Add a more intuitive explanation of the definition of angular velocity

[moorepants]

See the comment here https://github.com/moorepants/learn-multibody-dynamics/pull/171#issuecomment-1962958217

[Peter230655]

Eq (59) can be derived from er (54) as is stated there. I think at least the scalar (2D) version is likely known to a high school student. This is what I meant by 'intuitive clearer formulas can be derived from the definition (54)'.

[moorepants]

54 is the definition and this is 59 (for reference):

[Peter230655]

and (59) looks familiar, at least its 2D version (?)

[Peter230655]

I started to read the book Dynamics by Carlos M. Roithmayr and Dewey H. Hodges. (Kane's original book got lost in shipping and I got a refund). The authors state in the preface, that theirs is simply an updated version of Kane's original. They also give the the definition, eq(54) in your lecture, eq(1) page 20 in their book, and call in a formal, abstract definition, without giving any intuitive explnation. From what I have read so far, they use it to formally derive more 'intuitive' results.

[moorepants]

If you start with a direction cosine matrix and recognize that the columns are unit vectors expressed in the second frame and then you time differentiate those unit vectors, then you get three components of the angular velocity vector. I think those map directly to the equation we present.

[Peter230655]

...and keep in mind, that the entries are $\hat a_i \circ \hat b_j$ (?). Let me try this.

[Peter230655]

In the book Spacecraft Dynamics by T. Kane https://ecommons.cornell.edu/items/bdf70b22-3ff9-4ee8-9503-a603ed268a51 on pages 47 - 50 is spelled out what you mentioned about the direction cosine matrices. (I got this from C. Roithmayr)