Free Vibration Analysis of Thin Membranes Using EFGM

Vibration of thin membranes under tension is a topic of very high physical significance. One of its most common application is in a typical western percussion instrument where a uniform membrane under tension is used to produce sound. Some classical Indian percussion instruments use a variable density membrane to generate complex modes of vibrations. Present work is aimed at developing an efficient computer program based on Galerkin scheme, a particular case of Mesh Free Methods, to solve the basic wave propagation equation for the specific case of a thin rectangular membrane fixed along the perimeter. A MATLAB code is written to extract its first few modes of vibration. Starting from the basic wave equation, a weak form for the governing differential equation is formulated, followed by the formulation of discrete equations, as prescribed by Element Free Galerkin Method (EFGM). It is observed that the code can produce accurate mode shapes associated with the vibration. A convergence study is performed to determine the minimum number of nodes required to produce accurate results. It was observed that the results converged in a relatively low number of nodes in comparison to a traditional FEM technique. The error plots produced by EFGM method also proved that the results obtained from the meshfree method were much more reliable than the traditional FEM results. The work has future scope to be developed into a full-fledged 3-D capable tool to analyses complex geometries and composite cavities.