# Angular velocity of a 3D rigid body with Eulerian Angles

## Main Question or Discussion Point

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.

## Answers and Replies

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