Having hard time computing generalized forces Suppose we have a rigid body and point $O$ is fixed. It is clear then that it has three degrees of freedom and so we can choose Euler angles $\phi,\psi,\theta$ as generalized coordinates to describe its position.
But how to compute $Q_{\psi},Q_{\phi},Q_{\theta}$ -- generalized forces? Can you give any hint?
 A: 
The rotation matrix $~\mathbf R~$ is a function of the Euler angles $~\phi~,\theta~,\psi~$
from here you obtain the angular velocity
$$\mathbf \omega=\mathbf J_R(~\phi~,\theta~,\psi~)\,\mathbf{\dot{q}}$$
where $~\mathbf{\dot{q}}=[\dot\phi~,\dot\theta~,\dot\psi]^T~$ and $\mathbf J_R~$ $~3\times 3~$ distributor matrix
thus the generalized torque
$$\mathbf Q=  \mathbf J_R^T\,\mathbf \tau\\
\mathbf\tau=\mathbf r\times \mathbf F\\
\mathbf F=[F_1~,-F_2~,0]^T$$
Matrix $~\mathbf J_R~$
with
$$R=R_z(\psi)\,R_x(\phi)\,R_z(\theta)\quad \Rightarrow\\
 \left[ \begin {array}{ccc} 0&-\omega_{{3}}&\omega_{{2}}
\\ \omega_{{3}}&0&-\omega_{{1}}\\ 
-\omega_{{2}}&\omega_{{1}}&0\end {array} \right] 
=\mathbf R^T\,\frac{d}{dt}\,\mathbf R\quad \text{or}\quad
\mathbf\omega=\mathbf J_R\,\mathbf{\dot{q}}\\ 
J_R= \left[ \begin {array}{ccc} \cos \left( \theta \right) &0&\sin \left( 
\phi \right) \sin \left( \theta \right) \\ -\sin
 \left( \theta \right) &0&\sin \left( \phi \right) \cos \left( \theta
 \right) \\  0&1&\cos \left( \phi \right) 
\end {array} \right] 
$$

