Make terminology consistent regarding observation of change and expressing

[moorepants]

We say things like:

"the velocity of P in A means the change of a vector from a point fixed in A to point P possibly moving in A when observed from A"

"velocity of P with respect to A"

"express velocity of P in A"

I think we need to consistently use "when observed from" to characterize the derivative when observed from a frame and that "with respect to" should probably not be used for this because I think I may also use it for expressing a vector in the unit vectors of a frame.

[Peter230655]

As my native language is German, I never comment on wording, you are American.
I think you are right with this consistency: when observed from appeals to intuition.

[Peter230655]

Somewhere above eq(59) you write:
...if there is a unit vector k fixed in both A and B, then the angular speed is omega = | beta | x k....
(Copying the text did not work, I hope you can identify it).
Frankly, it took me a while to understand it. Of course the explanation is eq(59) together with the fact that (a x b = 0) < => (b = alpha*a, alpha a real number), which you give below figure 16
This means, the only vector omega does not change is (any multiple of) itself. So, if k is fixed, it must be a multiple of omega.
Also the next sentence about beta is clear from eq(59): if c = a x b, then c is perpendicular to both a and b, as you say below fig 16.
I have no idea, whether such a hint there would be useful.

[Peter230655]

An intuition for eq (67) maybe could be this:
Above you showed, that omega is the (only) vector fixed in both frame A and frame B.
So, no contribution to d/dt (omega) could come from the relative motion between A and B, after all, omega is fixed in both.
So, A_d/dt(omega) = B_d/dt(omega).

Of course, this is just many words for omega x omega = 0.

[Peter230655]

Definition (54) seems "arbitrary" at first glance, I believe I mentioned this last year - of course a definition is just that, a definition.
Equation (59), derived directly from definition (54), seems quite intuitive, to me anyway.
Now, if one keeps in mind what dot(b, a_hat) means, namely the projection of b onto a_hat, if one further keeps in mind, what eq(59) means, namely if in the "right hand picture" my thumb is omega, then omega turns my index finger towards my middle finger, then definition (54) makes sense also intuitively.
Of course, only my thinking, I do not know, what students need.

[Peter230655]

Your excercise above eq(59) concerning the gimbal lock.
If I understand it correctly, the motion of the T-handle is not affected by the possible gimbal lock at all, it is only if there were a gimbal mounted to the T-handle, it would not work right anymore?
( I could not see the video of the T-handle, in the link you gave me to see the lecture when you set it 'life', the video does not run, I believe I reported before)

[Peter230655]

https://moorepants.github.io/learn-multibody-dynamics/
This is the link I meant above.

[moorepants]

It is possible to select Euler angles that exhibit gimbal lock for the T-handle. Just try some others and check the rotation matrices or calculate the kinematical differential equations in explicit form to see the divide by zeros that can appear.

[Peter230655]

Wwhat I meant was this: the gimbal gets 'physically' locked, looses a degree of freedom. With the T-handle it is 'just' an unfortunate description of the system?

[moorepants]

I see you what you mean. Yes there is the physical manifestation in a real gimbal and the mathematical manifestation that results in singularities in our description of the system. I should probably separate those concepts in the text.