Free vibration analysis of an elliptical plate with eccentric elliptical cut-outs

The elaborated collocation multipole method is employed to obtain a semi-analytical solution, involving proper products of angular and radial Mathieu functions, for the free flexural vibrations of a fully clamped thin elastic plate of elliptical planform containing multiple elliptical cutouts of arbitrary size, location, and orientation. The problem boundary conditions are satisfied by uniformly collocating points on the boundaries, and exactly calculating the normal derivative of plate displacement at the collocation points through use of appropriate directional derivative in each coordinate system. The multipole expansion is truncated to yield a coupled algebraic linear system of equations that is then solved for the nontrivial eigensolutions. Extensive numerical simulations present the first three calculated natural frequencies and the associated deformed mode shapes of an elliptical plate with elliptical/circular cutouts, for a wide range of plate/cutout aspect ratios, and cutout location/orientation parameters. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data available in the existing literature.