压电智能结构分析的新方法研究及其应用
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摘要
由于智能结构具有自诊断、自适应、自修复等卓越功能,随着研究的不断深入和理论的成熟,现已从航空和军界应用,逐渐被拓展到土木工程、船舶、汽车等行业中,并且在航空、航天、潜艇、高速列车、汽车、桥梁、水坝、建筑等结构的健康监测、损伤自愈合及振动、噪声和形状控制等方面展现出良好的应用前景。近年来智能材料结构的研究应用已经引起世界各主要发达国家的极大重视,被列为优先发展领域和优先培育的21世纪高新技术产业之一。
压电材料具有频响范围宽、响应速度快、密实度大、精确度高、良好的线性行为等优点,既可制成传感器又可制成驱动器,通过正逆压电效应来实现智能控制。许多学者对压电材料智能结构开展了大量的基础和应用研究,并已取得了丰富的研究成果。
目前,对压电智能结构的研究内容主要包括：耦合理论；形状控制；振动控制；噪声控制；优化分析；故障诊断和监测；分析方法研究及试验研究等。本文基于广西大学秦荣教授创立的样条无网格方法和QR方法分别对压电智能复合材料层合板、压电框架结构建立新的分析模型,对压电智能结构静变形控制、形状最优控制、压电材料参数识别、振动主动控制展开研究。研究的主要工作和创新点如下：
1.基于高阶剪切变形理论和-Iamilton变分原理,采用样条无网格方法,建立了压电智能复合板静变形分析的新模型,推导了样条无网格法刚度矩阵；基于样条无网格离散模型对压电驱动器驱动电压的灵敏度和对驱动器铺设位置的灵敏度,建立了压电智能板形状最优控制模型。通过设计不同的压电驱动器对各种边界条件、不同基体材料的压电层合板进行静变形控制和形状最优控制的算例分析,结果表明本文建立的新模型正确,能够有效的控制结构变形。样条无网格法具有精度高、输入简单、运行效率高、处理边界条件简便等优点。
2.对压电智能层合板的变形控制进行了解析法研究,提出了符合压电材料正逆压电效应特性的压电层表面的电学边界条件。针对考虑一阶剪切影响的四边简支压电层合板,根据电学平衡方程及电学边界条件推导得到了压电层沿厚度方向电势分布为双曲函数的变化规律。算例讨论了压电层合板在机械荷载和电荷载作用下的变形、电势分布,并对结构变形进行了开环和闭环控制,结果表明推导的解析解与样条无网格解能够互相印证,吻合很好。由于单片压电驱动器控制力不足,为了提高驱动效率,对书本式压电驱动器驱动力的解析解进行了公式推导,建立了压电片层数n与压电驱动器的驱动力Mxp非线性的量化关系。算例分析表明,书本式压电驱动器的控制效果较好,能够应用于压电层合板及钢框架结构的变形控制中。
3.建立了压电材料参数识别分析的新模型。将材料参数识别的问题转化为极小化目标函数的问题,目标函数定义为测量位移与样条无网格法计算的相应位移之差的平方和；推导了基于样条无网格法求解位移值相对于材料各参数导数的灵敏度计算公式,采用基于信赖域技巧的Levenberg-Marquardt方法极小化目标函数；在参数识别过程中,以样条无网格方法计算的理论位移为真值,以给定方差下的随机正态分布数据模拟带误差的测量位移。研究了压电复合材料板分别在机械荷载及电荷载作用下,基体材料和压电材料的参数识别问题,算例表明本文提出的材料参数识别方法具有较高的精度和较好的稳定性,是行之有效的。
4.基于高阶剪切变形理论,推导了一种新的可以考虑剪切影响、压电效应、初始几何缺陷及P-△效应的压电智能梁柱单元。当不考虑剪切影响和压电效应时,新单元的刚度矩阵可以退化为线弹性情况下的单刚形式；通过引入初始几何缺陷影响系数的方法,可将初始几何缺陷与P-△效应联合分析,建立了相应的单元几何刚度矩阵。算例结果表明新单元模型正确,为压电智能框架结构建模奠定了理论研究基础。
5.基于新的压电智能梁柱单元刚度矩阵,采用样条QR方法建立了压电智能框架结构的动力分析新模型；利用模态控制理论,运用LQR最优控制方法,建立了压电智能框架结构振动主动控制计算模型；将压电堆式驱动器布置在框架结构柱上,对压电钢框架结构进行了振动主动控制仿真分析,讨论了结构的P-△效应和初始几何缺陷对结构自振频率及控制力的影响。算例结果表明,本文建立的分析模型能够有效的抑制结构的振动；考虑P-△效应和初始几何缺陷后,结构的自振频率减小,控制电压明显增加,说明这两个因素对结构振动主动控制的影响是不可忽略的,分析初始几何缺陷和P-△效应对结构振动控制的影响很有意义。
本文进行了大量的数值模拟计算,将基于新方法的分析结果与解析解和有限元解进行了比对,结果表明建立的新模型是正确和有效的,具有输入简单、计算精度高、稳定性好、物理概念清晰、处理边界条件方便、计算量少,运行速度快等优点。本文采用新方法对压电智能复合材料层合板及钢框架结构开展形状控制、振动控制、优化分析以及参数识别的理论研究和仿真分析,具有重要的理论和现实意义,提出的分析方法及得到的结论具有参考价值。
Smart structures are widely used in the military aviation and aerospace engineering due to their superior properties, such as self-diagnosis, adaptive, self-repair, etc. With the development of thorough investigation and improving theory, applications of smart structures have been extended to civil engineering, shipbuilding, automobile and other industries. A good application prospect of smart structures has been unfolded for the structural health monitoring, damage self-healing and vibration, noise and shape control in the fields of aviation, aerospace, submarines, high-speed trains, automobiles, bridges, dams and architecture. In recent years, a great deal of attention has been paid to the research and application of smart structures in the world's major developed countries, and therefore the research and application of smart structures has been classified as one of the priority development areas in terms of the21st century high-tech industries.
Piezoelectric materials have a lot of advantages such as wide frequency response range, fast response, large compactness, high accuracy, good linear behavior, etc. and they can be used as not only sensors but also actuators. The smart control in structures can be realized by way of the unique direct and inverse piezoelectric effect of piezoelectric materials. Over the past decades, extensive studies have been carried out in the basic and applied research of piezoelectric smart structures and a wealth of research achievements have been made.
At present, the studies of piezoelectric smart structures are mainly concerned with coupling theory, shape control, vibration control, noise control, optimization analysis, fault diagnosis and monitoring, analysis methods and experiment. Based on the spline meshless method and QR method founded by Professor Qin Rong in Guangxi University, new analysis models for piezoelectric smart laminated plates and piezoelectric frame structures are proposed to study the static deformation control, optimal shape control, active vibration control and parameter identification of piezoelectric material structures. The main work and innovations of the research in this paper are as follows:
1. Based on Reddy's third order plate theory and Hamilton variational principle, a new model of the static deformation control for piezoelectric smart composite plate is developed and the stiffness matrix is deduced by using spline meshless method. A model of the optimal shape control for piezoelectric smart composite plate is derived according to the sensitivity matrix which is based on the spline meshless discrete model with respect to the driving voltage and location of the actuator layer. By designing different piezoelectric actuators, various boundary conditions and different matrix materials, some numerical examples of open-loop control and closed-loop control are given. The numerical results show that the developed model is correct and effective for shape control. The spline meshless method has the advantages of high precision, simple input, high efficiency and concise boundary conditions.
2. The study of the deformation control for piezoelectric smart laminated plate by the analytical method and the find of the electrical boundary conditions of piezoelectric layer in line with the unique features of direct and inverse piezoelectric effect. Based on the electrical balance equation and electrical boundary conditions, for the simply supported piezoelectric laminated plates considering the first shear impact, the deduced electric potential distribution is expressed as a hyperbolic function along the thickness. Numerical examples are given to analyze the deformation, electric potential distribution, open-loop and closed-loop control of the piezoelectric laminated plate under mechanical loads and electric loads. The results illustrate that the analytical solutions are cross-checked with those of spline meshless method and the agreement between them is satisfactory. Due to the insufficient controlling power of monolithic piezoelectric actuator, the analytical formula of driving force for the book-block piezoelectric actuator is derived to improve the driving efficiency. Moreover, the non-linear quantitative relationship between piezoelectric film layers (n) and the driving force of the piezoelectric actuators (Mxp) is established. Examples show that better control is obtained by the book-block piezoelectric actuator. So, it can be applied to the deformation control of piezoelectric laminated plate and steel frame structure.
3. A new model for parameters identification of piezoelectric material. Based on the displacement modeled by the spline meshless method, the parameter identification problem is formulated as the problem of minimizing the objective function which is defined as a square sum of differences between the measured displacement and the computed displacement by the spline meshless method. The sensitivity matrix is calculated by the derivation of parameters to be identified. The calculation formula of sensitivity is deduced based on the displacement values obtained by the spline meshless method in line with the derivative of each material parameter. Levenberg-Marquardt method with the trust-region based techniques, is used to solve the minimization problem. In the process of parameter identification, the calculated results obtained by the spline meshless method using the true values of material parameters replace the measured data on displacements. However, normal distribution noises are added to simulate errors in measurement. For piezoelectric composite plate under the mechanical loads and electrical loads, the identification of matrix material parameter and piezoelectric parameter are investigated. Numerical examples show that the proposed method for parameter identification in this paper has high accuracy and good stability, and is effective.
4. Based on Reddy's third order theory, a new piezoelectric smart beam-column element is derived by considering the shear deformation effect, piezoelectric effect, initial geometric imperfection and P-A effect. When the shear effect and piezoelectric effect are not considered, the element geometric stiffness matrix derived in this paper for piezoelectric smart beam-column element derived can be degenerated to the one for the linear elastic case. With introducing its influence coefficient, the initial geometric imperfection is jointly analyzed with P-Δ effect and the corresponding element stiffness matrix is established. The correctness of this beam-column element is illustrated by some numerical examples. This work lays a theoretical foundation for the modeling of piezoelectric smart frame structure.
5. Based on the piezoelectric smart beam-column element stiffness matrix derived in the above, the new dynamic analysis model for piezoelectric frame structure is developed by the spline QR method. The mode control theory and LQR optimal control method are used to establish the calculation model of active vibration control for piezoelectric frame structure. The influence of the P-A effect and initial geometric imperfection on the structural natural frequency and control power is discussed by way of laying piezoelectric stack actuators on the column. The simulation analysis of active vibration control for piezoelectric laminated beam and piezoelectric-steel frame structure is given and the results demonstrate that the vibration of the structure can be effectively suppressed by using this analysis model. With considering the P-Δ effect and initial geometric imperfection, the natural frequency of the structure decreases and the control voltage significantly increases. It is obvious that the influence of these two factors on the active vibration control can't be ignored and its study is of great significance.
By comparing with analytical solution and finite element solution, numerical results illustrate that the new model established by the proposed methods in this paper is correct and effective and those methods have a lot of advantages, such as simple input, high accuracy, good stability, clear physical concept, concise boundary conditions, less computation, run fast, etc. In this paper, the new methods are used to study shape control, vibration control, optimization analysis, and parameter identification for piezoelectric smart composite laminates and steel frame structures, which is of great significance in theory and practice. The proposed methods as well as the obtained conclusions have an important reference value.
压电材料具有频响范围宽、响应速度快、密实度大、精确度高、良好的线性行为等优点,既可制成传感器又可制成驱动器,通过正逆压电效应来实现智能控制。许多学者对压电材料智能结构开展了大量的基础和应用研究,并已取得了丰富的研究成果。
目前,对压电智能结构的研究内容主要包括：耦合理论；形状控制；振动控制；噪声控制；优化分析；故障诊断和监测；分析方法研究及试验研究等。本文基于广西大学秦荣教授创立的样条无网格方法和QR方法分别对压电智能复合材料层合板、压电框架结构建立新的分析模型,对压电智能结构静变形控制、形状最优控制、压电材料参数识别、振动主动控制展开研究。研究的主要工作和创新点如下：
1.基于高阶剪切变形理论和-Iamilton变分原理,采用样条无网格方法,建立了压电智能复合板静变形分析的新模型,推导了样条无网格法刚度矩阵；基于样条无网格离散模型对压电驱动器驱动电压的灵敏度和对驱动器铺设位置的灵敏度,建立了压电智能板形状最优控制模型。通过设计不同的压电驱动器对各种边界条件、不同基体材料的压电层合板进行静变形控制和形状最优控制的算例分析,结果表明本文建立的新模型正确,能够有效的控制结构变形。样条无网格法具有精度高、输入简单、运行效率高、处理边界条件简便等优点。
2.对压电智能层合板的变形控制进行了解析法研究,提出了符合压电材料正逆压电效应特性的压电层表面的电学边界条件。针对考虑一阶剪切影响的四边简支压电层合板,根据电学平衡方程及电学边界条件推导得到了压电层沿厚度方向电势分布为双曲函数的变化规律。算例讨论了压电层合板在机械荷载和电荷载作用下的变形、电势分布,并对结构变形进行了开环和闭环控制,结果表明推导的解析解与样条无网格解能够互相印证,吻合很好。由于单片压电驱动器控制力不足,为了提高驱动效率,对书本式压电驱动器驱动力的解析解进行了公式推导,建立了压电片层数n与压电驱动器的驱动力Mxp非线性的量化关系。算例分析表明,书本式压电驱动器的控制效果较好,能够应用于压电层合板及钢框架结构的变形控制中。
3.建立了压电材料参数识别分析的新模型。将材料参数识别的问题转化为极小化目标函数的问题,目标函数定义为测量位移与样条无网格法计算的相应位移之差的平方和；推导了基于样条无网格法求解位移值相对于材料各参数导数的灵敏度计算公式,采用基于信赖域技巧的Levenberg-Marquardt方法极小化目标函数；在参数识别过程中,以样条无网格方法计算的理论位移为真值,以给定方差下的随机正态分布数据模拟带误差的测量位移。研究了压电复合材料板分别在机械荷载及电荷载作用下,基体材料和压电材料的参数识别问题,算例表明本文提出的材料参数识别方法具有较高的精度和较好的稳定性,是行之有效的。
4.基于高阶剪切变形理论,推导了一种新的可以考虑剪切影响、压电效应、初始几何缺陷及P-△效应的压电智能梁柱单元。当不考虑剪切影响和压电效应时,新单元的刚度矩阵可以退化为线弹性情况下的单刚形式；通过引入初始几何缺陷影响系数的方法,可将初始几何缺陷与P-△效应联合分析,建立了相应的单元几何刚度矩阵。算例结果表明新单元模型正确,为压电智能框架结构建模奠定了理论研究基础。
5.基于新的压电智能梁柱单元刚度矩阵,采用样条QR方法建立了压电智能框架结构的动力分析新模型；利用模态控制理论,运用LQR最优控制方法,建立了压电智能框架结构振动主动控制计算模型；将压电堆式驱动器布置在框架结构柱上,对压电钢框架结构进行了振动主动控制仿真分析,讨论了结构的P-△效应和初始几何缺陷对结构自振频率及控制力的影响。算例结果表明,本文建立的分析模型能够有效的抑制结构的振动；考虑P-△效应和初始几何缺陷后,结构的自振频率减小,控制电压明显增加,说明这两个因素对结构振动主动控制的影响是不可忽略的,分析初始几何缺陷和P-△效应对结构振动控制的影响很有意义。
本文进行了大量的数值模拟计算,将基于新方法的分析结果与解析解和有限元解进行了比对,结果表明建立的新模型是正确和有效的,具有输入简单、计算精度高、稳定性好、物理概念清晰、处理边界条件方便、计算量少,运行速度快等优点。本文采用新方法对压电智能复合材料层合板及钢框架结构开展形状控制、振动控制、优化分析以及参数识别的理论研究和仿真分析,具有重要的理论和现实意义,提出的分析方法及得到的结论具有参考价值。
Smart structures are widely used in the military aviation and aerospace engineering due to their superior properties, such as self-diagnosis, adaptive, self-repair, etc. With the development of thorough investigation and improving theory, applications of smart structures have been extended to civil engineering, shipbuilding, automobile and other industries. A good application prospect of smart structures has been unfolded for the structural health monitoring, damage self-healing and vibration, noise and shape control in the fields of aviation, aerospace, submarines, high-speed trains, automobiles, bridges, dams and architecture. In recent years, a great deal of attention has been paid to the research and application of smart structures in the world's major developed countries, and therefore the research and application of smart structures has been classified as one of the priority development areas in terms of the21st century high-tech industries.
Piezoelectric materials have a lot of advantages such as wide frequency response range, fast response, large compactness, high accuracy, good linear behavior, etc. and they can be used as not only sensors but also actuators. The smart control in structures can be realized by way of the unique direct and inverse piezoelectric effect of piezoelectric materials. Over the past decades, extensive studies have been carried out in the basic and applied research of piezoelectric smart structures and a wealth of research achievements have been made.
At present, the studies of piezoelectric smart structures are mainly concerned with coupling theory, shape control, vibration control, noise control, optimization analysis, fault diagnosis and monitoring, analysis methods and experiment. Based on the spline meshless method and QR method founded by Professor Qin Rong in Guangxi University, new analysis models for piezoelectric smart laminated plates and piezoelectric frame structures are proposed to study the static deformation control, optimal shape control, active vibration control and parameter identification of piezoelectric material structures. The main work and innovations of the research in this paper are as follows:
1. Based on Reddy's third order plate theory and Hamilton variational principle, a new model of the static deformation control for piezoelectric smart composite plate is developed and the stiffness matrix is deduced by using spline meshless method. A model of the optimal shape control for piezoelectric smart composite plate is derived according to the sensitivity matrix which is based on the spline meshless discrete model with respect to the driving voltage and location of the actuator layer. By designing different piezoelectric actuators, various boundary conditions and different matrix materials, some numerical examples of open-loop control and closed-loop control are given. The numerical results show that the developed model is correct and effective for shape control. The spline meshless method has the advantages of high precision, simple input, high efficiency and concise boundary conditions.
2. The study of the deformation control for piezoelectric smart laminated plate by the analytical method and the find of the electrical boundary conditions of piezoelectric layer in line with the unique features of direct and inverse piezoelectric effect. Based on the electrical balance equation and electrical boundary conditions, for the simply supported piezoelectric laminated plates considering the first shear impact, the deduced electric potential distribution is expressed as a hyperbolic function along the thickness. Numerical examples are given to analyze the deformation, electric potential distribution, open-loop and closed-loop control of the piezoelectric laminated plate under mechanical loads and electric loads. The results illustrate that the analytical solutions are cross-checked with those of spline meshless method and the agreement between them is satisfactory. Due to the insufficient controlling power of monolithic piezoelectric actuator, the analytical formula of driving force for the book-block piezoelectric actuator is derived to improve the driving efficiency. Moreover, the non-linear quantitative relationship between piezoelectric film layers (n) and the driving force of the piezoelectric actuators (Mxp) is established. Examples show that better control is obtained by the book-block piezoelectric actuator. So, it can be applied to the deformation control of piezoelectric laminated plate and steel frame structure.
3. A new model for parameters identification of piezoelectric material. Based on the displacement modeled by the spline meshless method, the parameter identification problem is formulated as the problem of minimizing the objective function which is defined as a square sum of differences between the measured displacement and the computed displacement by the spline meshless method. The sensitivity matrix is calculated by the derivation of parameters to be identified. The calculation formula of sensitivity is deduced based on the displacement values obtained by the spline meshless method in line with the derivative of each material parameter. Levenberg-Marquardt method with the trust-region based techniques, is used to solve the minimization problem. In the process of parameter identification, the calculated results obtained by the spline meshless method using the true values of material parameters replace the measured data on displacements. However, normal distribution noises are added to simulate errors in measurement. For piezoelectric composite plate under the mechanical loads and electrical loads, the identification of matrix material parameter and piezoelectric parameter are investigated. Numerical examples show that the proposed method for parameter identification in this paper has high accuracy and good stability, and is effective.
4. Based on Reddy's third order theory, a new piezoelectric smart beam-column element is derived by considering the shear deformation effect, piezoelectric effect, initial geometric imperfection and P-A effect. When the shear effect and piezoelectric effect are not considered, the element geometric stiffness matrix derived in this paper for piezoelectric smart beam-column element derived can be degenerated to the one for the linear elastic case. With introducing its influence coefficient, the initial geometric imperfection is jointly analyzed with P-Δ effect and the corresponding element stiffness matrix is established. The correctness of this beam-column element is illustrated by some numerical examples. This work lays a theoretical foundation for the modeling of piezoelectric smart frame structure.
5. Based on the piezoelectric smart beam-column element stiffness matrix derived in the above, the new dynamic analysis model for piezoelectric frame structure is developed by the spline QR method. The mode control theory and LQR optimal control method are used to establish the calculation model of active vibration control for piezoelectric frame structure. The influence of the P-A effect and initial geometric imperfection on the structural natural frequency and control power is discussed by way of laying piezoelectric stack actuators on the column. The simulation analysis of active vibration control for piezoelectric laminated beam and piezoelectric-steel frame structure is given and the results demonstrate that the vibration of the structure can be effectively suppressed by using this analysis model. With considering the P-Δ effect and initial geometric imperfection, the natural frequency of the structure decreases and the control voltage significantly increases. It is obvious that the influence of these two factors on the active vibration control can't be ignored and its study is of great significance.
By comparing with analytical solution and finite element solution, numerical results illustrate that the new model established by the proposed methods in this paper is correct and effective and those methods have a lot of advantages, such as simple input, high accuracy, good stability, clear physical concept, concise boundary conditions, less computation, run fast, etc. In this paper, the new methods are used to study shape control, vibration control, optimization analysis, and parameter identification for piezoelectric smart composite laminates and steel frame structures, which is of great significance in theory and practice. The proposed methods as well as the obtained conclusions have an important reference value.
引文
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