An empirical study of the stability of periodic motion in the forced spring-pendulum

Abstract

The elastic pendulum is a two-degree-of-freedom, nonlinear device in which the primary mass slides up and down the pendulum arm subject to the restoring force of a linear spring. In this study, radial motion (motion along the arm) is excited directly. Responses to this excitation include purely radial motion as well as swinging motion due to a 2:1 internal resonance. Changes in the behaviour of the nonlinear spring-pendulum occur when, under the control of a parameter. one response becomes unstable and is replaced by another. These bifurcations are explored analytically, numerically and experimentally, using the basic ideas of Floquet theory. Poincaré sampling is used to reduce the problem of describing the stability of a limit cycle to the easier task of defining the stability of the fixed point of a Poincaré map. Empirical estimates of characteristic multipliers in four-dimensional state space are obtained by examining transient behaviour after perturbations; the Karhunen-Loeve decomposition is used to identify dominant local modes in these transients.