Theoretical and Experimental Investigations of Nonlinear Vibrations for a Beam Structure

Abstract

This paper discusses theoretical and experimental investigations of the planar nonlinear vibrations and chaotic dynamics of a beam structure composed of two beams with right-angled L-shape. Differential equations of motion for the L-shape beam structure with two-degree-of-freedom are derived firstly by application of the substructure synthesis method and the Lagrangian equation. Then, the method of multiple scales is utilized to obtain a four-dimensional averaged equation of the structure. Based on the averaged equations, numerical simulations are presented to analyze the nonlinear responses and chaotic dynamics of the L-shape beam structure. The bifurcation diagram, phase portrait, amplitude spectrum and Poincare map are plotted to illustrate that the periodic solutions and chaotic motions occur in the planar nonlinear vibrations of the L-shaped beam structure under certain conditions. Finally, the experimental apparatus and schemes for measuring the amplitude of nonlinear vibrations for the L-shape beam structure are introduced briefly. Then, the detailed analysis for experimental data and signals which represent the nonlinear responses of the beam structure are given.