Accurate variational approach for free vibration of variable thickness thin and thick plates with edges elastically restrained against translation and rotation

An efficient and accurate variational formulation is developed to study the vibration problem of variable thickness thin and thick plates with edges elastically restrained against both rotation and translation. The weak formulation with Ritz method is first employed to reduce the governing partial differential equations of motion of the plate to a system of ordinary differential equations. An analog procedure is then used to incorporate the natural boundary conditions. Although the proposed procedure requires some mathematical manipulations to derive the required boundary analog equations, it can produce lower upper bound solutions compared to the conventional Ritz method where the geometric boundary conditions can only be satisfied. The fast convergence and high accuracy of the proposed method are validated through convergence and comparison studies. Accurate solutions are achieved via few Ritz terms for all the cases considered.

► Free vibration of variable thickness thin and thick plates with elastic edges is studied.
► An accurate variational formulation is developed.
► The method uses a procedure to exactly implement the natural boundary conditions.
► The method can produce lower upper bound solutions than the Ritz method.