@Article{cmes.2009.051.115,
AUTHOR = {S.Yu. Reutskiy},
TITLE = {Vibration Analysis of Arbitrarily Shaped Membranes},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {51},
YEAR = {2009},
NUMBER = {2},
PAGES = {115--142},
URL = {http://www.techscience.com/CMES/v51n2/26755},
ISSN = {1526-1506},
ABSTRACT = {In this paper a new numerical technique for problems of free vibrations of arbitrary shaped non-homogeneous membranes:∇<sup>2</sup>w + k<sup>2</sup>q(x)w = 0, x∈ Ω⊂R<sup>2</sup>, 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 <i>k</i>. 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.},
DOI = {10.3970/cmes.2009.051.115}
}