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 引用本文: 赵希宁, 杨晓东, 张伟. 含电学边界的压电层合梁的非线性弯曲波[J]. 力学学报, 2021, 53(4): 1124-1137.
Zhao Xining, Yang Xiaodong, Zhang Wei. NONLIEAR BENDING WAVES OF A PIEZOELECTRIC LAMINATED BEAM WITH ELECTRICAL BOUNDARY[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(4): 1124-1137.
 Citation: Zhao Xining, Yang Xiaodong, Zhang Wei. NONLIEAR BENDING WAVES OF A PIEZOELECTRIC LAMINATED BEAM WITH ELECTRICAL BOUNDARY[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(4): 1124-1137.

NONLIEAR BENDING WAVES OF A PIEZOELECTRIC LAMINATED BEAM WITH ELECTRICAL BOUNDARY

• 摘要: 非线性科学己成为近代科学发展的一个重要标志, 特别是非线性动力学和非线性波的研究对于解决自然科学各领域中遇到的复杂现象和问题有着极其重要的意义. 本文研究了含电学边界条件的压电层合梁的非线性弯曲波传播特性.首先, 考虑几何非线性效应和压电耦合效应, 利用哈密顿原理建立了一维无限长矩形压电层合梁弯曲波的非线性方程.其次, 采用Jacobi椭圆函数展开法对非线性弯曲波方程进行求解, 得到了非线性弯曲波动方程在近似情况下对应的冲击波解和孤波解.最后, 利用约化摄动法得到了非线性薛定谔方程, 进一步得到了亮孤子和暗孤子解.基于两种方法具体研究了外加电压、压电层厚度等参数对冲击波和孤立波以及亮孤子和暗孤子特性的影响. 研究结果表明, 在波速较小时, 外加电压对冲击波的影响较大, 波速较大时, 外加电压对孤立波影响减弱.通过调整作用在压电层合梁上的电压发现了存在亮孤子和暗孤子, 分析结果表明随着外加电压值的增大, 亮孤子和暗孤子的振幅都增大.

Abstract: Nonlinear science has been an important symbol in the development of modern science, especially the researches in nonlinear dynamics and nonlinear waves have extraordinary significance in solving the complex phenomena and problems encountered in various fields of natural science. In this paper, the nonlinear bending wave propagation of a piezoelectric laminated beam with electrical boundary conditions is studied. Firstly, considering the geometric nonlinear effect and piezoelectric coupling effect, the nonlinear equation of the one-dimensional infinite rectangular piezoelectric laminated beams is established by using Hamiltonian principle. Secondly, the Jacobi elliptic function expansion method is used to treat the nonlinear flexural wave equation, and the corresponding shock wave solution and solitary wave solution of the nonlinear flexural wave equation are obtained in the approximate case. Last, the nonlinear Schrodinger equation is obtained by using the reduced perturbation method, and the bright and dark soliton solutions are further obtained. Moreover, the effects of external voltage and the thickness of the piezoelectric layer on the characteristics of shock wave and solitary wave as well as bright and dark solitons are studied. The results show that when the wave velocity is small, the external voltage has a great influence on the shock wave, and when the wave velocity is large, the external voltage has no effect on the solitary wave. The bright solitons and the dark solitons can be obtained by adjusting the external voltage applied to the piezoelectric laminated beam. It is found that the amplitudes of bright and dark solitons increase with the increase of external voltages.

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