Free vibrations of circular and sectorial plates by using the shape functions with trigonometric and hyperbolic terms

In this paper, free vibration of different plates in the polar coordinate is studied by using the Rayleigh–Ritz method and calculating strain and kinetic energies. The investigated plates include annular sectorial and circular, sectorial and circular with fixed center, circular sectorial, and solid circular. In this work, trigonometric and hyperbolic terms are used to express the shape function of each plate so that the shape function approximately satisfies boundary conditions and also regularity conditions for the solid plates. The mentioned plates can have any different types of the boundary conditions and the presented method is versatile and has no limitations. For the solid plates, the shape function is calculated in such a way that it can be used to calculate strain energy of the plate. Some types of the plates are examined as the case studies and the results are also calculated by the ABAQUS CAE for validation of the proposed method.