A simple problem just for fun

[A.T.]

If you didn't start from rest motion of the system need not be periodic, the large cylinder and the small disk can acquire a uniform rotational drift or I made a grave error.

That makes sense. But a uniform rotational drift can be transformed away by adopting an uniformly rotating frame, where you still have periodic motion. And I don't think you can do that with a double pendulum for all initial conditions.

 

[bob012345]

If anyone is interested, you can tackle the problem of figuring out the highest angle $\theta$ that one can start with from rest in terms of $\mu_s$, the coefficient of static friction.

 

[bob012345]

If anyone is interested, you can tackle the problem of figuring out the highest angle $\theta$ that one can start with from rest in terms of $\mu_s$, the coefficient of static friction.

Well, I guess nobody is interested so I will just put the answer without going through the derivation here.

Spoiler

$$\tan\theta_{max}=\mu_s[1+\frac{mr^2}{I_w}+\frac{mR^2}{I_c}]$$