Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

Abstract

Analytical results are presented on chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. The L-shaped beam is composed of two beams which are a horizontal beam and a vertical beam. The two beams are firmly connected with a right angle at each end. The beams joint with the right angle is attached to a linear spring. The other ends are firmly clamped for displacement. The L-shaped beam is compressed horizontally via the spring at the beams joint. The L-shaped beam deforms to a post-buckled configuration. Boundary conditions are required with geometrical continuity of displacements and dynamical equilibrium with axial force, bending moment, and share force, respectively. In the analysis, the mode shape function proposed by the senior author is introduced. The coefficients of the mode shape function are fixed to satisfy boundary conditions of displacements and linearized equilibrium conditions of force and moment. Assuming responses of the beam with the sum of the mode shape function, then applying the modified Galerkin procedure to the governing equations, a set of nonlinear ordinary differential equations is obtained in a multiple-degree-of-freedom system. Nonlinear responses of the beam are calculated under periodic lateral acceleration. Nonlinear frequency response curves are computed with the harmonic balance method in a wide range of excitation frequency. Chaotic vibrations are obtained with the numerical integration in a specific frequency region. The chaotic responses are investigated with the Fourier spectra, the Poincare projections, the maximum Lyapunov exponents and the Lyapunov dimension. Applying the procedure of the proper orthogonal decomposition to the chaotic responses, contribution of vibration modes to the chaotic responses is confirmed. The following results have been found: The chaotic responses are generated with the ultra-subharmonic resonant response of the two-third order corresponding to the lowest mode of vibration. The Lyapunov dimension shows that three modes of vibration contribute to the chaotic vibrations predominantly. The results of proper orthogonal decomposition confirm that the three modes contribute to the chaos, which are the first, second, and third modes of vibration. Moreover, the results of the proper orthogonal decomposition are evaluated with velocity which is equivalent to kinetic energy. Higher modes of vibration show larger contribution to the chaotic responses, even though the first mode of vibration has the largest contribution ratio.