压电杆结构的弹性动力学分析
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摘要
压电材料由于其优越的力电双向性能被广泛的制成传感器和激励器等智能构件中的敏感元件,以薄层、薄膜、涂层等方式粘附着在基体上形成智能构件。压电复合材料层合杆结构已经是航天工程,智能工程,能源领域,海洋探测甚至生物技术等领域不可缺少的功能器件。压电层既可以埋入基体中也可以覆盖在基体表面上,从而组合成为一个复合层状结构。随着科技的进步和大量工程实践中的需要,对于该领域的研究已经从压电材料本身的研究扩展到考虑综合应用的智能结构的整体性能的研究。随着压电材料被更广泛的应用和在生产实践中面临更多的问题,对于智能结构的动力学性能研究必须不断加以深化。波在压电复合层合杆结构中传播特性的研究是智能结构的动力学性能研究的重要组成部分,有着重要的理论意义和实际应用价值。
本论文研究的主要工作包括:
(1)基于Hamilton原理的矩阵形式,并结合局部坐标函数形式,建立了弹性波在覆盖压电层的圆柱形杆(包括变截面圆柱杆)中传播的基本方程式,通过求解特征值得到了波传播的频散关系并对频散特征进行了讨论,通过数值算例讨论了影响频散特征曲线和位移、电势以及应力分布规律的因素。分析了变截面压电杆的截面变化程度对各个力学分量分布的影响。
(2)在平面正交曲线坐标系和空间正交曲线坐标系中建立了弹性波在平面弯曲压电杆、三维圆形弯曲压电杆和三维任意曲率压电杆中传播时的基本方程式,通过Galerkin积分求解特征方程式并讨论了不同曲率下波传播的频散特性。通过数值算例对比了压电材料曲杆和弹性材料曲杆的频散曲线,讨论了材料参数对频散特性的影响。分析了前三阶频散关系下位移分量和电势分量沿横截面以及沿曲线轴线的分布情况。
(3)在一维问题简化的基础上,通过Laplace变换和相关的复变函数变换,求得了压电曲杆的频散关系,进而推导了位移和电势的关系,先求出了轴向位移的响应函数,再求出电势响应函数,推导了在杆端受到应力脉冲荷载作用下,压电曲杆的动力响应表达式。通过数值计算得到了瞬态位移、应力和电势的响应曲线,分析了纵向冲击波作用下圆截面层合压电杆的动力响应的特性。
(4)建立了有限变形下压电层合杆中几何非线性弹性波传播的基本方程式,用逐步近似法假设位移和电势函数及其相互关系,采用摄动法求解了运动方程,得到了位移和电势函数。讨论了非线性波的波形畸变特性,分析了初始频率和初始幅值对波的非线性特征的影响。
本文对压电层合杆和纯压电杆的弹性动力学问题进行了系统的理论分析。其研究成果丰富了智能结构的弹性动力学理论,并可为压电层合杆的动态力学性能设计和应用提供有意义的理论依据。
Piezoelectric materials have been widely used to manufacture various sensors and actuators in the form of laminar layer, thin membrane and coat as sense organs in the areas of smart or intelligent structures because of its strong electriomechanical coupling effects. Piezoelectric composite layered rods structures have become one of the important functional devices in the application areas as aerospace engineering, smart structures, ocean exploration, energy fields and even biological engineering. Piezoelectric elements can be incorporated into a laminated composite structure, either by embedding it or by mounting it onto the surface of the host structure. With the developing of advanced technical and the growing demand for engineering practice, researches in this area have been further extended from pure piezoelectric properties into the properties of smart structures. By considering of broad application and encountering more problems in manufacture practice, researches of dynamical properties of piezoelectric materials have been further deepened. The study of characteristics of wave propagation in the laminated rods structures becomes one of the most important parts in dynamical property researches of smart structures and is very meaningful in theory and valuable in application.
The main work in dissertation includes:
(1)Based on the matrix formulation of Hamilton’s principle and co-ordinated fuctions, the basic equations of wave propagation in laminated piezoelectric cylindrical bars (including section-varying) are derived. The wave dispersion curves are obtained by solving an eigenvalue problem. By the numerical examples, the factors of influence on distributions of dispersion curves, displacements and electric potential are discussed through numerical results. The influence of changes of cross section on the distributions of displacements and electric potential is studied.
(2)The equations of wave propagation in plane bent piezoelectric rods, in 3D circular bent piezoelectric rods, and in 3D bent piezoelectric rods with arbitrary curvature are established in the orthogonal curvilinear coordinate system. The wave dispersion curves in the rods with different curvature are obtained by performing Galerkin integral and solving an eigenvalue problem. The differences of dispersion curves in piezoelectric bent rods and in elastic bent rods are discussed by the results of numerical examples. The influence of material parameters on dispersion relations is also discussed. And distributions of displacements and electric potential of the first three modes in the cross section and along the axes are studied.
(3) Based on simplified the problem to one dimension, by performing Laplace transform and some relevant transforms in functions of a complex variable, dispersion relations in bent piezoelectric rods are obtained, then the relations between displacements and electric potential, and the dynamical response expressions of displacements and electric potential under stress impulse acting on end of the rod are derived. The dynamical response curves of displacements, stress and electric potential are discussed in the numerical examples, and the dynamical response characteristics under longitudinal impulse wave act on laminated piezoelectric rods are analysed.
(4) Based finite deformation theory, the equations of geometrical nonlinear elastic wave propagation in the laminated piezoelectric rod are established. The method of successive approximation is used to assume the functions of displacement and electric potential and their relations are also obtained. By using perturbation method, the equations are solved and functions of displacements and electric potential are obtained. Distortion of shapes of nonlinear wave and effects of initial frequency and initial amplitude on nonlinear characteristics of wave propagation are discussed.
本论文研究的主要工作包括:
(1)基于Hamilton原理的矩阵形式,并结合局部坐标函数形式,建立了弹性波在覆盖压电层的圆柱形杆(包括变截面圆柱杆)中传播的基本方程式,通过求解特征值得到了波传播的频散关系并对频散特征进行了讨论,通过数值算例讨论了影响频散特征曲线和位移、电势以及应力分布规律的因素。分析了变截面压电杆的截面变化程度对各个力学分量分布的影响。
(2)在平面正交曲线坐标系和空间正交曲线坐标系中建立了弹性波在平面弯曲压电杆、三维圆形弯曲压电杆和三维任意曲率压电杆中传播时的基本方程式,通过Galerkin积分求解特征方程式并讨论了不同曲率下波传播的频散特性。通过数值算例对比了压电材料曲杆和弹性材料曲杆的频散曲线,讨论了材料参数对频散特性的影响。分析了前三阶频散关系下位移分量和电势分量沿横截面以及沿曲线轴线的分布情况。
(3)在一维问题简化的基础上,通过Laplace变换和相关的复变函数变换,求得了压电曲杆的频散关系,进而推导了位移和电势的关系,先求出了轴向位移的响应函数,再求出电势响应函数,推导了在杆端受到应力脉冲荷载作用下,压电曲杆的动力响应表达式。通过数值计算得到了瞬态位移、应力和电势的响应曲线,分析了纵向冲击波作用下圆截面层合压电杆的动力响应的特性。
(4)建立了有限变形下压电层合杆中几何非线性弹性波传播的基本方程式,用逐步近似法假设位移和电势函数及其相互关系,采用摄动法求解了运动方程,得到了位移和电势函数。讨论了非线性波的波形畸变特性,分析了初始频率和初始幅值对波的非线性特征的影响。
本文对压电层合杆和纯压电杆的弹性动力学问题进行了系统的理论分析。其研究成果丰富了智能结构的弹性动力学理论,并可为压电层合杆的动态力学性能设计和应用提供有意义的理论依据。
Piezoelectric materials have been widely used to manufacture various sensors and actuators in the form of laminar layer, thin membrane and coat as sense organs in the areas of smart or intelligent structures because of its strong electriomechanical coupling effects. Piezoelectric composite layered rods structures have become one of the important functional devices in the application areas as aerospace engineering, smart structures, ocean exploration, energy fields and even biological engineering. Piezoelectric elements can be incorporated into a laminated composite structure, either by embedding it or by mounting it onto the surface of the host structure. With the developing of advanced technical and the growing demand for engineering practice, researches in this area have been further extended from pure piezoelectric properties into the properties of smart structures. By considering of broad application and encountering more problems in manufacture practice, researches of dynamical properties of piezoelectric materials have been further deepened. The study of characteristics of wave propagation in the laminated rods structures becomes one of the most important parts in dynamical property researches of smart structures and is very meaningful in theory and valuable in application.
The main work in dissertation includes:
(1)Based on the matrix formulation of Hamilton’s principle and co-ordinated fuctions, the basic equations of wave propagation in laminated piezoelectric cylindrical bars (including section-varying) are derived. The wave dispersion curves are obtained by solving an eigenvalue problem. By the numerical examples, the factors of influence on distributions of dispersion curves, displacements and electric potential are discussed through numerical results. The influence of changes of cross section on the distributions of displacements and electric potential is studied.
(2)The equations of wave propagation in plane bent piezoelectric rods, in 3D circular bent piezoelectric rods, and in 3D bent piezoelectric rods with arbitrary curvature are established in the orthogonal curvilinear coordinate system. The wave dispersion curves in the rods with different curvature are obtained by performing Galerkin integral and solving an eigenvalue problem. The differences of dispersion curves in piezoelectric bent rods and in elastic bent rods are discussed by the results of numerical examples. The influence of material parameters on dispersion relations is also discussed. And distributions of displacements and electric potential of the first three modes in the cross section and along the axes are studied.
(3) Based on simplified the problem to one dimension, by performing Laplace transform and some relevant transforms in functions of a complex variable, dispersion relations in bent piezoelectric rods are obtained, then the relations between displacements and electric potential, and the dynamical response expressions of displacements and electric potential under stress impulse acting on end of the rod are derived. The dynamical response curves of displacements, stress and electric potential are discussed in the numerical examples, and the dynamical response characteristics under longitudinal impulse wave act on laminated piezoelectric rods are analysed.
(4) Based finite deformation theory, the equations of geometrical nonlinear elastic wave propagation in the laminated piezoelectric rod are established. The method of successive approximation is used to assume the functions of displacement and electric potential and their relations are also obtained. By using perturbation method, the equations are solved and functions of displacements and electric potential are obtained. Distortion of shapes of nonlinear wave and effects of initial frequency and initial amplitude on nonlinear characteristics of wave propagation are discussed.
引文
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