Free vibration of stepped thickness rectangular plates using spectral finite element method

Free vibration of stepped thickness rectangular plates is investigated in this paper using the spectral finite element method (SFEM). It is impossible to obtain closed-form solutions for a uniform rectangular plate with arbitrary boundary conditions except the Levy-type plates, i.e., at least simply-supported at two opposite edges. Moreover, the variation of plate thickness adds complexity to the problem. The Kantorovich method is employed to obtain an analytical approximation solution form. Then, a plate spectral finite element is developed accordingly in the frequency domain. Similar to the conventional finite element method (CFEM), after assembling elements and applying boundary conditions, each modal frequency and associated mode shape function can be determined by iteratively solving corresponding equations in plate x and y directions. Available literature results along with NASTRAN simulations are used to validate our SFEM predictions of stepped thickness plates with various boundary conditions. Compared to the CFEM, only a fraction of the mesh is needed to achieve comparable accuracy in each mode. Substantial computation cost can then be saved. Mode shape functions are extracted and presented in a semi-analytical form. Physical insights of wave propagation characteristics in a stepped thickness rectangular plate can be collected from these results. In summary, an efficient and accurate SFEM is developed to conduct free vibration analysis of stepped thickness rectangular plates with various boundary conditions.