How to add angular velocity vectors?

Question

I was reading David Morin's mechanics book and came across this problem.

And here is the solution provided:

I am just wondering why you can express the total angular velocity of the coin with respect to the lab frame by simply adding the different angular velocity vectors. In the earlier part of the book, there's is a theorem: Clearly in this case, angular velocity vector from the rotation about the centre of the contact-point circle ($\mathbf{\Omega}\mathbf{\hat{z}}$)and the angular velocity vector from the rotation about $\mathbf{\hat{x_3}}$ do not share a common origin (if you extend a line from the centre of the coin perpendicular to the surface of the coin, it clearly will not always pass through the centre of the contact-point circle).

I think my understanding is flawed. Why can you add angular velocities in this case?

Answer 1

Starting with the rotations matrix \begin{align*} &[\,_1^3\,\mathbf S\,]=[\,_1^2\,\mathbf S\,]\,[\,_2^3\,\mathbf S\,]\quad\Rightarrow\quad [\,_1^3\,\mathbf{\dot{S}}\,]=[\,_1^2\,\mathbf{\dot{S}}\,]\,[\,_2^3\,\mathbf S\,]+ [\,_1^2\,\mathbf S\,]\,[\,_2^3\,\mathbf{\dot{S}}\,]\\ &\text{with}\quad \mathbf{\dot{S}}=\mathbf{\tilde{\omega}}\,\mathbf S\quad \mathbf{\tilde{\omega}}= \left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}} \\ \omega_{{z}}&0&-\omega_{{x}}\\ -\omega_{{y}}&\omega_{{x}}&0\end {array} \right]\quad\Rightarrow \\\\ &\mathbf{\tilde{\omega}}_{13}[\,_1^3\,\mathbf{{S}}\,]= \mathbf{\tilde{\omega}}_{12}[\,_1^2\,\mathbf{{S}}\,]\,[\,_2^3\,\mathbf S\,]+ [\,_1^2\,\mathbf S\,]\,\mathbf{\tilde{\omega}}_{23}[\,_2^3\,\mathbf{{S}}\,]\\\\ &\text{multiply from the right with}\quad [\,_3^1\,\mathbf{{S}}\,]\\\\ &\mathbf{\tilde{\omega}}_{13}= \mathbf{\tilde{\omega}}_{12}\underbrace{[\,_1^2\,\mathbf{{S}}\,]\,[\,_2^3\,\mathbf S\,][\,_3^1\,\mathbf{{S}}\,]}_{I_3}+ [\,_1^2\,\mathbf S\,]\,\mathbf{\tilde{\omega}}_{23}\underbrace{[\,_2^3\,\mathbf{{S}}\,][\,_3^1\,\mathbf{{S}}\,]} _{ [\,_2^1\,\mathbf S\,]}\\ &\text{thus the angular velocity vector}\\ &\mathbf\omega_{13}=\mathbf\omega_{12}+[\,_1^2\,\mathbf S\,]\mathbf\omega_{23} \end{align*} \begin{align*} &\text{with}\\ &[\,_1^2\,\mathbf S\,]=\left[ \begin {array}{ccc} \cos \left( \psi \right) &-\sin \left( \psi \right) &0\\ \sin \left( \psi \right) &\cos \left( \psi \right) &0\\ 0&0&1\end {array} \right] \quad, [\,_2^3\,\mathbf S\,]=\left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \phi \right) &-\sin \left( \phi \right) \\ 0 &\sin \left( \phi \right) &\cos \left( \phi \right) \end {array} \right] \\ &\mathbf\omega_{12}=\begin{bmatrix} 0 \\ 0 \\ \omega_\psi \\ \end{bmatrix}\quad \mathbf\omega_{23}=\begin{bmatrix} \omega_\phi \\ 0 \\ 0 \\ \end{bmatrix} \end{align*}