Proof of angular velocity identity

Question

I've been going thru the book by Hodges and Roithmayr regarding dynamics theory and I can't seem to prove that:

I have tried using multiple substitutions, including the fact that every component of the direction cosine matrix is equal to its cofactor, in multiple, different ways. But I feel like I've been walking in circles. The last attempt I made was, when using the definition of angular velocity:

And then writing every derivative of the coordinate system by substituting its direction cosines and substituting the cofactors inside. This led to another interesting thing:

Which may or may not be related to the proof I'm looking for, but I cannot work out how. My guess is that this would be a pretty standard equation to use, since angular velocity is such a big thing in dynamics, but I could not find it anywhere.

I think the "Maxwell–Betti reciprocal work theorem" may help. https://en.wikipedia.org/wiki/Betti%27s_theorem — r13, Apr 09 '21 at 23:19
Could you please explain the $^A\vec\omega^B$ notation? I've never came across it. — NMech, Apr 10 '21 at 05:39
@NMech is the angular velocity of a body B in a reference frame A. — Engels Leonhardt, Apr 10 '21 at 13:42
I don't know if the rule permits that (answer the question of yourself) or not. But since nobody has provided a response with a pointed answer, I think it is considered a good service to the readers who have the same question. Also, it might draw further responses with great insight. So why not, I support posting your findings. — r13, Apr 10 '21 at 13:52
Or, better just add the finding to the original question to form a new question, and asking for review/confirmation. — r13, Apr 10 '21 at 13:56
@engels Leonardt yes you can post and accept your answer. There is no rule against that. — NMech, Apr 10 '21 at 14:07

score 0 · Accepted Answer · answered Apr 12 '21 at 15:43

Knowing that the derivative of a vector in frame A can be written as:

And similarly, the derivative of the same vector written in the B frame can be written as:

Substituting the second equation in the first one we have:

For the above to be true, either the vector being differentiated is a null vector (which is a trivial solution) or:

Proof of angular velocity identity

1 Answers1