Abstract
Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and flexural rigidity with the consideration of the effect of Poisson’s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.
Highlights
Circular plates are one of the most important structural elements widely used in civil, mechanical, aeronautical, and electronic engineering[1,2,3,4,5]
We extend our recent work on the axially inhomogeneous beam structures with variable cross sections[25], and propose an exact analytical method to solve the axisymmetric flexural free vibrations of the radially inhomogeneous circular Mindlin plates with variable thickness
We concentrate on the axisymmetric flexural free vibrations of the radially inhomogeneous circular Mindlin plates with non-uniform thickness, and propose an exact analytical method to reduce the two coupled governing differential equations with variable coefficients into a pair of uncoupled Sturm-Liouville equations under two restrictive conditions on the geometrical/material parameters of the Mindlin plates
Summary
Circular plates are one of the most important structural elements widely used in civil, mechanical, aeronautical, and electronic engineering[1,2,3,4,5]. The dynamical modal analysis for the inhomogeneous circular plates with non-uniform thickness has attracted much interest up to now, and many analytical solutions are available based on the classical Kirchhoff assumption. Xiang and Zhang[24] solved analytically the transverse free vibration of the circular Mindlin plates with multiple stepwise thickness variations. We extend our recent work on the axially inhomogeneous beam structures with variable cross sections[25], and propose an exact analytical method to solve the axisymmetric flexural free vibrations of the radially inhomogeneous circular Mindlin plates with variable thickness.
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