热环境中粘贴压电层功能梯度材料梁的自由振动  

Free Vibration of Functionally Graded Material Beams With Surface-Bonded Piezoelectric Layers in Thermal Environment

作　　者：李世荣[1] 苏厚德[1] 程昌钧[2] 

机构地区：[1]兰州理工大学理学院工程力学系,兰州730050 [2]上海大学力学系,上海200436

出　　处：《应用数学和力学》2009年第8期907-918,共12页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10872083;10602021);教育部博士点基金资助项目(200807310002)

摘　　要：研究了上下表面粘贴压电层的功能梯度材料Euler-Bernoulli梁在升温及电场作用下的屈曲和自由振动行为.在精确考虑轴线伸长基础上,建立了压电功能梯度材料层合梁在热-电-机载荷作用下的几何非线性动力学控制方程.其中,假设功能梯度材料性质沿厚度方向按照幂函数连续变化,上下压电层为各向同性均匀材料.在小振幅和谐振动假设下,上述非线性偏微分方程组被转化为两套相互耦合的常微分方程组,即过屈曲问题的控制方程和过屈曲构形附近的线性振动控制方程.采用打靶法数值求解上述两个耦合的常微分方程边值问题,获得了在均匀电场和横向非均匀升温场作用下两端固定压电-功能梯度材料层合梁在屈曲前和过屈曲构型附近的自由振动响应.绘出了梁的过屈曲平衡路径以及前3阶固有频率随热、电载荷及材料梯度参数变化的特性曲线.结果表明,梁的前3阶频率在屈曲前随着温度升高而减小,在进入过屈曲后它们却随着温度升高而增加.通过施加电压在压电层产生拉应力可有效地提高粱的热屈曲临界载荷,从而提高其固有频率.Free vibration of statically thermal post-buckled functionally graded material beams with surface-bonded piezoelectric layers subjected to both temperature rise and voltage is studied. By accu- rately considering the axial extension and based on Etfler-Bemoulli beam theory, geometrically nonlin- ear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subjected to thermo-electro-mechanical loadings were fotmulated. It was assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate and that the piezoelectric layers are isotropic and homogenous. By assunfing that the amplitude of beam＇ s vibration is small and its response harmonic, the above mentioned non-linear partial differential equations were reduced to two sets of coupled ordinary differential equations; the one for the post-buckling, and the other for linear vibration of the beam superimposed upon the post-buckled configuration. By using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, response of post-buckling and free vibration in the vicinity of the post-buckled configuration of the beam with fixed-fixed ends and subjected to transversely non-uniform heating and tmiform electric field were obtained. Thermo-electric post-buckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity and the material gradient parameters were plotted. It is found that the three lowest frequencies of the pre-buckled beam decrease with an increase in the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

关 键 词：功能梯度材料 压电层合梁 热屈曲 自由振动 固有频率 

分 类 号：O343[理学—固体力学]