Chaotic Vibrations on a Coupled Vibrating System of a Post-Buckled Cantilevered Beam and an Axial Vibrating Body Connected With a Stretched String

Abstract

Analytical results are presented on chaotic vibrations on a coupled vibrating system of a post-buckled cantilevered beam and an axial vibrating body connected with a stretched string. The string is stretched between the top end of the cantilevered beam and the axial vibrating body which consists of a mass and a spring. As an initial axial displacement is applied to the spring, the beam is buckled by the tensile force of the string. The main scope of this paper is to investigate the effects of the axial inertia of the vibrating body on the chaotic vibrations of the system. The dynamical model involves nonlinear geometrical coupling between the deformation and the axial force of the beam at the boundary. Furthermore, the problem includes the static buckling and the nonlinear vibration. By using the mode shape function, which was proposed by the senior author, as a coordinate function of the governing equations, nonlinear ordinary differential equations in multiple-degree-of-freedom system are derived by the modified Galerkin procedure. Periodic responses of the beam are calculated with the harmonic balance method, while chaotic responses are integrated numerically. Chaotic time responses are inspected with the Fourier spectra, the Poincare´ projections, the maximum Lyapunov exponents and the principal component analysis. Chaotic responses are generated from the sub-harmonic resonance responses of 1/2 and 1/3 orders. The results of the principal component analysis shows that the lowest mode of vibration contributes to the chaotic response dominantly, while the second mode of vibration also contribute to the chaos with small amount of amplitude. Inspection of the kinetic energy of each vibration mode shows that the vibration mode with large axial displacement is also dominant in the chaotic response.