The equivalence between the wave propagation method and Bolotin's method in the asymptotic estimation of eigenfrequencies of a rectangular plate

Abstract

The Kirchhoff thin plate equation is an important mathematical model in structural dynamics. Because this partial differential equation has order four, analysis of vibration eigenfrequencies is much more complicated than that of a vibrating membrane, whose governing PDE has order two. As a matter of fact, for the simplest rectangular geometry the plate boundary value problems do not even allow the separation of the two space variables, thus one must resort to alternative approaches. Bolotin's method, published in 1961, offers such an asymptotic alternative. More recently, the three authors generalized Keller-Rubinow's wave method to the thin plate equation, constituting a different approach. In this paper, we establish that for various combinations of boundary conditions (clamped, hinged, roller-supported and free), after some conversion the wave method and Bolotin's method yield an identical set of transcendental equations for asymptotic values of the eigenfrequencies of a rectangular plate, thus confirming the equivalence of these two methods.