How is angular velocity related to angles of spherical pendulum

Question

In a simple pendulum angular velocity $ω$ is related to angle $θ$ as $$θ = wΔt$$ But what would be the relationship between the angular velocity and angles comprising a spherical pendulum described by angles $θ$ and $\phi$?

Answer 1

So the $3\times 3$ Rotation Matrix R is a function of $\varphi$ and $\theta$

With

Where equation (1) and (2) are the pendulum differential equation . With the solution of $\varphi(t)$ and $\theta(t)$ you can obtain the component of the angular velocity $\vec \omega$ equation (3).

Example