Vibration analysis of irregular-shaped plates on simple supports

Abstract

In this work, we propose a general perturbative approach for modal analysis of irregular-shaped plates of uniform thickness with uniform boundary conditions. Given a plate of irregular boundary, first, a uniform circular plate of identical thickness and area, centred at the centroid, is determined. The irregular boundary is then treated as a perturbation with a suitable smallness parameter, and is expressed as a generalized Fourier series. The frequency parameter, shape function and boundary conditions are then perturbed in terms of the smallness parameter. The homogeneous zeroth-order equation corresponds to the circular plate, which is exactly solvable. We show that the inhomogeneous equations in the higher orders can also be solved exactly using a particular solution structure. We can then construct the exact perturbative solution up to any order. The proposed method is demonstrated through the modal analysis of simply supported super-circular plates. The results are validated using the numerical results obtained from ANSYS ® , which are an excellent match. Interestingly, the supposedly degenerate modes with an even number of nodal diameters of super-circular plates are found to split naturally.