Volterra级数理论及其应用研究
发布时间：2018-07-31 12:20
【摘要】：众所周知,工程中的实际系统几乎总含有各种各样的非线性因素,例如机械系统中的间隙、干摩擦、轴承油膜,结构系统的大变形、非线性材料本构关系,控制系统的非线性控制策略等等。线性系统是为了分析的方便对精度要求较低或系统非线性对系统性能影响不大的系统一种简化模型。通常,线性系统模型可对实际系统动力学行为进行很好的逼近。然而,近年来,随着科学技术的发展和进步,对系统性能要求的不断提高,使得这种线性逼近并非总是可靠的,被忽略的非线性因素有时会在分析和计算中引起无法接受的误差。而且,工程当中越来越多的非线性现象也引起了人们的重视,非线性问题已经成为当前研究的热点问题之一。因此,有必要对非线性系统进行非线性研究,揭示非线性系统的本质,这对进行非线性系统的分析与设计具有重要的意义。 近几十年来,经过众多学者的努力,已经发展出了许多分析非线性系统的方法,例如平均法、KBM法、摄动法、多尺度法、谐波平衡法。然而,利用Volterra级数理论对非线性系统进行分析还是比较新颖的,而且该方法拥有许多其它方法所没有的优点。基于此,本文将详细地介绍了如何利用Volterra级数分析方法来分析非线性系统。 Volterra级数是一种描述非线性系统输入与输出之间关系的数学泛函。它是研究非线性系统的一种重要数学工具,它可看作是线性系统中的卷积运算在非线性系统分析中的扩展。同时,Volterra级数可看作是具有存储(记忆)能力的Taylor级数,能够用来描述一类非线性系统。Volterra级数是一个基于核函数的无穷项级数,利用核函数与系统输入的高阶卷积级数来得到系统的输出。虽然Volterra级数是无穷阶级数,但研究表明,实际当中有一大类非线性系统均可通过有限阶次的Volterra级数来表示,所以如果表示非线性系统的Volterra级数是收敛的,那么可以用一个截断的Volterra级数来近似分析非线性系统。 本文的主要内容如下:第一章主要介绍了Volterra级数研究的意义、目的以及Volterra级数国内外的研究现状,结构损伤识别的研究现状及其存在的问题。第二章介绍了Volterra级数完整以及截断的表达形式、广义频率响应函数的定义、基于谐波探测法的广义频率响应函数求解方法以及广义频率响应函数的一般递推算法、非线性输出频率响应函数的定义及其数值求解方法、输出频率响应函数的定义及其求解方法,以及NARMAX模型的相关理论—主要包括NARMAX模型表达式、基于正交最小二乘算法对关键项进行选择、模型有效性验证等内容。第三章基于Volterra级数和广义频率响应函数研究了非线性系统的随机振动频率响应,它主要包括三个部分:第一,推导出了受非确定信号激励的非线性系统输出功率谱的一般表达式;第二,基于非线性系统输出功率谱的一般表达式,研究了激励强度对非线性系统输出功率谱的影响;第三,研究了非线性系统的非线性参数对系统输出功率谱的影响。第四章主要介绍了利用NARMAX模型以及非线性输出频响函数进行损伤检测的理论基础,并利用数值仿真研究和实验研究证实了该方法能够有效地检测出结构是否存在损伤,这对于工程结构系统的健康监测具有重要的意义。第五章主要介绍了基于非线性输出频响函数对周期结构当中的非线性部件进行定位的理论基础,并利用数值仿真研究以及实验研究证实了该非线性定位方法的可行性以及高效性。另外,由于结构系统发生损伤后,一般会产生非线性特征,因此可以根据检测出的结构产生非线性的位置判断结构产生损伤的位置,这对于实际工程应用具有重要的指导意义。第六章主要是全文工作的总结以及对未来工作的展望。
[Abstract]:It is well known that the actual system in the project almost always contains a variety of nonlinear factors, such as the gap in the mechanical system, the dry friction, the bearing oil film, the large deformation of the structural system, the nonlinear material constitutive relation, the nonlinear control strategy of the control system and so on. The linear system is for the convenience of analysis to the lower precision or the system. A simplified model of a system that has little effect on the performance of the system. Usually, the linear system model can make a good approximation to the dynamic behavior of the system. However, in recent years, with the development and progress of science and technology, the requirement of the system performance is constantly improved, which makes this linear approximation not always reliable and neglected. Sexual factors sometimes cause unacceptable errors in analysis and calculation. Moreover, more and more nonlinear phenomena in the project have attracted people's attention. The nonlinear problem has become one of the hot issues in current research. Therefore, it is necessary to study nonlinear systems and reveal the essence of nonlinear systems. It is of great significance to analyze and design nonlinear systems.
In the past few decades, many methods have been developed to analyze nonlinear systems, such as the mean method, KBM method, perturbation method, multiscale method, harmonic balance method. However, the nonlinear system is analyzed by the Volterra series theory or more new, and the method has many other methods. Based on this, this paper will introduce in detail how to use Volterra series analysis method to analyze nonlinear systems.
Volterra series is a mathematical functional that describes the relationship between input and output of nonlinear systems. It is an important mathematical tool for studying nonlinear systems. It can be regarded as an extension of convolution operations in linear systems in nonlinear system analysis. At the same time, Volterra series can be seen as a Taylor series with memory (memory) ability. It can be used to describe the.Volterra series of a class of nonlinear systems, which is an infinite series based on the kernel function, and the output of the system is obtained by the high order convolution series of the kernel function and the system input. Although the Volterra series is an infinite class number, the study shows that there is a large class of nonlinear systems in the reality which can pass the finite order of Volte The RRA series is expressed, so if the Volterra series of the nonlinear system is convergent, then a truncated Volterra number can be used to approximate the nonlinear system.
The main contents of this paper are as follows: the first chapter mainly introduces the significance of Volterra series research, the purpose of the study of Volterra series, the status of the research on structural damage identification and the existing problems. The second chapter introduces the complete and truncated expressions of Volterra series, the definition of the generalized frequency response function, and the harmonic based on the harmonics. The generalized frequency response function solution method of wave probe method, general recurrence algorithm of generalized frequency response function, definition of nonlinear output frequency response function and numerical solution method, definition and solution method of output frequency response function, and related theory of NARMAX model, mainly including NARMAX model expression, are based on The third chapter studies the random vibration frequency response of the nonlinear system based on the Volterra series and the generalized frequency response function. It mainly includes three parts: first, the output power spectrum of a nonlinear system excited by a non deterministic signal is derived. Second, based on the general expression of the output power spectrum of the nonlinear system, the influence of the excitation intensity on the output power spectrum of the nonlinear system is studied. Third, the influence of the nonlinear parameters of the nonlinear system on the output power spectrum of the system is studied. The fourth chapter mainly introduces the use of the NARMAX model and the nonlinear output frequency response function. The theoretical basis of damage detection is carried out, and the numerical simulation research and experimental research have proved that the method can effectively detect whether the structure has damage, which is of great significance for the health monitoring of the engineering structure system. The fifth chapter mainly introduces the nonlinearity of the nonlinear frequency response function to the periodic structure. The theoretical basis of the location of the component is carried out, and the feasibility and efficiency of the nonlinear positioning method are confirmed by numerical simulation and experimental research. In addition, the nonlinear characteristics are generally produced because of the damage of the structural system. Therefore, a nonlinear location judgment structure can be produced to produce damage according to the detected structure. The sixth chapter is the summary of the whole paper and the prospect of the future work.
【学位授予单位】：上海交通大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：TH165.3
[Abstract]:It is well known that the actual system in the project almost always contains a variety of nonlinear factors, such as the gap in the mechanical system, the dry friction, the bearing oil film, the large deformation of the structural system, the nonlinear material constitutive relation, the nonlinear control strategy of the control system and so on. The linear system is for the convenience of analysis to the lower precision or the system. A simplified model of a system that has little effect on the performance of the system. Usually, the linear system model can make a good approximation to the dynamic behavior of the system. However, in recent years, with the development and progress of science and technology, the requirement of the system performance is constantly improved, which makes this linear approximation not always reliable and neglected. Sexual factors sometimes cause unacceptable errors in analysis and calculation. Moreover, more and more nonlinear phenomena in the project have attracted people's attention. The nonlinear problem has become one of the hot issues in current research. Therefore, it is necessary to study nonlinear systems and reveal the essence of nonlinear systems. It is of great significance to analyze and design nonlinear systems.
In the past few decades, many methods have been developed to analyze nonlinear systems, such as the mean method, KBM method, perturbation method, multiscale method, harmonic balance method. However, the nonlinear system is analyzed by the Volterra series theory or more new, and the method has many other methods. Based on this, this paper will introduce in detail how to use Volterra series analysis method to analyze nonlinear systems.
Volterra series is a mathematical functional that describes the relationship between input and output of nonlinear systems. It is an important mathematical tool for studying nonlinear systems. It can be regarded as an extension of convolution operations in linear systems in nonlinear system analysis. At the same time, Volterra series can be seen as a Taylor series with memory (memory) ability. It can be used to describe the.Volterra series of a class of nonlinear systems, which is an infinite series based on the kernel function, and the output of the system is obtained by the high order convolution series of the kernel function and the system input. Although the Volterra series is an infinite class number, the study shows that there is a large class of nonlinear systems in the reality which can pass the finite order of Volte The RRA series is expressed, so if the Volterra series of the nonlinear system is convergent, then a truncated Volterra number can be used to approximate the nonlinear system.
The main contents of this paper are as follows: the first chapter mainly introduces the significance of Volterra series research, the purpose of the study of Volterra series, the status of the research on structural damage identification and the existing problems. The second chapter introduces the complete and truncated expressions of Volterra series, the definition of the generalized frequency response function, and the harmonic based on the harmonics. The generalized frequency response function solution method of wave probe method, general recurrence algorithm of generalized frequency response function, definition of nonlinear output frequency response function and numerical solution method, definition and solution method of output frequency response function, and related theory of NARMAX model, mainly including NARMAX model expression, are based on The third chapter studies the random vibration frequency response of the nonlinear system based on the Volterra series and the generalized frequency response function. It mainly includes three parts: first, the output power spectrum of a nonlinear system excited by a non deterministic signal is derived. Second, based on the general expression of the output power spectrum of the nonlinear system, the influence of the excitation intensity on the output power spectrum of the nonlinear system is studied. Third, the influence of the nonlinear parameters of the nonlinear system on the output power spectrum of the system is studied. The fourth chapter mainly introduces the use of the NARMAX model and the nonlinear output frequency response function. The theoretical basis of damage detection is carried out, and the numerical simulation research and experimental research have proved that the method can effectively detect whether the structure has damage, which is of great significance for the health monitoring of the engineering structure system. The fifth chapter mainly introduces the nonlinearity of the nonlinear frequency response function to the periodic structure. The theoretical basis of the location of the component is carried out, and the feasibility and efficiency of the nonlinear positioning method are confirmed by numerical simulation and experimental research. In addition, the nonlinear characteristics are generally produced because of the damage of the structural system. Therefore, a nonlinear location judgment structure can be produced to produce damage according to the detected structure. The sixth chapter is the summary of the whole paper and the prospect of the future work.
【学位授予单位】：上海交通大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：TH165.3
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