Numerical Analysis on Chaotic Vibrations of a Cantilevered Beam under Vibroimpact

Abstract

Numerical results are presented on chaotic vibrations of a cantilevered beam under vibroimpact. The analytical model consists of a cantilevered beam, subjected to periodic excitation, and a bar that restrains the amplitude of the beam. Equation of the motion of the beam is discretised by the finite element method and impacts of the beam are computed by using the coefficient of restitution rule. Time responses of the beam are calculated with direct integration by the Newmark-β method. Then the responses are inspected with the frequency response curves, the Fourier spectra, the Lyapunov exponents and the principal component analysis. The numerical results are compared with the experimental results that are previously presented by the authors, which verify our numerical results. Effects of the location of the bar and of the clearance between the bar and the beam on the chaotic responses are examined by the numerical results. As the location of the bar becomes farther from the clamped end and the clearance becomes smaller, the frequency region of the impact vibration is enlarged. The chaotic responses of the beam generally have contribution ratio of higher vibration modes less than 5%. However, when the location of the impact is close to a node of a higher mode or super-harmonic resonance of a higher mode is excited, contribution of the higher mode increases up to 10% to 30%.