Exact Solutions for Nonaxisymmetric Vibrations of Radially Inhomogeneous Circular Mindlin Plates With Variable Thickness

Abstract

The nonaxisymmetric transverse free vibrations of radially inhomogeneous circular Mindlin plates with variable thickness are governed by three coupled differential equations with variable coefficients, which are quite difficult to solve analytically in general. In this paper, we discover that if the geometrical and material properties of the plates vary in generalized power form along the radial direction, then the complicated governing differential equations can be reduced into three uncoupled second-order ordinary differential equations which are very easy to solve analytically. Most strikingly, for a class of solid circular Mindlin plates with absolutely sharp edge, the natural frequencies can be expressed explicitly in terms of elementary functions, with the corresponding mode shapes given in terms of Jacobi polynomials. These analytical expressions can serve as benchmark solutions for various numerical methods.