弹性地基上四边自由矩形薄板的解析解

将弹性地基用Winkler模型来代替，并首先把弹性地基上薄板弯曲问题的控制方程表示成为Hamilton正则方程，然后利用辛几何方法对全状态相变量进行分离变量，求出其本征值后，再按本征函数展开的方法求出弹性地基上四边自由矩形薄板的解析解．由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性地基上薄板弯曲的基本方程出发，直接利用数学的方法求出可以满足四边自由边界条件的解析解，使得问题的求解更加理论化．还给出了计算实例来验证所采用的方法以及所推导出的公式的正确性．

Theoretic Solution of Rectangular Thin Plate on Foundation With Four Edges Free by Symplectic Geometry Method

The theoretic solution for rectangular thin plate on foundation with four edges free was derived by symplectic geometry method. In the analysis proceeding, the elastic foundation was presented by the Winlder model. Firstly, the basic equations for elastic thin plate were transferred into Hamilton canonical equations. The symplectic geometry method was used to separate the whole variables and eigenvalues were obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees were developed. Since the basic elasticity equations of thin plate is only used and it is not need to select the deformation function arbitrary. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example was given to demonstrate the accuracy and convergence of the current solution.