压电功能梯度层合梁的力-电-热耦合梁单元及最优振动控制

PIEZOTHERMOELASTIC COUPLING BEAM ELEMENT AND OPTIMAL VIBRATION CONTROL OF PIEZOELECTRIC FUNCTIONALLY GRADED BEAMS

给出了一个压电功能梯度层合梁振动分析的两节点力-电-热耦合梁单元,并将其用于功能梯度层合梁的振动最优控制。在这个多场耦合梁单元中,功能梯度材料的等效力学性能用Voigt或Mori-Tanaka模型表征;梁的位移场用Shi改进的三阶剪切变形板理论描述;压电层的电势场用Layer-wise理论分层表征,且呈高阶非线性电势场的压电层可离散成数个子层。用Hamilton原理推导了压电功能梯度梁的力-电-热耦合单元列式,用拟协调元法给出了多场耦合梁单元的高计算效率的显式单元刚度矩阵,以及采用线性二次型(LQR)最优控制算法进行压电功能梯度层合梁的最优振动控制。使用所得力-电-热耦合梁单元进行了压电功能梯度层合梁的静力和动力分析。数值算例表明,所得力-电-热耦合梁单元可靠、准确和高效,LQR最优控制算法得到最优控制电压可有效抑制功能梯度梁的振动且实现控制系统能量的优化。

A two-noded piezothermoelastic coupling beam element is presented for the dynamic analysis and optimal vibration control of piezoelectric functionally graded beams.The equivalent mechanical properties of functionally graded materials are modeled by Voigt model or Mori-Tanaka model.The kinematics of the beam is characterized by Shi’s third-order shear deformation theory;whereas the electric potential function for piezoelectric laminae is modeled using layer-wise theory,and a piezoelectric layer can be divided into sublayers and modeled using discrete layers to characterize nonlinear voltage distribution.Hamilton’s principle is used to derive the piezothermoelastic finite element equations,and the quasi-conforming element technique is adopted to evaluate the explicit element stiffness matrix of the piezothermoelastic coupling beam element.Linear Quadratic Regulator(LQR)control scheme is employed for the optimal active vibration control.The proposed beam element and active control scheme are validated through static and dynamic analyses of piezoelectric functionally graded beams.The numerical results show that the present beam element is reliable,accurate and efficient,the optimal control voltage evaluated from LQR control scheme can efficiently control the vibration of functionally graded beams,and the LQR control scheme can optimize the energy input for vibration control.