Global bifurcations and chaos in the forced oscillations of buckled structures

Abstract

We study the sinusoidally forced vibrations of a buckled beam. Experimental work indicates that the beam's response is 'chaotic', being a nonperiodic motion which contains appreciable energy at all frequencies. The governing nonlinear partial differential equation is shown to generate a dynamical system on a suitable function space and, since the excitation is periodic, a global Poincaré map, P¿, can be defined and the problem recast as one involving bifurcations of this map. We study the behavior as physical parameters such as force amplitude, ¿, are varied. We argue that much of the behavior can be captured by a single degree of freedom nonlinear oscillator, the Poincaré map of which is a diffeomorphism of the plane, and we indicate the importance of homoclinic orbits arising in global bifurcations of this map.