$\dot\psi$ 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 :

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

$\dot\psi$ 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 $$E=\frac{1}{2}\,(I_\phi\,\dot{\phi}^2+I_\psi\,\dot{\psi}^2)$$

With

$$E=E_0$$ you can solve this equation and obtain

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