Vibration Control Using Parametric Excitation

A simple pendulum whose pivot executes harmonic motion in the vertical direction 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 whether, for specified values of the natural frequency and the excitation frequency, it is possible to select an amplitude and frequency for the parametric excitation such that the pendulum’s vibratory rotation is reduced. The analysis supplements numerical integration of the equation of motion with a Fourier series analysis suitable to situations where the parametric frequency is a multiple of the forcing frequency. Studies of the behavior for excitation frequencies close to, and far from, the natural frequency lead to a general guideline for selecting the parametric excitation. It is shown that, with judicious selection of the parametric amplitude, a parametric frequency that is very high relative to the highest contemplated excitation frequency can substantially reduce the forced response at any lower excitation frequency.