Vibration control using parametric excitation

A simple pendulum whose pivot executes harmonic motion in the vertical dire ction is a prototype for systems subjected to parametric excitation. Forced excitation of this system is represented as a harmonically varying torque whose frequency is taken to be arbitrary. The investigation explores whethe r, for specified values of the natural frequency and the excitation frequen cy, it is possible to select an amplitude and frequency for the parametric excitation such that the pendulum's vibratory rotation is reduced. The anal ysis supplements numerical integration of the equation of motion with a Fou rier series analysis suitable to situations where the parametric frequency is a multiple of the forcing frequency. Studies of the behavior for excitat ion frequencies close to, and far from, the natural frequency lead to a gen eral guideline for selecting the parametric excitation. It is shown that, w ith judicious selection of the parametric amplitude, a parametric frequency that is very high relative to the highest contemplated excitation frequenc y can substantially reduce the forced response at any lower excitation freq uency.