Contribution of Multiple Vibration Modes to Chaotic Vibrations of a Post-buckled Beam with an Axial Elastic Constraint

Abstract

Analytical results are presented on contribution of multiple modes of vibration to chaotic responses of a post-buckled clamped beam constrained by an axial spring. Introducing the mode shape function proposed by the senior author and applying the Galerkin procedure to the governing equation of the beam, a set of nonlinear ordinary differential equations in amultiple-degree-of-freedom system is obtained. Chaotic time responses are integrated numerically. Responses of the beam subjected to periodic lateral acceleration are investigated by comparing with the relevant experimental results. Dominant chaotic responses are generated within the frequency ranges of the subharmonic resonance of 1/2 and 1/3 orders. The maximum Lyapunov exponent of the chaotic response corresponding to the sub-harmonic resonance of 1/2 order is greater than that of the chaos with the sub-harmonic resonance of 1/3 order. The analytical results of the chaotic responses have remarkable agreement with that of the experimental results. The Lyapunov dimension and the Poincare projection of the chaotic responses predict that more than three modes of vibration contribute to the chaos based on the calculation from the equation of multiple-degree-of-freedom system. The principal component analysis shows that the lowest vibration mode contributes dominantly. Higher modes of vibration contribute to the chaos with small amount of amplitude.