Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs

In this paper, a semi-analytical approach is proposed to solve natural frequencies and natural modes for circular plates with multiple circular holes by using the indirect formulation in conjunction with degenerate kernels and Fourier series. All the kernels in the indirect formulation are expanded into degenerate form. By uniformly collocating points on the boundary, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through singular value decomposition (SVD). After determining the unknown Fourier coefficients, the corresponding mode shape is obtained by using the indirect boundary integral formulations. The results of the annular plate, as a special case, are compared with the analytical solution to verify the validity of the present method. For the cases of circular plates with multiple circular holes, the results are also compared with those of finite element method (FEM) using ABAQUS. Besides, the effect of eccentricity of the hole on the natural frequencies is also considered. Good accuracy, high rate of convergence and computational efficiency are the main features of the present method due to the semianalytical procedure. r 2007 Elsevier Ltd. All rights reserved.

[1]  R. Kanwal.  Linear Integral Equations , 1925, Nature.

[2]  Rainer Kress,et al.  On the numerical solution of a hypersingular integral equation in scattering theory , 1995 .

[3]  Jeng-Tzong Chen,et al.  Null-Field Integral Equation Approach for Plate Problems With Circular Boundaries , 2006 .

[4]  P.A.A. Laura,et al.  Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge , 2006 .

[5]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[6]  K. Nagaya,et al.  Method for solving eigenvalue problems of the Helmholtz equation with a circular outer and a number of eccentric circular inner boundaries , 1989 .

[7]  Li Cheng,et al.  Vibration analysis of annular-like plates , 2003 .

[8]  S. Rawtani,et al.  Vibration Analysis of Circular Plates With Eccentric Hole , 1978 .

[9]  Jeng-Tzong Chen,et al.  Analysis of Circular Torsion Bar with Circular Holes Using Null-field Approach , 2006 .

[10]  P.A.A. Laura,et al.  TRANSVERSE VIBRATIONS OF A FREE-FREE CIRCULAR ANNULAR PLATE , 1999 .

[11]  Jeng-Tzong Chen,et al.  Mathematical analysis and treatment for the true and spurious eigenequations of circular plates by the meshless method using radial basis function , 2004 .

[12]  An-Chien Wu,et al.  Null-field Integral Equations for Stress Field around Circular Holes under Antiplane Shear , 2006 .

[13]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[14]  J. Sládek,et al.  Regularization Techniques Applied to Boundary Element Methods , 1994 .

[15]  Principal Investigator,et al.  BOUNDARY INTEGRAL EQUATION METHOD , 1997 .

[16]  P.A.A. Laura,et al.  TRANSVERSE VIBRATIONS OF CIRCULAR, ANNULAR PLATES WITH SEVERAL COMBINATIONS OF BOUNDARY CONDITIONS , 1998 .

[17]  E. Ventsel.  AN INDIRECT BOUNDARY ELEMENT METHOD FOR PLATE BENDING ANALYSIS , 1997 .

[18]  Ying-Te Lee,et al.  Mathematical analysis and numerical study to free vibrations of annular plates using BIEM and BEM , 2006 .

[19]  J. R. Hutchinson.  Analysis of Plates and Shells by Boundary Collocation , 1991 .

[20]  Dynamic Analysis of Plates by Boundary Elements , 1999 .

[21]  H. Hong,et al.  Review of Dual Boundary Element Methods With Emphasis on Hypersingular Integrals and Divergent Series , 1999 .

[22]  Yoshihiro Narita,et al.  Natural frequencies of simply supported circular plates , 1980 .

[23]  S. M. Vogel,et al.  Natural Frequencies of Transversely Vibrating Uniform Annular Plates , 1965 .

[24]  H. Saunders,et al.  Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates , 1985 .

[25]  D. A. Vega,et al.  Transverse vibrations of circular, annular plates with a free inner boundary , 1998 .

[26]  Snehashish Chakraverty,et al.  Vibration of Plates , 2008 .