Analytical and experimental studies on dynamic instability of simply supported rectangular plates with arbitrary concentrated masses

Abstract

This paper presents innovative analytical and experimental investigations of transverse dynamic instability of a simply supported rectangular plate attached with arbitrary concentrated masses owing to parametric resonance excited by an in-plane uniformly distributed periodic load along two opposite edges, which has not been reported in the literature. Based on the von-Kármán large deflection theory of thin plate, differential equations of transverse motion are established by using the Galerkin method. The analytical results of dynamic instability regions and nonlinear response curves of the plate having arbitrary concentrated masses for various instability modes are obtained. Tests investigating the influence of concentrated mass on the out-of-plane dynamic instability of various modes of a plate under the in-plane periodical loading presented in the paper are first time reported in the literature. Independent swept frequency tests are carried out to investigate instability frequency-amplitude regions for transverse dynamic instability of the plate and the test results agree well with the theoretical counterparts. It is shown that concentrated masses significantly influence out-of-plane dynamic instability of the plate. The investigation leads to the following novel findings: (1) When the concentrated masses are located at positions corresponding to non-zero modal displacements, the masses increase the rate of energy dissipation and damping ratio of the plate, leading to an increase of the critical excitation amplitude for dynamic instability of the plate, but to a decrease of the natural frequencies of the plate and the critical excitation frequencies for dynamic instability of the plate. (2) Under the same excitation amplitude, the excitation frequencies and the widths of excitation frequency region for dynamic instability of the plate as well as excitation frequency intervals for nonlinear dynamic instability of the plate decrease with an increase of the concentrated masses, which are located at positions corresponding to non-zero modal displacements. (3) When the concentrated masses are located at positions corresponding to zero modal displacements, the concentrated masses almost do not affect the critical excitation frequency and the widths of excitation frequency region for dynamic instability of the plate as well as excitation frequency intervals for nonlinear dynamic instability of the plate. (4) The widths of excitation frequency regions for dynamic instability of the plate as well as excitation frequency intervals for nonlinear dynamic instability of the plate increases when masses move away from the position of the largest modal displacement of the plate.