中文摘要：

目前板结构振动特性的计算分析大多数都是基于经典板理论或者其他高阶近似理论,为了深入理解其振动特性,对于一些厚度比较大的板,采用三维弹性理论来进行分析十分必要。在空间坐标系中沿3个坐标轴方向的位移场使用三维改进傅里叶级数法来建模,通过计算系统拉格朗日函数的极值即可计算出描述位移场的所有未知系数。将本文计算结果与文献结果和NASTRAN软件计算的有限元结果进行比较,最大偏差不超过1.4%,结果吻合良好。结果表明：本文方法能够准确预测三维弹性矩形厚板结构的振动特性,且无论矩形板的边界条件是否对称,都能使用该方法来简便地获得固有频率、振型以及强迫响应。

英文摘要：

Most current studies of vibration characteristics of plate structures are based on the classical plate theory or other high-order approximation theories. In order to get a deep understanding of thick plate vibration characteris- tics, it is of critical importance to analyze the vibrations using the three-dimensional elasticity theory. In this paper, a three-dimensional improved Fourier series method is first proposed to construct three vibrating displacement fields in a thick rectangular plate with various boundary conditions. All the unknown coefficients were then determined from the extrema of system Lagrangian constructed completely from 3-D elasticity theory. Finally, numerical exam- ples were given to validate the current method through comparison with results from other approachs, and finite ele- ment analysis （FEA） using NASTRAN. Good agreement was observed between the two sets of solutions, the largest deviation was less than 1.4%. The results show that the proposed method is able to accurately capture the vibration characteristics for 3-D rectangular plate structures. Natural frequencies, mode shapes, and forced response of thick plates can be easily obtained whether the boundary condition is symmetrical or not.