功能梯度压电圆板轴对称自由振动问题精确解

Exact Solution for Axisymmetric Free Vibration of a Functionally Graded Piezoelectric Circular Plate

将功能梯度压电圆板的位移变量和电势变量写为分离变量的形式,由压电动力学平衡方程导出以位移、电势及其一阶导数为状态变量的状态方程,考虑周边固支接地的边界条件,导出了求解功能梯度压电圆板自振频率精确解的方程。将方程退化至一般的非梯度纯弹性圆板的形式,求解其自振频率,得到的结果与相应的理论解完全吻合,从而验证了本文方法的正确性。更进一步地对梯度函数沿板厚以指数形式变化的功能梯度压电圆板的自振频率进行了计算,并得到了梯度化对板自振频率的影响规律。

The displacement and electric potential functions of a functionally graded (FG) piezoelectric circular plate are written as separate variables. The state equations are established based on the governing e-quations, in which the displacements, electric potential and their first order differentials are assumed as state variables. The exact frequency equation of axisymmetric free vibration is derived under clamped and earthed boundary conditions. It's noted that the presented method gives exact results and shows very nice convergence as compared with the results obtained from other analytical methods when reduced to non-FG plates. The frequencies of circular plate, in which the material coefficients are assumed to have the same exponent law dependence on the thickness-coordinate, are calculated. The effect of A on the frequency is presented.