Driven Pendulums

Abstract

One of the functions of the escapement in a pendulum clock is to transfer energy to the pendulum in order to replace energy lost due to friction etc. It does this by impulses whose timing is determined by the pendulum. By contrast, in a driven pendulum, energy is transferred to a pendulum by periodic forces whose period is not controlled by the pendulum. A driven pendulum is sometimes called a forced pendulum. There are three distinct ways in which a simple rod pendulum can be driven. In rotary driving the pendulum is subjected to prescribed varying torques about the suspension point. Theoretical investigation of a damped simple rod pendulum under sinusoidally varying torque has shown that a wide range of chaotic behaviour sometimes occurs, especially for large amplitudes and low values of the pendulum quality, Q. Experimental results, using a specially designed real pendulum with an electromagnetic drive have confirmed the existence of both periodic and chaotic behaviour. A simple rod pendulum can be started from the rest position by using rotary driving. A simple string pendulum cannot be driven by rotary driving. In horizontal driving the suspension point of a simple rod pendulum is subjected to prescribed varying horizontal displacements. In vertical driving, sometimes called lifting, the suspension point is subjected to prescribed varying vertical displacements. In both cases there are two forces on the simple rod pendulum. One is that due to gravity on the point mass, m, and is constant. The other is that needed to impose the prescribed force or displacement, and this force varies with time. A simple rod pendulum can be started from the rest position by using horizontal driving, but it cannot be started from the rest position by using vertical driving. A simple string pendulum can be driven by horizontal driving, and by vertical driving. A prescribed force or a prescribed displacement can be either periodic, such as a sine wave or random. In the latter case random process theory is needed for understanding the effects of driving. Driven damped simple harmonic motion provides a useful approximation to some aspects of the behaviour of driven pendulums. Some aspects of horizontal driving are described.