newtonian mechanics - mathematical model of a system of rigid bodies

In my quest for getting comfortable with the mathematical modelling of a system of rigid bodies, I am posing the following problem:

Problem:

Suppose in outer space (no gravity) there are two rigid bodies as in the following figure: The rigid bodies are connected with one degree of freedom rotational joint about $\vec{i}_1$ axis. There is a coordinate system attached in the center of mass of each rigid body. Suppose the following: $\begin{bmatrix} \vec{i}_1\\ \vec{j}_1\\ \vec{k}_1\end{bmatrix} = R_{i\psi} R_{j\chi} R_{k\theta} \cdot \begin{bmatrix} \vec{i}_0\\ \vec{j}_0\\ \vec{k}_0\end{bmatrix}$ and $\begin{bmatrix} \vec{i}_2\\ \vec{j}_2\\ \vec{k}_2\end{bmatrix} = R_{i\phi} \cdot \begin{bmatrix} \vec{i}_1\\ \vec{j}_1\\ \vec{k}_1\end{bmatrix}$, where $R = R_{i\psi} R_{j\chi} R_{k\theta}$ is a rotation matrix and $R_{i\phi}$ is the rotation matrix about $\vec{i}$ axis with angle $\phi$. On the first rigid body is a attached a motor in point $S$ in the figure. The motor's shaft is attached by the second rigid body, thus the motor's bearings are forming that "1 d.o.f" rotational joint. There are no other forces or torques acting on the system. I want to model the dynamics of this system.

Modelling:

I am using D'Alembert principle. The system has $3 + 3 + 1$ degrees of freedom. The variables I am choosing are $x,y,z$ the coordinates (in world frame) of the center of mass of first rigid body, $\psi, \chi, \theta$ the angles giving the orientation of the first rigid body and finally the angle $\phi$. Suppose the first rigid body has the inertia matrix $I_{01}$ in it's coordinate system and the second rigid body has the inertia matrix $I_{02}$ in it's coordinate system. D'Alembert's equations for a system of $N$ rigid bodies with $q_1, ... , q_n$ degrees of freedom are: \begin{align} \begin{cases} \sum_{i=1}^N \left( F_i^a(t) - \dot{P}_i(t)\right) \cdot \frac{\partial \bar{v}_i}{\partial \dot{q}_1} + \sum_{i=1}^m \left( T_i^a(t) - \dot{L}_i(t)\right) \cdot \frac{\partial \omega_i}{\partial \dot{q}_1} = 0 \\ \vdots \\ \sum_{i=1}^N \left( F_i^a(t) - \dot{P}_i(t)\right) \cdot \frac{\partial \bar{v}_i}{\partial \dot{q}_n} + \sum_{i=1}^m \left( T_i^a(t) - \dot{L}_i(t)\right) \cdot \frac{\partial \omega_i}{\partial \dot{q}_n} = 0 \end{cases} \end{align} where $\bar{v}_i$ and $\omega_i$ are the linear respectively angular velocity of each rigid body and $F_i^a$ is the total applied force on the $i'th$ rigid body whereas $T_i^a$ is the total applied torque (about the center of mass) on the $i'th$ rigid body, all in world frame.

Question:

Suppose the motor in point $S$ applies the torque $R^T \cdot \begin{bmatrix} T_m\\0\\0\end{bmatrix}$ on the second rigid body. Am want to know if in the above presented case:

1. $F_1^a = F_2^a = \begin{bmatrix} 0\\0\\0 \end{bmatrix}$ ?


2. $T_1^a = -R^T \cdot \begin{bmatrix} T_m\\0\\0\end{bmatrix} = -T_2^a$ ?

Conclusion:

Therefore the equations of the motion will result upon replacing everything in D'Alembert's equations ... am I correct? I've seen (I think) solutions using the parallel axis theorem, which I do not understand. Why is the parallel axis theorem needed? Where is it used? It seems that every thing can be modeled without it ... Am I wrong?