Chaotic Vibrations of a Rectangular Plate with In-plane Asymmetric Constraints

This paper presents analytical results on chaotic vibrations of a rectangular plate under in-plane asymmetric constraints. The rectangular plate with an initial deflection is simply supported at all edges. The plate is subjected to gravitational and periodic acceleration laterally. The boundary of the plate is attached with an elastic material represented with distributed linear-springs in the in-plane direction. The outer sides of the distributed springs are constrained by both uniform and asymmetric in-plane displacements as the in-plane constraints. Neglecting the effect of inertia force along in-plane direction, the Marguerre type equations modified with lateral inertia are applied as the governing equations of the plate. The response of lateral deflection is assumed with multiple modes of vibration including unknown time functions. Stress function related with nonlinear coupling of the deflection is derived to satisfy the compatibility equation. The stress function satisfies both equilibrium conditions of in-plane forces and in-plane moments of forces at the boundaries. Applying the Galerkin procedure, the equation is reduced to a set of nonlinear ordinary differential equations. Characteristics of restoring force, linear vibrations and nonlinear responses are calculated on the plate with the static deformation due to initial deflection, gravity and initial in-plane compressive displacement at the boundary. The characteristics of restoring force show the type of a softening-and-hardening spring with a negative gradient. Nonlinear periodic responses are calculated with the harmonic balance method. Non-periodic responses are integrated numerically with the Runge-Kutta-Gill method. Frequency-response curves are calculated with multiple modes of vibration. It is found that chaotic responses are generated in two specific frequency regions: lower and higher frequency ranges. The chaotic responses are inspected with the Fourier spectra, the Poincare projections and the maximum Lyapunov exponents. The chaotic responses involve the sub-harmonic resonance response of 1/2 order corresponding to the lowest mode of vibration. Increasing the in-plane asymmetric constraints at the boundaries, both frequency regions of the chaotic responses shift to the higher frequency ranges. Moreover, the lower frequency range of the chaotic responses becomes narrower, while the higher frequency range of the chaotic responses becomes wider by the in-plane asymmetric constraints.