A perturbation approach, coupled with the adjoint concept, is used to derive an analytic expression for the natural frequencies of a nearly rectangular membrane. The method is applied for a rectangular membrane with a semicircle at one of the boundaries. The fundamental natural frequency results for this configuration are presented and compared with results from a finite-element method and results from an approximate Galerkin method. The agreement between the fundamental natural frequencies calculated with the perturbation approach and those calculated with the finite-element method improves as the radius of the semicircle decreases and as the semicircle location becomes more eccentric.
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2.
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5.
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Laura, P.A.A., Avalos, D.R., and Carnicer, R.S., 1986, "Fundamental frequency of rectangular membranes with a boundary perturbation," Applied Acoustics19, 309-313.
7.
Meinke, H.H., Lange, K.P., and Rurger, J.F., 1963, "TE and TM waves in waveguides of very general cross-section ," Proceedings of the Institution of Electrical and Electronic Engineers 51, 1436-1443.
8.
Nayfeh, A.H., 1981, Introduction to Perturbation Techniques, Wiley, New York.
9.
Nayfeh, A.H., Mook, D.T., Lobitz, D.W., and Sridhar, S., 1976, "Vibrations of nearly annular and circular plates ," Journal of Sound and Vibration47, 75-84.
10.
Steele, C.W., 1968, "Numerical computation of electric and magnetic field in a uniform waveguide of arbitrary cross section." Journal of Computational Physics3, 148-153.
11.
Yee, H.Y., 1965, "On determination of cut-off frequencies of waveguides with arbitrary cross section," Proceedings of the Institution of Electrical and Electronic Engineers54, 64.
