Volterra級數(shù)理論及其應(yīng)用研究
發(fā)布時間：2018-07-31 12:20
【摘要】：眾所周知,工程中的實(shí)際系統(tǒng)幾乎總含有各種各樣的非線性因素,例如機(jī)械系統(tǒng)中的間隙、干摩擦、軸承油膜,結(jié)構(gòu)系統(tǒng)的大變形、非線性材料本構(gòu)關(guān)系,控制系統(tǒng)的非線性控制策略等等。線性系統(tǒng)是為了分析的方便對精度要求較低或系統(tǒng)非線性對系統(tǒng)性能影響不大的系統(tǒng)一種簡化模型。通常,線性系統(tǒng)模型可對實(shí)際系統(tǒng)動力學(xué)行為進(jìn)行很好的逼近。然而,近年來,隨著科學(xué)技術(shù)的發(fā)展和進(jìn)步,對系統(tǒng)性能要求的不斷提高,使得這種線性逼近并非總是可靠的,被忽略的非線性因素有時會在分析和計算中引起無法接受的誤差。而且,工程當(dāng)中越來越多的非線性現(xiàn)象也引起了人們的重視,非線性問題已經(jīng)成為當(dāng)前研究的熱點(diǎn)問題之一。因此,有必要對非線性系統(tǒng)進(jìn)行非線性研究,揭示非線性系統(tǒng)的本質(zhì),這對進(jìn)行非線性系統(tǒng)的分析與設(shè)計具有重要的意義。 近幾十年來,經(jīng)過眾多學(xué)者的努力,已經(jīng)發(fā)展出了許多分析非線性系統(tǒng)的方法,例如平均法、KBM法、攝動法、多尺度法、諧波平衡法。然而,利用Volterra級數(shù)理論對非線性系統(tǒng)進(jìn)行分析還是比較新穎的,而且該方法擁有許多其它方法所沒有的優(yōu)點(diǎn)�；诖�,本文將詳細(xì)地介紹了如何利用Volterra級數(shù)分析方法來分析非線性系統(tǒng)。 Volterra級數(shù)是一種描述非線性系統(tǒng)輸入與輸出之間關(guān)系的數(shù)學(xué)泛函。它是研究非線性系統(tǒng)的一種重要數(shù)學(xué)工具,它可看作是線性系統(tǒng)中的卷積運(yùn)算在非線性系統(tǒng)分析中的擴(kuò)展。同時,Volterra級數(shù)可看作是具有存儲(記憶)能力的Taylor級數(shù),能夠用來描述一類非線性系統(tǒng)。Volterra級數(shù)是一個基于核函數(shù)的無窮項(xiàng)級數(shù),利用核函數(shù)與系統(tǒng)輸入的高階卷積級數(shù)來得到系統(tǒng)的輸出。雖然Volterra級數(shù)是無窮階級數(shù),但研究表明,實(shí)際當(dāng)中有一大類非線性系統(tǒng)均可通過有限階次的Volterra級數(shù)來表示,所以如果表示非線性系統(tǒng)的Volterra級數(shù)是收斂的,那么可以用一個截斷的Volterra級數(shù)來近似分析非線性系統(tǒng)。 本文的主要內(nèi)容如下:第一章主要介紹了Volterra級數(shù)研究的意義、目的以及Volterra級數(shù)國內(nèi)外的研究現(xiàn)狀,結(jié)構(gòu)損傷識別的研究現(xiàn)狀及其存在的問題。第二章介紹了Volterra級數(shù)完整以及截斷的表達(dá)形式、廣義頻率響應(yīng)函數(shù)的定義、基于諧波探測法的廣義頻率響應(yīng)函數(shù)求解方法以及廣義頻率響應(yīng)函數(shù)的一般遞推算法、非線性輸出頻率響應(yīng)函數(shù)的定義及其數(shù)值求解方法、輸出頻率響應(yīng)函數(shù)的定義及其求解方法,以及NARMAX模型的相關(guān)理論—主要包括NARMAX模型表達(dá)式、基于正交最小二乘算法對關(guān)鍵項(xiàng)進(jìn)行選擇、模型有效性驗(yàn)證等內(nèi)容。第三章基于Volterra級數(shù)和廣義頻率響應(yīng)函數(shù)研究了非線性系統(tǒng)的隨機(jī)振動頻率響應(yīng),它主要包括三個部分:第一,推導(dǎo)出了受非確定信號激勵的非線性系統(tǒng)輸出功率譜的一般表達(dá)式;第二,基于非線性系統(tǒng)輸出功率譜的一般表達(dá)式,研究了激勵強(qiáng)度對非線性系統(tǒng)輸出功率譜的影響;第三,研究了非線性系統(tǒng)的非線性參數(shù)對系統(tǒng)輸出功率譜的影響。第四章主要介紹了利用NARMAX模型以及非線性輸出頻響函數(shù)進(jìn)行損傷檢測的理論基礎(chǔ),并利用數(shù)值仿真研究和實(shí)驗(yàn)研究證實(shí)了該方法能夠有效地檢測出結(jié)構(gòu)是否存在損傷,這對于工程結(jié)構(gòu)系統(tǒng)的健康監(jiān)測具有重要的意義。第五章主要介紹了基于非線性輸出頻響函數(shù)對周期結(jié)構(gòu)當(dāng)中的非線性部件進(jìn)行定位的理論基礎(chǔ),并利用數(shù)值仿真研究以及實(shí)驗(yàn)研究證實(shí)了該非線性定位方法的可行性以及高效性。另外,由于結(jié)構(gòu)系統(tǒng)發(fā)生損傷后,一般會產(chǎn)生非線性特征,因此可以根據(jù)檢測出的結(jié)構(gòu)產(chǎn)生非線性的位置判斷結(jié)構(gòu)產(chǎn)生損傷的位置,這對于實(shí)際工程應(yīng)用具有重要的指導(dǎo)意義。第六章主要是全文工作的總結(jié)以及對未來工作的展望。
[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.
【學(xué)位授予單位】：上海交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2012
【分類號】：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.
【學(xué)位授予單位】：上海交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2012
【分類號】：TH165.3
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