THE LEGENDRE-TAU METHOD APPLIED TO THE KIRCHHOFF THIN-PLATE EQUATION

The Kirchhoff thin plate equation is an important mathematical model i n 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 al low separation of the space variables. Although some asymptotic method s (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 Lege ndre-tau method of Lanczos, a spectral method with infinite order of a ccuracy when applied to problems with infinitely smooth solutions, to approximate the spectrum and corresponding eigenfunctions of a vibrati ng rectangular thin plate subject to all standard combinations of ener gy-conserving and energy-dissipative boundary conditions. We then appl y Bolotin's Method to several typical cases and give a comparison of t he results. The numerical results of this paper lead to some interesti ng questions which are currently being investigated.