Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T10:26:23.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Method of Fundamental Solutions for Plate Vibrations in Multiply Connected Domains

Published online by Cambridge University Press:  05 May 2011

C. C. Tsai ,
D. L. Young  and
C. M. Fan
C. C. Tsai*
Affiliation:
Department of Information Technology, Toko University, Chia-Yi County, Taiwan 61363, R.O.C.
D. L. Young*
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
C. M. Fan*
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
* Assistant Professor
**Professor
***Postdoctoral Fellow
Get access

Abstract

This paper develops the method of fundamental solutions (MFS) to solve eigenfrequencies of plate vibrations of multiply connected domains. The complex-valued MFS combined with the mix potential method are utilized in order to avoid the spurious eigenvalues. The benchmarked problems of annular plates with clamped, simply supported and free boundary conditions are studied analytically as well as numerically. Wherein the results demonstrate that all true eigenvalues are contained and no spurious eigenvalues are included. In the analytical studies, the continuous version of the MFS is utilized to obtain the eigenequation by applying the degenerate kernels and Fourier series. The proposed numerical method is free from singularities, meshes, and numerical integrations and thus can be easily utilized to solve plate vibrations free from spurious eigenvalues in multiply connected domains.

Keywords

Method of fundamental solutionsMix potential methodPlate vibrationSpurious eigenvaluesAnnular plateClampedSimply supportedFree boundary conditions
Type
Articles
Information
Journal of Mechanics , Volume 22 , Issue 3 , September 2006 , pp. 235 - 245
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Kupradze, V. D. and Aleksidze, M. A., “The Method of Functional Equations for the Approximate Solution of Certain Boundary Value Problem,” USSR Computational Mathematics and Mathematical Physics, 4, pp. 82–126 (1964).CrossRefGoogle Scholar
2.Fairweather, G. and Karageorghis, A., “The Method of Fundamental Solutions for Elliptic Boundary Value Problems,” Advances in Computational Mathematics, 9, pp. 69–95 (1998).CrossRefGoogle Scholar
3.Golberg, M. A. and Chen, C. S., “The Method of FundaMental Solutions for Potential, Helmholtz and Diffusion Problems,” Boundary Integral Methods: Numerical and Mathematical Aspects, Golberg, M. A., Ed., pp. 103–176 (1998).Google Scholar
4.Fairweather, G., Karageorghis, A. and Martin, P. A., “The Method of Fundamental Solutions for Scattering and Radiation Problems,” Engineering Analysis with Boundary Elements, 27, pp. 759–769 (2003).CrossRefGoogle Scholar
5.Cho, H. A., Golberg, M. A., Muleshkov, A. S. and Li, X., “Trefftz Methods for Time Dependent Partial Differential Equations,” CMC: Computers, Materials, and Continua, 1, pp. 1–37(2004).Google Scholar
6.Young, D. L., Tsai, C. C. and Fan, C. M., “Direct Approach to Solve Nonhomogeneous Diffusion Problems Using Fundamental Solutions and Dual Reciprocity Methods,” Journal of the Chinese Institute of Engineers, 27, pp. 597–609 (2004).CrossRefGoogle Scholar
7.Young, D. L., Tsai, C. C., Murugesan, K., Fan, C. M. and Chen, C. W., “Time-Dependent Fundamental Solutions for Homogeneous Diffusion Problems,” Engineering Analysis with Boundary Elements, 28, pp. 1463–1473 (2004).CrossRefGoogle Scholar
8.Hu, S. P., Fan, C. M., Chen, C. W. and Young, D. L., “Method of Fundamental Solutions for Stokes’ First and Second Problems,” Journal of Mechanics, 21, pp. 25–32 (2005).CrossRefGoogle Scholar
9.Tsai, C. C., Young, D. L., Chen, C. W. and Fan, C. M., “The Method of Fundamental Solutions for Eigenproblems in Domains with and Without Interior Holes,” Proceedings of the Royal Society of London, Series A, 462, pp. 1443–1466 (2006).CrossRefGoogle Scholar
10.Leissa, A. W., Laura, P. A. A. and Gutierrez, R. H., “Transverse Vibrations of Circular Plates Having Non-Uniform Edge Constraints,” Journal of Acoustical Society of American, 66, pp. 180–184 (1979).CrossRefGoogle Scholar
11.Kitahara, M., Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates, Elsevier, Amsterdam (1985).Google Scholar
12.Hutchinson, J. R., “Vibration of Plates,” Boundary ElementsX, Brebbia, C. A., Ed., Springer, Berlin (1988).Google Scholar
13.Chen, J. T., Lin, S. Y., Chen, K. H. and Chen, I. L., “Mathematical Analysis and Numerical Study of True and Spurious Eigenequations for Free Vibration of Plates Using Real-part BEM,” Computational Mechanics, 34, pp. 165–180 (2004).CrossRefGoogle Scholar
14.Kondapalli, P. S. and Shippy, D. J., “Analysis of Acoustic Scattering in Fluids and Solids by the Method of Fundamental Solutions,” Journal of Acoustical Society of American, 91, pp. 1844–1854 (1992).CrossRefGoogle Scholar
15.Kang, S. W. and Lee, J. M., “Application of Free Vibration Analysis of Membranes Using the Non-Dimensional Dynamic Influence Function,” Journal of Sound and Vibration, 234, pp. 455–470 (2000).CrossRefGoogle Scholar
16.Chen, J. T., Kou, S. R., Chen, K. H. and Cheng, Y. C., “Comments on ‘Vibration Analysis of Arbitrary Shaped Membranes Using the Non-Dimensional Dynamic Influence Function',” Journal of Sound and Vibration, 235, pp. 156–171 (2000).CrossRefGoogle Scholar
17.Chen, J. T., Chang, M. H., Chen, K. H. and Lin, S. R., “The Boundary Collocation Method with Meshless Concept for Acoustic Eigenanalysis of Two-Dimensional Cavities Using Radial Basis Function,” Journal of Sound and Vibration, 257, pp. 667–711 (2002).CrossRefGoogle Scholar
18.Chen, J. T., Chang, M. H., Chen, K. H. and Chen, I. L., “Boundary Collocation Method for Acoustic Eigenanalysis of Three-Dimensional Cavities Using Radial Basis Function,” Computational Mechanics, 29, pp. 392–408 (2002).CrossRefGoogle Scholar
19.Kang, S. W. and Lee, J. M., “Free Vibration Analysis of Arbitrary Shaped Plates with Clamped Edges Using Wave-Type Functions,” Journal of Sound and Vibration, 242, pp. 9–26(2001).CrossRefGoogle Scholar
20.Karageorghis, A., “The Method of Fundamental Solutions for the Calculation of the Eigenvalues of the Helmholtz Equation,” Applied Mathematics Letters, 14, pp. 837–842 (2001).CrossRefGoogle Scholar
21.Young, D. L., Hu, S. P., Chen, C. W., Fan, C. M. and Murugesan, K., “Analysis of Elliptical Waveguides by Method of Fundamental Solutions,” Microwave and Optical Technology Letters, 44, pp. 552–558 (2005).CrossRefGoogle Scholar
22.Chen, C. W., Fan, C. M., Young, D. L., Murugesan, K. and Tsai, C. C., “Eigenanalysis for Membranes with Stringers Using the Methods of Fundamental Solutions and Domain Decomposition,” CMES: Computer Modeling in Engineering and Sciences, 8, pp. 29–44 (2005).Google Scholar
23.Burton, A. J. and Miller, G. F., “The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary-Value Problems,” Proceeding of the Royal Society A, 323, pp. 201–210 (1971).CrossRefGoogle Scholar
24.Panich, O. I., “On the Question of the Solvability of the Exterior Boundary Problem for the Wave Equation and Maxwell's Equation,” Uspekhi Matematicheskikh Nauk, 20, pp. 221–226(1965).Google Scholar
25.Brakhage, H. and Werner, P., “Über das Dirichletsche Aussenraumproblem Für Die Helmholtzsche Schwingungsgleichung,” Archiv Der Mathematik, 16, pp. 325–329 (1965).CrossRefGoogle Scholar
26.Tai, G. R. G. and Shaw, R. P., “Helmholtz Equation Eigenvalues and Eigenmodes for Arbitrary Domains,” Journal of Acoustical Society of American, 56, pp. 796–804 (1974).CrossRefGoogle Scholar
27.Chen, J. T., Liu, L. W. and Hong, H. K., “Spurious and True Eigensolutions of Helmholtz BIEs and BEMs for a Multiply Connected Problem,” Proceeding of the Royal Society Series A, 459, pp. 1897–1924 (2003).Google Scholar