Experiments on chaotic vibrations of a post-buckled beam with an axial elastic constraint

Abstract

To clarify chaotic behavior of thin walled beams, detailed experimental results are presented on chaotic vibrations of a post-buckled beam subjected to periodic lateral acceleration. A thin steel beam of thickness 0.198 mm, breadth 12.7 mm and length 106 mm is used as a test beam. Both ends of the beam are clamped for deflection. One end of the beam is elastically constrained by an axial spring. The beam is compressed to the post-buckled configuration by the axial spring. First, characteristics of restoring force and natural frequency of the beam are obtained. Dynamic nonlinear responses of the beam are measured under periodic acceleration. In specific frequency regions, chaotic responses are generated. The chaotic responses are examined carefully with the Poincaré maps, the Fourier spectra, the maximum Lyapunov exponents and the principal component analysis.The post-buckled beam shows the soften-and-hardening characteristics of restoring force. The dominant chaotic responses of the beam are bifurcated from the sub-harmonic resonances of 1 2 and 1 3 orders with the lowest mode of vibration. Changing the exciting frequency gradually, dynamical transition behaviors from these steady-state sub-harmonic response to the chaotic responses are precisely inspected by the Poincaré projection. The maximum Lyapunov exponent of the former chaotic response of 1 2 order is larger than that of the latter chaotic response of 1 3 order. The principal component analysis predicts that the contribution of the lowest mode of vibration to the chaos is dominant among other contributions of multiple vibration modes.