Exact free vibration analysis for membrane assemblies with general classical boundary conditions

Abstract

This paper proposes exact dynamic stiffness formulations for membranes and their assemblies under any arbitrary classical boundary conditions. First, by taking exact solutions in one direction satisfying all possible opposite edge supports, we can derive exact general solutions of the Helmholtz equation for membrane vibration. Then, generic force and displacement boundary conditions in the other direction are expressed in terms of the general solutions. Finally, the dynamic stiffness matrices of rectangular membrane elements are formulated, which can be assembled directly and allows applications of arbitrary boundary conditions. As an accurate and efficient modal solution technique, the Wittrick-Williams (WW) algorithm is applied onto the global dynamic stiffness matrix of the final structure. The most important issue of the WW algorithm, J0 count, has been resolved with the analytical expressions derived for all possible cases. The proposed dynamic stiffness method (DSM) is then applied to several examples including individual membranes and their assemblies. High accuracy and exactness of the proposed method within the whole frequency range is demonstrated by comparison with the finite element method. Besides, interesting findings have been observed on repeated eigenvalues with distinct mode shapes corresponding to certain aspect ratio and tension ratio, where physical and mathematical understanding has been provided.