Chaotic Oscillations of a Post-Buckled Beam Constrained by an Axial Spring: Part 2 — Analysis

Abstract

To compare with corresponding experiment, analytical results are presented on chaotic oscillations of a post-buckled beam constrained by an axial spring. The beam with an initial deflection is clamped at both ends. The beam is compressed to a post-buckled configuration by the axial spring. Then, the beam is subjected to both accelerations of gravity and periodic lateral excitation. Basic equations of motion includes geometrical nonlinearity of deflection and in-plane displacement. Applying the Galerkin procedure to the basic equation and using the mode shape function proposed by the author, a set of nonlinear ordinary differential equations is obtained with a multiple-degree-of-freedom system. Linear natural frequency due to the axial compression and restoring force of the post-buckled beam are obtained. Next, periodic responses of the beam are inspected by the harmonic balance method. Chaotic responses are obtained by the numerical integration of the Runge-Kutta-Gill method. Chaotic time responses are inspected by the Fourier spectra, the Poincare´ projections, the maximum Lyapunov exponents. Contribution of the number of modes of vibration to the chaos is also discussed by the principal component analysis. Chaotic response is generated within the sub-harmonic resonance responses of 1/2 and 1/3 orders. The maximum Lyapunov exponent corresponded to the sub-harmonic response of 1/2 order is greater than that of the sub-harmonic response of 1/3 order. By the inspection of the Lyapunov exponent on the chaotic response and the analysis with the multiple-degree-of-freedom system, more than three modes of vibration contribute to the chaos. Using the principal component analysis to the chaotic responses at multiple positions of the beam, the lowest mode of vibration contributes dominantly. Higher modes of vibration contribute to the chaos with small amount of amplitude.