矩形 本 值问题的 闭解 解法综述

The eigenvalue problems of rectangular plates, including the free vibration and eigenbuckling problems, have been attracting considerable interest of researchers. This paper reviews the available closed-form solution methods for the eigenvalue problems of rectangular plates, which are the Navier, Levy, Separation-of-Variable (SOV) and Kantorovich-Krylov methods. For each method, the basic idea, development history and application scopes are introduced first, and the free vibration problem of a rectangular first-order shear deformation plate is taken as the example to illustrate the solution procedures of each method. Especially, this paper focuses on various SOV methods developed in recent 20 years, including the direct, variational, iterative, improved and extended SOV methods. Finally, all the reviewed methods are summarized and compared from various perspectives with the help of numerical result. For Levy-type of plates, all methods can provide the exact solutions. For plates with other homogeneous boundary conditions, both Kantorovich-Krylov method and SOV methods can produce highly-accurate solutions.