# Euler's equation of motion for rigid bodies rotating with one rotation axis not through the body's center of mass [closed]

this is my first question in this forum. Thanks for all the knowledge and support shared throughout the whole website!

I have a body with rotations around 3 axes. I am looking for the external torques on this body. Two of the rotation axes go through the center of mass of the body, but one doesn't (let this be the y-axis). I am also interested in the angular acceleration around axis y. I could need some guidance about my understanding regarding axis y.

My understanding is that this offset between the body’s center of mass and axis y doesn't matter for the angular velocity vector, because the angular velocity is independent of any distances. Thus, I can set the rotational reference frame's origin aligned with the principle axes of the body and derive the angular velocity vector directly from the three rotations. Correct?

How about the moments of inertia in the angular momentum vector? I believe for the calculation of the external torques on the body, the distance between the body and axis y doesn't matter as well, because I am in the local rotational reference frame. Correct?

The next step is the angular acceleration around axis y. Here I need to consider the distance between the body and axis y. However, I have already considered the geometry of the body for the moment of inertia in the Euler equations, so it feels wrong to consider the geometry for the calculation of the moment of inertia around axis y again. Do I just consider the mass of the body and the distance of its center of mass to axis y?

Let me give an example, so the axes, velocities and bodies have some names:

The system in the screenshot is rotating with angular velocity $$\dot{\beta}$$ around $$y_s$$. The rotational reference frame of the system is frame S ($$x_s$$, $$y_s$$, $$z_s$$). Rotation around $$x_s$$ and $$z_s$$ is locked, all components are rigid.

Another rotational reference frame G ($$x_G$$, $$y_G$$, $$z_G$$) is in distance d to the origin of frame S. Frame G together with a black gimbal and a white gyroscope rotates with angular velocity $$\dot{\alpha}$$ around $$z_G$$ and together with frame S around $$y_s$$. Gyro and gimbal cannot rotate around $$y_G$$. The gimbal cannot rotate around $$x_G$$. The gyro rotates with angular velocity $$\dot{\phi}$$ dot around $$x_G$$.

I am interested in the external torque tensor $$\vec{M}_s$$ on the gyro and the angular acceleration $$\ddot{\beta}$$.

If my understanding is correct, then I calculate the external torque tensor $$\vec{M}_G$$ as follows. The moments of inertia $$I_{xG}$$, $$I_{yG}$$, $$I_{zG}$$ are all related to frame G. $$I_{xG}$$ considers only mass and geometry of the gyro, $$I_{yG}$$ and $$I_{zG}$$ consider the mass and geometry of both the gyro and the gimbal. Correct? (at $$t=0$$ the axes $$x_s$$ and $$x_G$$ point into the same direction)

$$\vec{M}_G = \frac{d\vec{K}_G}{dt} + \vec{\omega} \times \vec{K}_G$$ $$\text{ with } \vec{K}_G=\begin{pmatrix} I_{xG}\dot{\phi}+I_{xG}\dot{\beta}sin(\dot{\alpha}t)\\ I_{yG}\dot{\beta}cos(\dot{\alpha}t)\\ I_{zG}\dot{\alpha} \end{pmatrix} \text{ and } \vec{\omega}= \begin{pmatrix} \dot{\beta}sin(\dot{\alpha}t)\\ \dot{\beta}cos(\dot{\alpha}t)\\ \dot{\alpha} \end{pmatrix}$$

I can translate $$\vec{M}_G$$ to $$\vec{M}_s$$ with the transformation matrix $$\vec{Q}$$:

$$\vec{M}_s= \vec{Q}\vec{M}_G = \begin{bmatrix} cos(\dot{\alpha}t) & -sin(\dot{\alpha}t) & 0 \\ sin(\dot{\alpha}t) & cos(\dot{\alpha}t) & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$

I can calculate the angular acceleration with $$\ddot{\beta} = - \vec{M}_{s,y} / I_{ys}$$. How do I calculate $$I_{ys}$$? It is composed of the mass and geometry of the yellow lever. I have already considered the mass and geometry of the gyro and gimbal before in frame G, so this time do I only need to consider the distance d to the center of gravity, which would lead to $$I_{ys} = I_{y,lever} + d^2(m_{gyro} + m_{gimbal})$$?

Thanks for any support!

• you might have some luck with engineers for systems like this. Commented Mar 21, 2023 at 13:07
• I am not interested in the particular solution of the given system. It is an example. I am looking for the methodology to solve coupled rotational systems. The answer before the question got closed used the Lagrangian Method, which seems to be the way to go. I would like to give the answer a +1 and choose it as the correct answer if possible. Thanks Commented Apr 16, 2023 at 18:37