The modified weighted residual formulation in the wave based method for plate bending problems: A general formulation for different types of edge restraints

Wave based method is an efficient numerical technique that can be used to solve the structural dynamic problems, especially the vibration of plates. The approach expands the dynamic field variables to a weighted sum of wave functions, and the contribution factors are determined by minimizing the errors on the boundary conditions. While the governing equation is satisfied a priori, the boundary conditions are satisfied in the integral sense, using preferably a weighted residual formulation. However, the weighted residual formulation is not always suitable for all kinds of boundary conditions. In plate bending problems, the plate edges can be modelled as clamped, simply supported, free, symmetric or (visco) elastically restrained against translation or/and rotation, depending on how they are restrained in reality. But the currently available weighted residual formulations are suitable either only for the non-elastic restraints (clamped, simply supported, free and symmetric) or only for the general (visco) elastic restraints with both translational and rotational elasticities. In this paper, a general formulation is proposed for all the aforementioned types of edge restraints. It is equivalent to the conventional one when the plate edges are non-elastically restrained, but performs better with the introduction of corner residuals for the plate with general elastically restrained edges. Apart from the ideal non-elastic and the general full-elastic restraints, this modified formulation also works well for the partial-elastic ones with only either translational or rotational elasticity. Numerical cases are studied on a rectangular plate for all the restraint types, where the proposed formulation is validated. The restraint types can be identical or different among the plate edges, and how to apply the formulation to the non-uniformed situation is also demonstrated. Throughout the considered cases, this paper shows the accuracy and efficiency of the wave based method by comparing with the finite element method in computing the forced vibration of the thin plate. With the modified weighted residual formulation, the method further shows its great convenience in changing the restraint types or stiffness (and damping) parameters of plate edges.