Chaotic Vibrations of a Post-Buckled Beam with a Variable Cross Section under Periodic Excitation.

Abstract

This paper presents the numerical simulation for the chaotic vibrations of a post-buckled beam with a variable cross section. The beam is clamped at both ends and is compressed in an axial direction. When the beam is deformed to a post-buckled configuration, chaotic vibration will be excited easily under periodic lateral acceleration. Applying Galerkin method, the basic equation of the beam is reduced to the ordinary differential equation of multiple-degrees-of-freedom systems. To get the steady-state response of the beam, the harmonic balance method is used. In a typical region of frequency, the chaotic response is excited. The chaotic time progress is calculated by the numerical integration. The chaotic response is examined carefully by the Poincare projection, the Lyapunov exponent and the Lyapunov dimension. The results of tapered beams are compared with that of the straight beam. The chaotic response with the asymmetric mode of vibration appears remarkably. The response shows the complicated projection to the chaotic attracter, moreover, the Lyapunov dimension takes higher value. The chaos of the beam with variable cross section shows the more complicated behavior to that of straight beam.