中文摘要：

变量的分离被采用解决 Hamiltonian 为薄盘子的横向的免费颤动的特征值问题的双形式，和在关上的形式的自然模式的明确的表达被执行。靠近形式的自然模式确切满足薄板的特征值问题的管理方程并且为边界条件的任何类型是适用的。与简单地支持的(S) 和夹钳的(C) 的所有联合，边界条件适用于自然模式，模式形状特别地被获得，二个特征值方程关于二个空间坐标，正常模式和频率确切在的帮助下被解决被导出。为盒子 SSCC， SCCC 和 CCCC 的准确 eigensolutions 是不能的然而，被获得，这被相信他们成功地在这篇论文被发现。在现在的结果和女性的结果之间的比较验证现在的准确答案，它能因此为验证不同近似途径作为基准被拿。

英文摘要：

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.