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Vibration Analysis of Arbitrarily Shaped Membranes

S.Yu. Reutskiy1
Science and Technology Center of Magnetism of Technical Objects. The National Academy of Science of Ukraine, Industrialnaya St.,19, 61106, Kharkov, Ukraine

Computer Modeling in Engineering & Sciences 2009, 51(2), 115-142.


In this paper a new numerical technique for problems of free vibrations of arbitrary shaped non-homogeneous membranes:∇2w + k2q(x)w = 0, x∈ Ω⊂R2, B[w] = 0, x∈∂Ω is presented. Homogeneous membranes of a complex form are considered as a particular case. The method is based on mathematically modeling of physical response of a system to excitation over a range of frequencies. The response amplitudes are then used to determine the resonant frequencies. Applying the method, one gets a sequence of boundary value problems (BVPs) depending on the spectral parameter k. The eigenvalues are sought as positions of the maxima of some norm of the solution. In the particular case of a homogeneous membrane the method of fundamental solutions (MFS) is proposed as an effective solver of such BVPs in domains of a complex geometry. For non-homogeneous membranes the combination of the finite difference method and conformal mapping is used as a solver of the BVPs. The results of the numerical experiments justifying the method are presented.


Free vibration, eigenvalue problem, membrane, irregular domain, non-homogeneous membrane, conformal mapping, nonlinear eigenvalue problem.

Cite This Article

Reutskiy, S. (2009). Vibration Analysis of Arbitrarily Shaped Membranes. CMES-Computer Modeling in Engineering & Sciences, 51(2), 115–142.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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