非均匀电弹性板横向自由振动的精确解

摘要

非均匀电弹性材料通过调整材料的组分可以使其力电特性按照优化设计进行分配,从而满足结构元件不同部位对材料特性的不同要求.由于材料参数的非均匀性,其相应结构的控制方程通常为变系数的偏微分方程,使得相应的力学分析具有很大的挑战性.本文假设电弹性材料参数沿厚度方向以同一指数形式变化,给出了非均匀电弹性矩形板的基本方程.应用分离变量方法给出了四边简支和四边固支非均匀电弹性板的横向自由振动的精确解,分析了非均匀电弹性板在不同边界条件下的不均匀系数与振动频率之间的变化规律.通过算例,验证了精确解的正确性.%A design can be optimized by adjusting the material composition to meet different material property re -quirements of different component positions;based on this,the mechanical and electrical properties of the inhomo-geneous electroelastic material can be assigned.Due to the inhomogeneous of material parameters,the governing e-quations of corresponding structure are usually partial differential equations with variable coefficients,which making the mechanics study a tough challenge.In this study,these properties were assumed to have the same exponent-law dependence in the direction of plate thickness; a basic equation of the inhomogeneous rectangular electroelastic plate was also presented.Using the separation of variables,the exact solutions for free vibrations of the inhomoge-neous electroelastic rectangular plate were obtained under the conditions of simply supported on four sides and fixed on four sides.The relation between the inhomogeneity coefficient and vibration frequency under different boundary conditions were then investigated,and the correctness of the exact solution was verified using examples.