弹性地基上四边自由矩形薄板的自由振动

将弹性地基用Winkler模型来代替。首先把弹性地基上矩形薄板的动力学方程表示成为Hamilton正则方程，然后采用辛几何方法对全状态相变量进行分离变量，并利用得到的共扼辛正交归一关系，求出弹性地基上四边自由矩形薄板的固有频率和振型的解析解表达式。由于在求解过程中不需要事先人为的选取挠度函数，而是从弹性地基上矩形薄板的动力学基本方程出发，直接利用数学的方法求出可以满足四边自由边界条件的固有频率和振型的解析解表达式，使得问题的求解更加合理化。文中的最后还给出了计算实例来验证本文所采用的方法以及所推导出公式的正确性。

Free vibration of rectangular thin plate on elastic foundation with four edges free

In this paper, the analytical solution of eigenfrequncies and vibration model shapes of a thin rectangular plate on foundation with four free edges derived by symplectic geometry method. Firstly, the basic dynamic equations for the elastic thin plate on Winkler foundation are transferred into Hamilton canonical equations. And then the whole variables are separated and the eigenvalues are also obtained by the symplectic geometry method. Finally, according to the method of eigenfunetion expansion in the symplectic geometry, the explicit solutions of eigenfrequncies and vibration model shapes of the thin rectangular plate on Winkler foundation with four free edges are presented. Since only the basic dynamic equations of the thin plate are used and it is need not assume the deformation function prior to solving, the presented solution is reasonable and theoretical. In order to proof the correction of the formulations, numerical results are also presented to comparing with that by the another methods.