Interaction between Internal Resonance and Dynamic Snap-through in Chaotic Vibrations of a Post-buckled Beam with Variable Cross Section.

Abstract

Chaotic responses are investigated for a post-buckled beam under the interaction between internal resonance and dynamic snap-through. The beam with variable cross section is clamped at both ends and the beam is axially compressed to a post-buckled configuration. The buckled beam is excited by a periodic acceleration. Applying the Galerkin method to the governing equation of the beam, nonlinear differential equations of a multi-degree-of-freedom system are reduced. Periodic solutions of steady-state responses are calculated by the harmonic balance method. In a typical frequency region, chaotic response bifurcates from the periodic response. Time progress of the chaotic motion is calculated by the numerical integration. The chaotic response is examined in detail by the Fourier spectrum, the Poincare section, the Lyapunov exponent and the Lyapunov dimension, respectively. Under the condition of the one-to-two internal resonance, the chaos is generated by a small amplitude of excitation. Two modes of the vibration induced in the chaos. Increasing the amplitude of excitation, the chaotic response transits to the complicated response coupled with the dynamic snap-through and the internal resonance. Induced modes in the chaos are counted as nearly four.