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Research Papers

Free Vibration Analysis of a Circular Plate With Multiple Circular Holes by Using Indirect BIEM and Addition Theorem

[+] Author and Article Information
W. M. Lee

Department of Mechanical Engineering, China University of Science and Technology, Taipei, 11581, Taiwanwmlee@cc.cust.edu.tw

J. T. Chen1

Department of Harbor and River Engineering, Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwanjtchen@mail.ntou.edu.tw

1

Corresponding author.

J. Appl. Mech 78(1), 011015 (Oct 22, 2010) (10 pages) doi:10.1115/1.4001993 History: Received April 28, 2009; Revised March 13, 2010; Posted June 16, 2010; Published October 22, 2010; Online October 22, 2010

In this paper, natural frequencies and natural modes of a circular plate with multiple circular holes are theoretically derived and numerically determined by using the indirect boundary integral formulation, the addition theorem, and the complex Fourier series. Owing to the addition theorem, all kernel functions are expanded into degenerate forms and further expressed in the same polar coordinates centered at one circle where the boundary conditions are specified. Not only the computation of the principal value is avoided but also the calculation of higher-order derivatives can be easily determined. By matching boundary conditions, a coupled infinite system of linear algebraic equations is derived as an analytical model for the free vibration of a circular plate with multiple circular holes. The direct-searching approach is utilized in the truncated finite system to determine the natural frequency through singular value decomposition. After determining the unknown Fourier coefficients, the corresponding mode shapes are obtained by using the indirect boundary integral formulations. Some numerical eigensolutions are presented and then utilized to explain some physical phenomenon such as the beating and the dynamic stress concentration. Good accuracy and fast rate of convergence are the main features of the present method, thanks to the analytical approach.

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Figures

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Figure 3

Notation of Graf's addition theorem for Bessel functions

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Figure 4

A circular plate with an eccentric hole subject to clamped-free boundary conditions

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Figure 5

Natural frequency parameter versus the number of terms of Fourier series for a circular plate with an eccentric hole (a=1.0, b=0.25, e/a=0.45)

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Figure 6

The minimum singular value versus the frequency parameter for a circular plate with one eccentric hole (a=1.0, b=0.25, e=0.45)

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Figure 7

The lower six natural frequency parameters and modes of a circular plate with an eccentric hole (a=1.0, b=0.25, e=0.45)

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Figure 8

Contour of residual DMSs around the hole with eccentricity e from 0 to 0.6: (a) the present method and (b) the finite element method (8)

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Figure 9

Effect of the eccentricity e on the natural frequencies for the free-clamped annular-like plate: (a) the present method and (b) the global discretization method (2)

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Figure 10

A circular plate with two holes subject to clamped-free boundary conditions

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Figure 11

Natural frequency parameter versus the number of terms of Fourier series for a circular plate with two holes (a=1.0, b=0.25, c=0.15)

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Figure 12

The minimum singular value versus the frequency parameter for a circular plate with two holes (a=1.0, b=0.25, c=0.15)

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Figure 13

The lower five natural frequency parameters and modes of a circular plate with two holes (a=1.0, b=0.25, c=0.15)

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Figure 14

The first natural frequency parameters and modes of a circular plate with two holes: (a) L/a=2.1 and (b) L/a=4.0

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Figure 1

Problem statement for an eigenproblem of a circular plate with multiple circular holes

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Figure 2

Degenerate kernel for U(s,x)

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