Study of Forced Vibrations of a Two-Layer Plate Under Harmonic Load

Abstract

We propose a technique for the numerical study of the dynamic response under forced harmonic vibrations of a rectangular two-layer plate consisting of a main rigid layer and a low-rigid damping layer. The material of the rigid layer is considered to be isotropic and perfectly elastic. The dynamic deformation of the damping layer material is described by linear physical equations of a viscoelastic solid, which represent a generalization of the Kelvin-Voigt hypothesis for the case of a complex stress state. It is believed that a plate with a damping layer deforms according to the classical Kirchhoff-Love hypotheses. A rectangular two-layer finite element with twenty degrees of freedom has been developed to model the inertial, stiffness and damping properties of the plate layers. A system of differential motion equations of a finite element model of the plate under harmonic load is obtained. A solution of the resulting system is sought in the form of decomposition of nodal displacement vector of the finite-element model according to eigenforms, which leads to a system of equations of a significantly smaller order relative to the vector of modal coordinates of the plate. Numerical experiments were carried out to determine the dynamic response of a rectangular hingedly supported two-layer plate under the action of a surface harmonic load with a frequency varying from zero to 200 Hz for the cases with and without taking into account the damping properties of the viscoelastic layer material. The plate was divided into 144 elements (12 elements in the direction of each side). It is shown that taking into accont of the damping properties significantly limits the vibration amplitudes and dynamic stress of the plate only under vibrations in the resonance zone. At the same time, far from this zone the damping properties of the material can be practically ignored. It is noted that in the case of the uniform harmonic pressure applied over the whole area of the plate, the resonance oscillations are excited only at those frequencies which correspond to the eigenforms with an odd number of half-waves in the direction of each side of the plate. To determine the lower part of the spectrum of eigenforms and frequencies the subspace iteration method has been used.