New analytical solutions for vibration problem of thick plates

Abstract

Exact solutions for the vibrations and stability problems in the mechanics of solids are sufficiently rare. For rectangular thick plates, exact solutions in the form of trigonometric series were constructed only for the case when all or two opposite sides of the plate are simply-supported. The discussion about the possibility of constructing exact solutions continues to this day. As a rule, an approximate solution is constructed in the analytical form on the basis of a variational approach. It should be noted, that as frequency increases, the number of basic functions involved in the solution also has to be increased, consequently the solutions of such type are inefficient for describing a structural element within the framework of such methods as the Continuous Element method, the Spectral Element method and the Dynamic Stiffness method. In the present research, for the first time, the analytical solutions for the vibration problem of thick orthotropic plates are obtained. The modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in a series, permitting the derivation of an infinite system of linear algebraic equations. Cases of practically important boundary conditions of completely free sides and fully clamped sides are considered. It should be noted that the presented analytical solutions for FFFF and CCCC boundary allows to completely describe the structural element in the form of a plate by means of a dynamic stiffness matrix and their use for modeling more complex structures, also in the framework of methods such as Continuous Element method, Spectral Element method and Dynamic Stiffness method. The obtained results could also be applied in projecting constructions, in developing new devices and in the optimization of their parametersin projecting of constructions, in developing new devices and in the optimization of their parameters.