Transverse Vibration Analysis of an Arbitrarily-Shaped Membrane by the Weak-Form Quadrature Element Method

The recently proposed weak-form quadrature element method (QEM) is applied to transverse vibration analysis of membranes. It differs considerably from the strong-form quadrature element method where the differential equations are tackled directly. The variational description of the governing Holmholtz equations is used to establish discrete equations. The membrane is partitioned into a few large quadrilateral quadature elements. In each element, the integrands involved in the variational description of the problem are approximated by high order interpolations at Gauss-Lobatto sampling points and the derivatives in the intergrands are approximated using differential quadrature analog at the same sampling points. Two examples are studied to demonstrate the present quadarture element method. In the first example, a benchmark problem is examined to validate the convergence of the present method. Computed results in the transverse vibration analysis of a square membrane are compared with the analytical solution. In the second example, transverse vibrations of circular membranes with an eccentric cutout are considered for various eccentricity ratios. Comparison with other available results is made and good agreement is reached. It is shown that the weak-form quadrature element method is highly efficient and applicable to vibration analysis of membranes of arbitrary shape.