Abstract
This paper presents analytical results on chaotic vibrations of a rectangular shell-panel under in-plane asymmetric constraints. The shell-panel with initial deflections is simply supported at all edges and also subjected to gravitational and periodic acceleration laterally. The boundary of the shell-panel is attached with an elastic material represented with distributed linear-springs in the in-plane directions. The outer sides of the springs are constrained by both uniform and asymmetric in-plane displacements. Neglecting the effect of inertia force along in-plane direction, the Donnell type equations modified with lateral inertia are applied as the governing equations of the shell-panel. 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, equation of motion is reduced to a set of nonlinear ordinary differential equations. Characteristics of restoring force, linear vibrations and nonlinear responses are calculated on the shell-panel. The characteristics of restoring force show the type of a sofitening-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. It is found that chaotic responses are generated in specific frequency regions. 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 1/3 order corresponding to the lowest mode of vibration. Increasing the in-plane asymmetric constraints at the boundaries, contribution ratio of the mode of vibration that has a nodal line perpendicularly to the asymmetric constraints increases on the chaotic response.
