7.5: rotating ring

2026-03-13T15:38:24Z

✏️ EXAMPLE D7.5: rotating ring

← Older revision		Revision as of 15:38, 13 March 2026
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	[[File:D7-Ex5-2-eng.png\|thumb\|center\|250px\|link=]]		[[File:D7-Ex5-2-eng.png\|thumb\|center\|250px\|link=]]

	<center><math>{\vvec}_{\textrm{E}}(\textrm{J}) = {\vvec}_{\textrm{E}}(\textrm{G})+\vec{\Omega}_{\textrm{E}}^\text{ring} \times \vec{\textrm{GJ}} = (\odot L \dot{\psi})+[(\Uparrow\dot{\psi})+(⇒\dot{\varphi_0})] \times (↓R) = (\odot L \dot{\psi}) + (\otimes R \dot{\varphi_0})</math></center>		<center><math>{\vvec}_{\textrm{E}}(\textrm{J}) = {\vvec}_{\textrm{E}}(\textrm{G})+\vec{\Omega}_{\textrm{E}}^\text{ring} \times \vec{\textrm{GJ}} = (\odot \textrm{L} \dot{\psi})+[(\Uparrow\dot{\psi})+(⇒\dot{\varphi_0})] \times (↓R) = (\odot L \dot{\psi}) + (\otimes R \dot{\varphi_0})</math></center>

2026-03-13T15:37:38Z

✏️ EXAMPLE D7.5: rotating ring

← Older revision		Revision as of 15:37, 13 March 2026
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	[[File:D7-Ex5-2-eng.png\|thumb\|center\|250px\|link=]]		[[File:D7-Ex5-2-eng.png\|thumb\|center\|250px\|link=]]

	<center><math>{\vvec}_{\textrm{E}}(\textrm{J}) = {\vvec}_{\textrm{E}}(\textrm{G})+\vec{\Omega}_{\Es}^\text{ring} \times \vec{\textrm{GJ}} = (\odot L \dot{\psi})+[(\Uparrow\dot{\psi})+(⇒\dot{\varphi_0})] \times (↓R) = (\odot L \dot{\psi}) + (\otimes R \dot{\varphi_0})</math></center>		<center><math>{\vvec}_{\textrm{E}}(\textrm{J}) = {\vvec}_{\textrm{E}}(\textrm{G})+\vec{\Omega}_{\textrm{E}}^\text{ring} \times \vec{\textrm{GJ}} = (\odot L \dot{\psi})+[(\Uparrow\dot{\psi})+(⇒\dot{\varphi_0})] \times (↓R) = (\odot L \dot{\psi}) + (\otimes R \dot{\varphi_0})</math></center>

2026-03-13T15:37:22Z

✏️ EXAMPLE D7.5: rotating ring

← Older revision		Revision as of 15:37, 13 March 2026
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	[[File:D7-Ex5-2-eng.png\|thumb\|center\|250px\|link=]]		[[File:D7-Ex5-2-eng.png\|thumb\|center\|250px\|link=]]

	<center><math>{\vvec}_{\textrm{E}}(\textrm{J}) = {\vvec}_{\textrm{E}}(\textrm{G})+\vec{\Omega}_{\Ts}^\text{ring} \times \vec{\textrm{GJ}} = (\odot L \dot{\psi})+[(\Uparrow\dot{\psi})+(⇒\dot{\varphi_0})] \times (↓R) = (\odot L \dot{\psi}) + (\otimes R \dot{\varphi_0})</math></center>		<center><math>{\vvec}_{\textrm{E}}(\textrm{J}) = {\vvec}_{\textrm{E}}(\textrm{G})+\vec{\Omega}_{\Es}^\text{ring} \times \vec{\textrm{GJ}} = (\odot L \dot{\psi})+[(\Uparrow\dot{\psi})+(⇒\dot{\varphi_0})] \times (↓R) = (\odot L \dot{\psi}) + (\otimes R \dot{\varphi_0})</math></center>