Study of the Dynamic Behaviour of Circular Membranes with Low Tension

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Abstract
The dynamic behaviour of membranes has been widely studied by well-known authors for a long time. A clear distinction can be made between the behaviour of membranes without tension (plate case) and membranes subjected to large tension or pre-strain in their plane (membrane case). In classical theories, less attention has been paid to membranes subjected to a low level of tension, which solution is between both extreme cases. Recently, certain fields of research are demanding solutions for this intermediate behaviour. It is the case of membranes present in MEMS and sensor or the response of the tympanic membrane in mammals hearing system. In this paper, the behaviour of plates and circular membranes with boundary conditions clamped in the edges has been studied. The natural frequencies for both cases (plate and membrane) have been calculated using the solutions of the traditional theories and these have been compared with the numerical frequencies calculated by finite element analysis. The dynamic response of membrane with low tension, corresponding to a transition between these extreme behaviours, has also been calculated. A theoretical solution has been used complemented with a wide set of numerical finite elements calculations. The analytical and numerical solutions are very close, being the error made using both methods very low; nevertheless, there are no analytical solutions for the entire transition zone between the plate and membrane behaviour. Therefore, this range has been completed using finite element analysis. Broad ranges of geometric configurations have been studied. The transition behaviour of the membrane has been clearly identified. The main practical consequences of these results have been discussed, in particular focused on the response of the tympanic membrane.

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