Vibration Analysis of Arbitrarily Shaped Membranes

• Published in: Computer modeling in engineering & sciences 2009, Vol.51 (2), p.115-141
• Main Author: Reutskiy, S Y
• Format: Article
• Language: English
• Subjects: Boundary value problems; Conformal mapping; Eigenvalues; Finite difference method; Free vibration; Membranes; Resonant frequencies; Vibration analysis
• ISSN: 1526-1492; 1526-1506
• DOI: 10.3970/cmes.2009.051.115

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.