A system \(\Sigma\) comprised of two rigid bodies 1 and 2 is in motion in a Newtonian referential 0\((O, \boldsymbol{\hat{x}}_0, \boldsymbol{\hat{y}}_0, \boldsymbol{\hat{z}}_0 )\). Axis \((O,\boldsymbol{\hat{z}}_0)\) is directed upward.
Body 1 \((O, \boldsymbol{\hat{x}}_1,\boldsymbol{\hat{y}}_1,\boldsymbol{\hat{z}}_1= \boldsymbol{\hat{z}}_0)\) is in the shape of a torus (hoop). It is of homogeneous density, of mass center \(O\), radius \(R\), and mass \(m_1\). It is connected to frame 0 by an ideal pivot of axis \((O,\boldsymbol{\hat{z}}_0)\), and its position is defined by angle \(\psi = (\boldsymbol{\hat{x}}_0,\boldsymbol{\hat{x}}_1)= (\boldsymbol{\hat{y}}_0 ,\boldsymbol{\hat{y}}_1)\). Denote by \(I_1\) the moment of inertia of 1 about axis \((O,\boldsymbol{\hat{z}}_0)\).
Body 2 \((G,\boldsymbol{\hat{x}}_2 ,\boldsymbol{\hat{y}}_2 = \boldsymbol{\hat{y}}_1 ,\boldsymbol{\hat{z}}_2 )\) is a truncated hollow torus-shaped body of mass center \(G\), mass \(m_2\), with \(\boldsymbol{r}_{OG}= R \boldsymbol{\hat{z}}_2\). The exterior surface of 1 is in contact with the interior surface of 2: body 2 is free to ``slide’’ relative to 1, and its motion relative to 1 is entirely parametrized by angle \(\theta = (\boldsymbol{\hat{x}}_1 , \boldsymbol{\hat{x}}_2) = (\boldsymbol{\hat{z}}_0, \boldsymbol{\hat{z}}_2)\). Its inertia operator about \(G\) is represented by: \[ [{\cal I}_{G, 2}] = \left[ \begin{array}{ccc} A_2 & 0 & 0 \\ 0 & B_2 & 0 \\ 0 & 0 & C_2 \\ \end{array} \right] _{(\boldsymbol{\hat{x}}_2,\boldsymbol{\hat{y}}_2,\boldsymbol{\hat{z}}_2)} \] The joint between bodies 1 and 2 is not ideal. The corresponding contact action screw \(\{{\cal A}^c_{1\to 2} \}\) is characterized by \[ \boldsymbol{M}_{O, 1\to 2}^c \cdot \boldsymbol{\hat{y}}_1 = -k \dot{\theta}\qquad (k>0) \] A motor \(\cal M\) is mounted between 0 and 1, and imposes a prescribed couple \({\cal C}\boldsymbol{\hat{z}}_0\). All other effects are neglected.
1. (20 pts) Draw the necessary rotation diagrams, then find the velocity of point \(G\) using a method of your choice. Give the expressions of kinematic screws \(\{{\cal V}_{1/0}\}\) and \(\{{\cal V}_{2/0}\}\). Conclude that \(\boldsymbol{v}_{O\in 2/0}= \boldsymbol{0}\).
2. (10 pts) Find the kinetic screws of bodies 1 and 2.
3. (20 pts) Derive a single equation extracted from the Fundamental Theorem of Dynamics which relates couple \({\cal C}\) to the unknown angles $(t) $ and \(\theta (t)\) (and no other unknowns).
4. (20 pts) Apply the KET to system \(\Sigma\) to find another equation.
5. (10 pts) Assume that the motor’s action is such that the angular speed \(\dot{\psi}\) reaches a constant value \(\dot{\psi}_0\). Show that angle \(\theta\) reaches an equilibrium value \(\theta= \theta_e\). Find \(\theta_e\) in terms of \(\dot{\psi}_0\) and other constants.