Natural frequencies of rectangular membranes with partial intermediate supports

Abstract

The natural vibrations of rectangular membranes with partial intermediate supports are solved by a direct variational method known as whole element method (WEM). It is based on the use of extended trigonometrical series of uniform convergence. Fortunately, for the case of membranes supported on the perimeter, which is the case that interests us, the simplest series that we will use is reduced, in the unitary domain, to a Fourier series of sines in both coordinate axes. The characteristic that the supports are internal and partial (instead of complete) in the membrane, gives the work one of its conditions of singularity. To the authors’ knowledge, the analysis of the aforementioned case is not reported elsewhere in the literature. The proposed methodology guarantees that the frequencies found are only those related to the problem, eliminating spurious frequencies. It is demonstrated how, depending on the characteristic algorithm, it is possible to identify in an unmistakable way, spurious parameters that result when adopting this approach. It is proved that, in general, the frequency parameter of polygonal membranes does not match the square root of the parameter for frequency simple supported plates of the same shape. Evidently, this is due to the addition of intermediate supports. As has been known for the last century, without the presence of the analogy of the quadratic ratio between corresponding parameters is verified.