How to find the velocity and accelaration in a 3d space with 6 degrees of freedom?

Question

I have the following rigid body:

I assume that the body is a symmetric cylinder.x,y,z are the axes of the reference frame resulting from a transformation involving three orthogonal rotations θ1,θ2,θ3 about the axes of an orignal reference frame X,Y,Z.With first order small angle approximations and neglecting the terms of higher order in the Taylor expansion,the Euler-angle transformation the original and transformed reference frame is:

Until here everything is understantable.My question is how to find the velocities and accelerations in the transformed reference frame?

Answer 1

Differentiating both sides with respepct to time, you get $\begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix} = A \begin{bmatrix} v_X \\ v_Y \\ v_Z \end{bmatrix}$ where $A$ is the matrix containing the thetas. Similarly, taking second derivatives with respect to time on both sides gets you the accelerations.