I have a 1D Euler beam whose vibrations are to be controlled by piezoelectric actuators. The equation and the weak form that I derived are as follows (Boundary conditions are shown too).

The boundary condition and equations are shown. The beam is fixed at the left end. The moment only exists from x1 to x2 as that is the length of the piezo actuator.

I tried converting the standard demo example for biharmonic equation into this but I am not able to incorporate the source term. The time derivative is discretized using FDM. How should I formulate this in fenics?