The Legendre-tau method applied to the Kirchhoff thin plate equation

Abstract EN

The Kirchhoff thin plate equation is an important mathematical model in structural dynamics.

Because this equation has order four, analysis of the vibration eigenfrequencies and corresponding eigenfunctions is much more complicated than is the case for, say, a vibrating membrane.

For rectangular geometry the plate boundary value problem does not allow separation of the space variables.

Although some asymptotic methods (e.g., Bolotin’s Method and the Wave propagation Method of Chen and Zhou) for the approximation of the high frequency part of the spectrum exist, it appears that numerical methods are the best way to reliably obtain information on the lower end.

In this paper we employ the Iegendre-tau method of Lanczos, a spectral method with infinite order of accuracy when applied to problems with infinitely smooth solutions, to approxi-mate the spectrum and corresponding eigenfunctions of a vibrating rectangular thin plate subject to all standard combinations of energy-conserving and energy-dissipative boundary conditions.

We then apply Bolotin’s Method to several typical cases and give a comparison of the results.

The numerical results of this paper lead to some interesting questions which are currently being investigated.