四边固支弹性矩形薄板的自由振动

首先弹性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，求出四边固支弹性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固支边界条件下薄板的固有频率和振型的解析解表达式，使得问题的求解更加理论化和合理化．此外，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 with completed clamped supported

The analytic solution of the eigenfrequencies and models of a rectangular thin plate with completed clamped supported was derived by the symplectic geometry method. Firstly, the basic dynamic equations for the elastic thin plate were transferred into Hamilton canonical equations. And then the whole variables were separated. Finally, according to the eigenfunction expansion method in the symplectic geometry, the explicit solutions of the eigenfrequencies and models of the rectangular thin plate with completed clamped supported were obtained. Because only the basic dynamic elasticity equations of the thin plate were used, it does not need to select the deformation function arbitrarily. Therefore, the solution is reasonable and theoretical. Moreover, some numerical results were presented to demonstrate the correction of formulations.