On the Dynamics of a Conservative Elastic Pendulum

Abstract

This paper presents a finite element model of the elastica, without dissipation, in a form realizable in the laboratory: a set of rigid links connected by torsion springs. The model is shown to reproduce the linear elastic behavior of beams. The linear beam, and most nonlinear beams are not periodic. (The linear eigenfrequencies are incommensurate.) They do exhibit a basic cyclic behavior, the beam waving back and forth with a measurable period. Extensive exploration of the behavior of a fourlink model reveals windows of periodicity—isolated points in parameter space where the motion is nearly periodic. (The basic phase plane diagrams are asymmetric, and the time evolution of the motion distributes this asymmetry symmetrically in time.) The first such window shows a period twice the basic cycle time, the next, less well observed one, four times the basic cycle time.