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.

1.
Khurasia
,
H. B.
, and
Rawtani
,
S.
, 1978, “
Vibration Analysis of Circular Plates With Eccentric Hole
,”
ASME J. Appl. Mech.
0021-8936,
45
, pp.
215
217
.
2.
Tseng
,
J. G.
, and
Wickert
,
J. A.
, 1994, “
Vibration of an Eccentrically Clamped Annular Plate
,”
ASME J. Appl. Mech.
0021-8936,
116
, pp.
155
160
.
3.
Leissa
,
A. W.
, and
Narita
,
Y.
, 1980, “
Natural Frequencies of Simply Supported Circular Plates
,”
J. Sound Vib.
0022-460X,
70
, pp.
221
229
.
4.
Vogel
,
S. M.
, and
Skinner
,
D. W.
, 1965, “
Natural Frequencies of Transversely Vibrating Uniform Annular Plates
,”
ASME J. Appl. Mech.
0021-8936,
32
, pp.
926
931
.
5.
Vega
,
D. A.
,
Vera
,
S. A.
,
Sanchez
,
M. D.
, and
Laura
,
P. A. A.
, 1998, “
Transverse Vibrations of Circular, Annular Plates With a Free Inner Boundary
,”
J. Acoust. Soc. Am.
0001-4966,
103
, pp.
1225
1226
.
6.
Vera
,
S. A.
,
Sanchez
,
M. D.
,
Laura
,
P. A. A.
, and
Vega
,
D. A.
, 1998, “
Transverse Vibrations of Circular, Annular Plates With Several Combinations of Boundary Conditions
,”
J. Sound Vib.
0022-460X,
213
(
4
), pp.
757
762
.
7.
Vera
,
S. A.
,
Laura
,
P. A. A.
, and
Vega
,
D. A.
, 1999, “
Transverse Vibrations of a Free-Free Circular Annular Plate
,”
J. Sound Vib.
0022-460X,
224
(
2
), pp.
379
383
.
8.
Cheng
,
L.
,
Li
,
Y. Y.
, and
Yam
,
L. H.
, 2003, “
Vibration Analysis of Annular-Like Plates
,”
J. Sound Vib.
0022-460X,
262
, pp.
1153
1170
.
9.
Laura
,
P. A. A.
,
Masia
,
U.
, and
Avalos
,
D. R.
, 2006, “
Small Amplitude, Transverse Vibrations of Circular Plates Elastically Restrained Against Rotation With an Eccentric Circular Perforation With a Free Edge
,”
J. Sound Vib.
0022-460X,
292
, pp.
1004
1010
.
10.
Lee
,
W. M.
,
Chen
,
J. T.
, and
Lee
,
Y. T.
, 2007, “
Free Vibration Analysis of Circular Plates With Multiple Circular Holes Using Indirect BIEMs
,”
J. Sound Vib.
0022-460X,
304
, pp.
811
830
.
11.
Providakis
,
C. P.
, and
Beskos
,
D. E.
, 1999, “
Dynamic Analysis of Plates by Boundary Elements
,”
Appl. Mech. Rev.
0003-6900,
52
(
7
), pp.
213
236
.
12.
Kitahara
,
M.
, 1985,
Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates
,
Elsevier
,
Amsterdam
.
13.
IMSL
, 1999, Math/Library Volumes 1 and 2, Version 4.01, Visual Numerics, Inc.
14.
ABAQUS 6.5, 2004, Hibbitt, Karlsson and Sorensen, Inc., RI.
15.
Nagaya
,
K.
, and
Poltorak
,
K.
, 1989, “
Method for Solving Eigenvalue Problems of the Helmholtz Equation With a Circular Outer and a Number of Eccentric Circular Inner Boundaries
,”
J. Acoust. Soc. Am.
0001-4966,
85
, pp.
576
581
.
16.
Watson
,
G. N.
, 1995,
A Treatise on the Theory of Bessel Functions
, 2nd ed.,
Cambridge Library
,
Cambridge
.
17.
Martin
,
P. A.
, 2006,
Multiple Scattering Interaction of Time-Harmonic Wave With N Obstacles
,
Cambridge University Press
,
Cambridge
.
18.
Chandra
,
A.
,
Huang
,
Y.
,
Wei
,
X.
, and
Hu
,
K. X.
, 1995, “
A Hybrid Micro-Macro BEM Formulation for Micro-Crack Clusters in Elastic Components
,”
Int. J. Numer. Methods Eng.
0029-5981,
38
(
7
), pp.
1215
1236
.
19.
Lee
,
W. M.
, and
Chen
,
J. T.
, 2008, “
Scattering of Flexural Wave in Thin Plate With Multiple Holes by Using the Null-Field Integral Equation Approach
,”
Comput. Model. Eng. Sci.
1526-1492,
1124
(
1
), pp.
1
30
.