Unsteady vibrations of a three-layer plate with an asymmetric structure

Abstract

This article discusses the problem of symmetric vibrations of a three-layer plate pivotally supported along the edges in the longitudinal direction is solved. The materials of the plate layers are assumed to be elastic and isotropic. It is believed that the plate is not symmetrical in thickness, i.e. different thicknesses of layers. A brief review of scientific papers on theories of calculation of three-layer and multilayer plates is given. The equations of vibration of a three-layer plate, previously developed by the authors, are accepted as resolving equations. The stress and displacement components are expressed, as well as the oscillation equations, through the main parts of the longitudinal and transverse displacements of the points of the “intermediate” plane of the middle layer, which is at some distance from the middle plane of the plate. The conditions of the articulated plumage are formulated with respect to the displacements of the points of the middle layer. The initial conditions are assumed to be zero. The formulated problem is solved by an analytical-numerical method using a software package. The obtained numerical results are presented in the form of graphs of the dependences on time and the longitudinal coordinate of the displacements and voltages of the points, both the middle and the bearing layers of the plate. The calculations were performed for steel materials of the bearing layers, and for the middle layer the physic-mechanical parameters of the polymer material were adopted. The corresponding conclusions are made according to the results of numerical calculations. It was established that slight transverse displacements of the plate points appear due to the action of longitudinal external loads. Moreover, due to the insignificance of these displacements with symmetric vibrations of a three-layer plate, transverse displacements can be neglected. With symmetrical vibrations of the plate, longitudinal normal voltages are also excited despite the fact that the edges of the plate are free of external loads. These voltages are generated by the action of external tangential and normal voltages on the boundary surfaces.