各种典型边界FGM矩形板面内自由振动的二维弹性分析

为获得功能梯度材料（FGM）矩形板面内自由振动的动力学响应，基于二维线弹性理论建立了功能梯度材料矩形板面内自由振动的控制微分方程.采用微分求积法（DQM）数值研究了9种典型边界下FGM矩形板面内自由振动的频率特性，分析了边界条件、长宽比及梯度指数对自振频率的影响.分析结果表明:通过设置梯度指数为0，将FGM矩形板退化为各向同性矩形板，与已有各向同性矩形板的文献结果进行比较，表明了DQM的适用性和精确性；9种边界下长宽比对FGM矩形板基频的影响不同，基频随长宽比的增大而增大的板分别为:C-C-C-C板、SS2-C-SS2-C板、C-C-C-F板、SS1-C-SS1-C板、C-C-F-F板和SS1-SS1-SS2-SS2板；基频随长宽比的增大而减小的板分别为:F-F-F-F板与C-F-C-F板；SS1-SS1-SS1-SS1板发生剪切自锁现象，基频随长宽比的增大而基本保持不变；基频随梯度指数的增大而快速减小，梯度指数p 10时，基频变化不再明显. 

Two-Dimensional Elasticity Solutions for In-Plane Free Vibration of FGM Rectangular Plates under Different Boundary Conditions

In order to obtain the dynamic responses on in-plane free vibration of functionally graded material (FGM) rectangular plates, based on the two-dimensional linear elasticity theory, the governing partial differential equations for the in-plane free vibration of FGM rectangular plates were derived. Using differential quadrature method (DQM), the frequency characteristics for in-plane free vibration of FGM rectangular plates under 9 different boundary conditions were investigated. The effects of boundary conditions, geometrical parameters and material gradient indexes on the dimensionless frequencies of the FGM rectangular plates were analyzed. The material gradient index was set as zero to take FGM rectangular plates as isotropic rectangular plates. Then, the applicability and accuracy of the DQM were demonstrated by comparing the in-plane free vibration of the obtained isotropic rectangular plates with those in literature. The effect of the length-width ratio on the fundamental frequency of the FGM rectangular plates varies under different boundary conditions. The fundamental frequency increases with the length-width ratios for the plates C-C-C-C, SS2-C-SS2-C, C-C-C-F, SS1-C-SS1-C, C-C-F-F and SS1-SS1-SS2-SS2, and decreases with the increase of the length-width ratios for the plates F-F-F-F and C-F-C-F, but has no significant changes for SS1-SS1-SS1-SS1 plate because of shear locking. The fundamental frequency decreases rapidly with the increase of material gradient indexes, but it has no obvious change when the material gradient index p is more than 10.