According to Newton's laws the inertia force (i.e., mass times acceleration) has to be equal to the applied forces. In our case, the applied forces are : - the gravity force  - the damping force  - the stress  - a sinusoidal excitation force

where q is the angle with respect to the vertical, M is the mass, L is the length, g is the acceleration of gravity and g is the damping.

The driving force has amplitude A and driving frequency omega, w_f.
 
 

The equality of forces is :

The projection on u_q gives :

Finally, the equation of motion for a damped, sinusoidally driven pendulum in the earth's gravitational field is :

For the calculations, we choose values for M and L so that the products ML and g/L are equal to unity, it simplifies the equation to:

The chaotic behavior of the pendulum which can be observed, of course, come from the sinusoidal non-linear term.