On the rational derivation of a hierarchy of dynamic equations for a homogeneous, isotropic, elastic plate

Flexural equations of motion for a homogeneous, isotropic, elastic plate are derived by an antisymmetric expansion in the thickness coordinate of the displacement components. All but the lowest-order expansion functions are eliminated with the help of the three-dimensional equations of motion. and are plugged into the boundary conditions. Eliminating between these, an equation is obtained for the mean-plane vertical displacement which also includes arbitrary loading on the plate surface. This equation can be truncated to any order in the thickness and it is in particular noted that the corresponding dispersion relation seems to correspond to a power series expansion of the exact Rayleigh-lamb dispersion relation to all orders. Various truncations of the equation are discussed and are compared numerically with each other, the exact three-dimensional solution and Mindlin's plate equation. Both the dispersion relation and the corresponding displacement components as well as an excitation problem are used for the comparisons. The theories are reasonably close to each other and in order to be on the safe side none of them should in fact be used for frequencies above the cutoff of the second flexural mode. (C) 2001 Elsevier Science Ltd. All rights reserved.