Vibrations of Membranes and Plates with Cutouts

Abstract eng:
The research refers to free vibrations non-homogeneous (weakened with holes) isotropic and nonisotropic membranes and elastic thin plates. The purpose of the study is to examine the effect of shape, area, position, number and proportions of the holes, ratio of the membrane and plate sides and boundary conditions on free vibrations of membrane and plates. The plates are considered to be thin enough to apply 2D Kirchhoff-Love theory. Mathematically vibration problems for membranes and plates with cutouts are reduced to solution of the eivenvalue problems for nonsimply connected domains, which are solved in the research with analitic (when possiple), asymptotic and numerical methods. The most important and interesting is the effect of the area of a hole on natural frequencies and modes of free vibrations. It appeared that the natural frequencies may either increase or decrease with the hole area since the cutout affects both the stiffness and the mass of a solid. Special attention is devoted to frequencies those are doubled for homogeneous plates, for example, frequencies of the square plates with similar boundary conditions on all edges. Depending on the wave numbers frequencies may split (or may not) as the hole area increases. For small cutout area the asymptotic formula for natural frequencies has been obtained and the asymptotic results have been compared with the results of numerical analysis. The values of the natural frequencies appear to be very sensitive to proportion of the hole and membrane/plate. The change of the ratio may cause the switch of the vibration modes. References [1] A.L. Smirnov, S.N. Mikryukov, A.V. Lebedev, Numerical analysis of free vibrations and stability of cylindrical shell with cutouts. Review of applied and industrial mathematics 12(1), 162-163, 2005 (in Russian). [2] A.L. Smirnov, Vibrarions and stability of non-isotropic plates with cutouts. CD IV European Congress on Computational Mechanics (ECCM IV): Solids, Structures and Coupled Problems in Engineering, Paris (France), 2010.

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