|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
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, wf.
The equality of forces is :
The projection on uq 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.