Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap6:6.3_dynamics_with_one_degree_of_freedom

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book:chap6:6.3_dynamics_with_one_degree_of_freedom [2022/02/14 13:57] – created abrilbook:chap6:6.3_dynamics_with_one_degree_of_freedom [2022/04/12 13:53] (current) – abril
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 for every given set of initial conditions $\bigl( \theta(t_0), \dot\theta(t_0) \bigr)$, for every given set of initial conditions $\bigl( \theta(t_0), \dot\theta(t_0) \bigr)$,
  
-<WRAP 120pt right #fig_mathPendulum_phaseSpace>+<WRAP 140pt right #fig_mathPendulum_phaseSpace>
 {{EOM_mathPendulum_phaseSpace.png}}\\ {{EOM_mathPendulum_phaseSpace.png}}\\
 Figure 6.5: The potential $U(\theta)$ (top) and the phase-space plot (bottom) for the EOM [[#equation_EOMmathPendulum | EOM 6.3.1]] of the mathematical pendulum. The numbers marked on the contour lines indicated the energy of a trajectory in units of $MgL$. Figure 6.5: The potential $U(\theta)$ (top) and the phase-space plot (bottom) for the EOM [[#equation_EOMmathPendulum | EOM 6.3.1]] of the mathematical pendulum. The numbers marked on the contour lines indicated the energy of a trajectory in units of $MgL$.
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 Such a trajectory is called a homocline. Such a trajectory is called a homocline.
  
-<WRAP 120pt right #fig_mathPendulumHomocline>+<WRAP 140pt right #fig_mathPendulumHomocline>
 {{EOM_mathPendulum_homocline.png}}\\ {{EOM_mathPendulum_homocline.png}}\\
 Figure 6.6: Anticlockwise moving heterocline for the mathematical pendulum. Figure 6.6: Anticlockwise moving heterocline for the mathematical pendulum.
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   -  Trajectories with initial conditions lying below the clockwise moving heterocline will persistently rotate clockwise and never reverse their motion.   -  Trajectories with initial conditions lying below the clockwise moving heterocline will persistently rotate clockwise and never reverse their motion.
  
-<WRAP 120pt left #fig_steps2phase-space-plot> +<WRAP 160pt right #fig_steps2phase-space-plot> 
-{{identify-fixed-points.png}}]+{{identify-fixed-points.png}}
 Figure 6.7: Step by step sketch of a phase-space plot. Figure 6.7: Step by step sketch of a phase-space plot.
 </WRAP> </WRAP>
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 This EOM can once be integrated by the same strategy adopted for the swing and the spherical pendulum. This EOM can once be integrated by the same strategy adopted for the swing and the spherical pendulum.
 Thus, one finds the effective potential  Thus, one finds the effective potential 
 +<wrap #eq_EOMpearlRing-Ueff></wrap>
 \begin{align*}  \begin{align*} 
   U_{\text{eff}}(\theta) = -\omega^2\; \cos\theta \; \left[ 1 - \left( \frac{\Omega}{\omega}  \right)^2  \; \cos\theta \right] \tag{6.3.5}   U_{\text{eff}}(\theta) = -\omega^2\; \cos\theta \; \left[ 1 - \left( \frac{\Omega}{\omega}  \right)^2  \; \cos\theta \right] \tag{6.3.5}
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 For $E > U_{\text{eff}}(\theta=\pi)$ they rotate around the ring in clockwise or counter-clockwise direction for $\dot\theta < 0$ or $\dot\theta > 0$, respectively. For $E > U_{\text{eff}}(\theta=\pi)$ they rotate around the ring in clockwise or counter-clockwise direction for $\dot\theta < 0$ or $\dot\theta > 0$, respectively.
  
-<WRAP 120pt #fig_EOM_rotGovernor>+<WRAP #fig_EOM_rotGovernor>
 {{EOM_rotGovernor_Ueff.png}} {{EOM_rotGovernor_Ueff.png}}
 Figure 6.9: The left panel shows the effective potential for the pearl on a ring for parameter values $(\Omega/\omega) \in \{ 0, 2^{-1/2}, 1, 1.2, 1.5, 2, 5 \}$ from bottom to top. The subsequent panels show phase-space portraits of the motion for $\Omega/\omega = 2^{-1/2}$, $1$, and$2$, respectively. Figure 6.9: The left panel shows the effective potential for the pearl on a ring for parameter values $(\Omega/\omega) \in \{ 0, 2^{-1/2}, 1, 1.2, 1.5, 2, 5 \}$ from bottom to top. The subsequent panels show phase-space portraits of the motion for $\Omega/\omega = 2^{-1/2}$, $1$, and$2$, respectively.
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 is shown in [[#fig_EOM_rotGovernorDissipation |Figure 6.12]].  is shown in [[#fig_EOM_rotGovernorDissipation |Figure 6.12]]. 
  
-<WRAP 120pt left #fig_EOM_rotGovernorDissipation>+<WRAP 400pt center #fig_EOM_rotGovernorDissipation>
 {{EOM_rotGovernor_gamma_02.png}} {{EOM_rotGovernor_gamma_02.png}}
 Figure 6.12: Phase-space plot for a rotational governor with rotation frequency $\Omega = 2\omega$. Figure 6.12: Phase-space plot for a rotational governor with rotation frequency $\Omega = 2\omega$.
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 Henceforth we the focus on the motion of one of the beats.  Henceforth we the focus on the motion of one of the beats. 
  
-<WRAP 120pt #fig_carousel>+<WRAP #fig_carousel>
 {{Carousel.png}} {{Carousel.png}}
 Figure 6.13: Experimental setup and description of configurations for a toy carousel. Figure 6.13: Experimental setup and description of configurations for a toy carousel.
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 and beyond the heteroclinic trajectories that connect the maxima of the potential one finds trajectories that keep rotating in the same direction.  and beyond the heteroclinic trajectories that connect the maxima of the potential one finds trajectories that keep rotating in the same direction. 
  
-<WRAP 120pt left #fig_carussell-phaseSpace>+<WRAP #fig_carussell-phaseSpace>
 {{EOM_carussell_potential.png}} {{EOM_carussell_potential.png}}
 Figure 6.14: The effective potential (left) and phase space plots for the parameter values Figure 6.14: The effective potential (left) and phase space plots for the parameter values
book/chap6/6.3_dynamics_with_one_degree_of_freedom.1644843456.txt.gz · Last modified: 2022/02/14 13:57 by abril