Natural frequencies of non-homogeneous membranes with arbitrary interfaces

CARLOS P. FILIPICH; MARTA B. ROSALES

Buenos Aires, Argentina

Congreso; VIII Congreso Argentino de Mecánica Computacional (MECOM 2005); 2005

AMCA, UADE

The study deals with the generalized solution of the title problem. The free vibration problem of a rectangular membrane with partial domains each of uniform density and arbitrary shape is tackled. Previous studies by other researches include straight parallel to the borders and oblique interfaces, and bent ones. The solution is found by means of a direct variational method with a complete space of trigonometric functions. In consequence the solution is convergent to the exact one, i.e. the eigenvalues are exact and the mode shapes have uniform convergence towards the exact ones. The approach is straightforward and very efficient from the computational viewpoint. Diverse illustrations are included. The cases of an oblique straight line interface and an open curve line which divide the membrane in two domains each of different density are first presented. Also a rectangular plate with an interior closed domain is stated and the numerical example of a rectangular interior zone is also included. The curve line and the interior zone cases have not been, to the authors? knowledge, dealt with yet. Since the well-known analogy between the frequency parameter of membranes and simply supported plates is usually employed to solve membranes, here it is shown that this analogy is not valid when the density is not uniform. Consequently, the natural frequencies of simply supported, rectangular non-homogeneous plates may not be derived from this solution and viceversa. As is known the differential problem of the vibrating membrane is governed by the Helmholtz equation. This also represents the eigenvalue problem of cavities.  Comparison of the results is made with frequencies found using a numerical solution found by solving the direct problem with power series allows an additional comparison and with other authors results. In all cases the results are excellent and the computational cost is very low.