The behaviour of a forced spherical pendulum operating in a weightless environment

Abstract

Summary In this article, we show that by subjecting the pivot of a simple inextensible pendulum to small amplitude high frequency rectilinear oscillations it is possible to make it operate in a weightless environment. The axis of vibration of the pivot defines a preferred direction in space and a consequential dynamical structure which is completely absent when the pivot is fixed. Using spherical polar coordinates centred at the pivot, we show that the motion of such a pendulum has fast and slow-scale components which we analyse using the method of multiple scales. The slow scale equation for the polar angle is autonomous, and a phase plane analysis reveals the essential orbital structure including the existence of conical solutions analogous to the terrestrial fixed pivot conical pendulum. In the absence of an azimuthal velocity component, its behaviour can provide a direct simulation of a plane terrestrial simple fixed pivot pendulum with a correspondingly simple form for the small amplitude period. We can also use a two-scale analysis to examine the effects of damping. Here, the slow scale polar equation has two asymptotically stable states, and we employ a combination of numerical and asymptotic analyses to elicit the slow scale orbital trajectories.