Free Precession animation - body frame to space frame

[potatocar]

I'm creating an animation of free precession of a cuboid in GeoGebra. The axis of rotation is not one of the principal axes (but does go through center of mass).

Since it's much easier to find the angular velocity and L in the body frame, I defined the e1, e2 and e3 axes (as opposed to the xyz coordinate system that already exists in GeoGebra as the space frame) and defined the omega and L vectors via the body frame axes. So I get the motion shown in (a). e3 is fixed, omega and L precess about it, which is correct in the body frame.

I don't know how to get from (a) to (b) now. I know I have to rotate the cuboid (the body frame axes, omega and L will also rotate accordingly since they're defined via the cuboid). QUESTION: which vector do I rotate them about and with what angular velocity?

I might also be going about this a completely wrong way, I'll appreciate any comment.

 

[lippyka]

Answer:To move from (a) to (b), you need to rotate the body frame axes, omega and L vectors about the axis of rotation with the angular velocity of the precession. Since the axis of rotation is not one of the principal axes, you'll need to define the axis of rotation in terms of the body frame axes (e1, e2, e3). For example, if the axis of rotation is defined by a unit vector u = (u1,u2,u3), then you can use the formula for a rotation matrix to calculate the rotation matrix R that will rotate the body frame axes, omega and L vectors about the axis of rotation u: R = [u1u2-u3 0 -u1; -u2u3 0 u2; u3^2-u1^2-u2^2 1 -u3] where u1, u2, u3 are the components of u. Once you have the rotation matrix R, you can use it to rotate the body frame axes, omega and L vectors with the angular velocity of the precession.