功能梯度压电圆板自由振动问题的三维精确分析

本文对周边为广义刚性滑动和广义简支两种边界条件下的功能梯度压电材料圆板自由振动问题进行分析。根据轴对称横观各向同性压电材料基本方程，并利用有限Hankel变换得到了功能梯度压电材料圆板的状态空间方程。假设材料的机械和电学性质均沿板厚方向按统一的指数函数形式梯度分布，从而获得了周边为广义刚性滑动和广义弹性简支两种边界条件下功能梯度压电圆板自由振动问题的三维精确频率方程，该方程是一个关于自由振动频率的超越方程，通过求解该超越方程可得到在不同板厚以及不同的材料性质梯度变化情况下的圆板自由振动频率值，结果表明在相同的材料性质梯度变化情况下频率均随着板厚增加而增大，而在相同的板厚情况下频率则随材料性质梯度变化指数的增大而减小的结论。

Three Dimensional Exact Solution for Free Vibration of Functionally Gradient Piezoelectric Circular Plate

An exact three dimensional analysis was presented for the free vibration of a functionally gradient piezoelectric material circular plate that is generalized rigid slipping supported and generalized elastic simply supported. According to the basic equations of axisymmetric transfer isotropic piezoelectric materials, the state space equations of the functionally gradient piezoelectric circular plate were developed by the method of finite Hankel transformation. Assuming that the mechanical and electric properties of the materials have the same exponential law dependence on the thickness-coordinate, an exact three dimensional frequency equations of the free vibration for generalized rigid slipping supported and generalized elastic simply supported circular plate was obtained. This equation is a transcendental equation. And the frequency values of free vibration for the circular plate with different thickness and different graded variation can be obtained by the solution of this transcendental function. The conclusion is that the frequency increases with the increasing thickness of plate, and the decreased frequency will be got with the increase of the material properties grads for the plate with the same thickness.