Angular velocity of a 3D rigid body with Eulerian Angles

[Curl]

Say I have an odd shaped rigid body in pure rotation. I fix an orthogonal x-y-z coordinate system to the body, and coincide the origin of this coordinate system to the origin of my global coordinate system X-Y-Z.

I have 3 Eulerian angles (as functions of time) which I can use to describe the orientation of my rigid body at any instant. I also have a transformation matrix A which I can use to convert any vector r' in the x-y-z system to a vector r in the X-Y-Z system.

I am trying to find out how I can express the angular velocity, ω of my rigid body using this information. I know that r x ω=dr/dt for any position vector r describing the position of a point on the rigid body (correct?).
I also know that r=Ar', so that r=dA/dt*r' (since r' is constant, the point stationary in the x-y-z frame).
I cannot "solve" for ω in the r x ω=dr/dt since there are infinitely many possibilities (it yields a skew-symmetric 3x3 matrix). It seems like if I have the position vector of a point on the body as a function of time, I should be able to express the angular velocity but I'm stuck.

It's probably a very easy question but I'd appreciate any help. Thanks.

 

[A.T.]

http://www.kwon3d.com/theory/euler/avel.html

 

[D H]

Say I have an odd shaped rigid body in pure rotation.

Clarify this, please. Do you mean that the body is undergoing a forced rotation about some fixed axis, or that the body is floating freely in space and has some constant angular momentum in an inertial frame, or neither of these two?

 

[Curl]

Odd shaped rigid body - imagine a rock. Not a 2D/1D object (like a lamina or rod).
It is undergoing forced rotation about the origin (we don't know the axis of rotation) by some external torque, but no net force on the body, and it is not translating. The origin of x-y-z always coincides with the origin of X-Y-Z. For now, let's say that the external torque is such that the body is rotating at a constant angular velocity.

My idea was if I have r(t) and dr(t)/dt I could express the angular velocity. A.T., I've seen that method but I wanted to see if it is possible to do this problem my way.

 

[PJC01]

I would suggest using the rotational matrix to calculate the angular velocity of your rigid body. The rotational matrix can be derived from the Eulerian angles and represents the orientation of the body in 3D space. Once you have this matrix, you can use it to calculate the angular velocity using the formula ω = dA/dt * A^T, where A^T is the transpose of the rotational matrix. This formula will give you a 3x3 skew-symmetric matrix, but you can extract the angular velocity components from it.

Another approach could be to use the time derivative of the Eulerian angles to calculate the angular velocity. The Eulerian angles represent the orientation of the body relative to the global coordinate system, so the time derivative of these angles can give you the rate of change of the orientation. You can then use this information to calculate the angular velocity vector.

It is important to note that the angular velocity of a rigid body is a vector quantity, so it has both magnitude and direction. Therefore, it is necessary to use a vector representation, such as the rotational matrix or the angular velocity vector, to fully describe the angular velocity of the body.

In summary, to express the angular velocity of a 3D rigid body with Eulerian angles, you can use the rotational matrix or the time derivative of the Eulerian angles. Both approaches will give you the angular velocity vector, which represents the rotational motion of the body.