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Research Papers

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

[+] Author and Article Information
Jianghong Yuan

School of Mechanics and Engineering,
Southwest Jiaotong University,
Chengdu, Sichuan 610031, China;
AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China

Xiaona Wei

School of Astronautics,
Beihang University,
Beijing 100191, China

Yin Huang

AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China
e-mail: huangyin_thu@163.com

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 30, 2016; final manuscript received May 4, 2017; published online May 19, 2017. Assoc. Editor: George Kardomateas.

J. Appl. Mech 84(7), 071003 (May 19, 2017) (9 pages) Paper No: JAM-16-1632; doi: 10.1115/1.4036696 History: Received December 30, 2016; Revised May 04, 2017

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.

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Figures

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Fig. 1

Geometry and coordinate system for a circular plate with radial inhomogeneity and nonuniformity

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Fig. 2

The mode shapes in terms of W associated with Ωi (i=1,2,3,4,5) for the parameter combinations: (a) p=−0.4, n=1 and (b) p=−0.4, n=2

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Fig. 3

The influence of parameter p on the mode shape associated with Ω1 for n=1, 2

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Fig. 4

The inhomogeneous and nonuniform circular plate is approximated by a stepped plate with K steps. Each step is considered as an annular segment (or a solid circular segment for the first step) with equal width.

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Fig. 5

Dimensionless natural frequencies of the solid circular plate with an absolutely sharp edge for the different plate models

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