[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/ltjnX.png

$\dot\psi$ is the rotation of the rotor , $\vartheta=\pi/2$ is your configuration.

you can obtain the solution  of your problem from the  conservation of the energy :

$$E=T+U=~\text{constant}$$
where T is the kinetic energy  and U is the potential energy 

for $\vartheta=0$ is $$E_0=m\,g\,h$$ and for $vartheta=\pi/2$ is the
energy $$E=\frac{1}{2}\,(I_\phi\,\dot{\phi}^2+I_\psi\,\dot{\psi}^2)$$

thus:

$$E=E_0$$
solve this equation for $\dot{\psi}$ you obtain :

$$\dot{\psi}=\frac{\sqrt{I_\psi\,(2\,m\,g\,h-I_\phi\,\dot{\phi}^2})}{I_\psi}$$