Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness  

作　　者：Jianghong YUAN Weiqiu CHEN 

机构地区：[1]Center for Mechanics and Materials and Key Laboratory of Applied Mechanics(AML),Department of Engineering Mechanics,Tsinghua University,Beijing 100084,China [2]State Key Laboratory of Traction Power,Southwest Jiaotong University,Chengdu 610031,China [3]School of Mechanics and Engineering,Southwest Jiaotong University,Chengdu 610031,China [4]Department of Engineering Mechanics,Zhejiang University,Hangzhou 310027,China [5]Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province,Zhejiang University,Hangzhou 310027,China

出　　处：《Applied Mathematics and Mechanics(English Edition)》2017年第4期505-526,共22页应用数学和力学（英文版）

基　　金：Project supported by the National Natural Science Foundation of China(Nos.11402133,11620162,11321202,and 11532001)

摘　　要：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 fiexural 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.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 fiexural 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.

关 键 词：free vibration circular Mindlin plate variable thickness inhomogeneous material Heun-type equation 

分 类 号：O327[理学—一般力学与力学基础]