结构故障诊断的几种方法
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摘要
本文将微分方程反问题理论应用于结构故障诊断。从微分方程反问题的角度,本文给出了关于原始刚度的微分方程(正问题)的求解方法,和关于由裂纹等故障引起的附加刚度的积分方程(反问题)的一组识别模型,由此来识别由疲劳裂纹等故障引起的结构的微小变化,从而可对裂纹的发生、发展进行在线检测。
本文通过Fourier变换,将求解微分方程参数的问题从时域转换到频域上,然后利用小扰动理论将这一反问题进行求解,转化为求解第一类Fredholm积分方程的问题。首先将第一类积分方程进行离散化,然后对Landweber方法进行改进,因为Landweber方法虽然是比较稳定的求解方法,但是计算量大,本文利用矩阵方面的知识对离散得到的矩阵进行处理使得Landweber方法的计算量大大减少,另外,本文模仿Landweber方法构造了一种适用于具有核对称的第一类积分方程的正则化策略并对其进行了有效的改进,使其计算效率大大增加。最后本文利用正则化同伦方法对第一类积分方程进行求解,同伦方法最主要的优点就是求解算子方程时是一种大范围收敛的方法,但是对于不适定的问题,它又克服不了算子方程的不适定性,而在处理反问题的不适定性方面,Tikhonov正则化方法无疑是最著名、最有效的一种方法,用正则化同伦方法求解第一类积分方程可以结合两者的优点。
The theory of inverse problems of a partial differential equation is used for the structure fault diagnosis in this paper. A set of identification models of a differential stiffness and of an integral equation on the additional stiffness are given to identify the small stiffness of a variational structure which is caused by the fault in the structure, thus may to the crack of occurrence, development carry on the online examination.
This paper through the Fourier transformation, transforms the differential equation of parameter question from the time domain to the frequency domain, then utilizes the small perturbance theory to solve inverse problem. The question is transformed to solve the first kind of Fredholm integral equation. At first, this paper takes the first kind of integral equation to discrete, and takes the Landweber method to improve. Because Landweber method is a well stable solution method, but the computational burden is big. This paper utilizes the knowledge of matrix to reduce greatly the computational burden of the Landweber method. Secondly, this paper imitates Landweber method to structure one kind regularization strategy which is suit in having the symmetric Kernel about the first kind of integral equation and has made the effective improvement to it, which improve computational efficiency obviously. Finally, this paper uses the regularization homotopy method to solve the first kind of integral equation. The most main merit of homotopy method is a wide range convergence method, but regarding to the ill-posed question, it can not overcome ill-posedness of the operator equation. In conquering ill-posed aspect of inverse problem, Tikhonov regularization method is without doubt most famous and the most effective method. The first kind of integral equation with the regularization homotopy method to solve is possible to unify both the merits.
本文通过Fourier变换,将求解微分方程参数的问题从时域转换到频域上,然后利用小扰动理论将这一反问题进行求解,转化为求解第一类Fredholm积分方程的问题。首先将第一类积分方程进行离散化,然后对Landweber方法进行改进,因为Landweber方法虽然是比较稳定的求解方法,但是计算量大,本文利用矩阵方面的知识对离散得到的矩阵进行处理使得Landweber方法的计算量大大减少,另外,本文模仿Landweber方法构造了一种适用于具有核对称的第一类积分方程的正则化策略并对其进行了有效的改进,使其计算效率大大增加。最后本文利用正则化同伦方法对第一类积分方程进行求解,同伦方法最主要的优点就是求解算子方程时是一种大范围收敛的方法,但是对于不适定的问题,它又克服不了算子方程的不适定性,而在处理反问题的不适定性方面,Tikhonov正则化方法无疑是最著名、最有效的一种方法,用正则化同伦方法求解第一类积分方程可以结合两者的优点。
The theory of inverse problems of a partial differential equation is used for the structure fault diagnosis in this paper. A set of identification models of a differential stiffness and of an integral equation on the additional stiffness are given to identify the small stiffness of a variational structure which is caused by the fault in the structure, thus may to the crack of occurrence, development carry on the online examination.
This paper through the Fourier transformation, transforms the differential equation of parameter question from the time domain to the frequency domain, then utilizes the small perturbance theory to solve inverse problem. The question is transformed to solve the first kind of Fredholm integral equation. At first, this paper takes the first kind of integral equation to discrete, and takes the Landweber method to improve. Because Landweber method is a well stable solution method, but the computational burden is big. This paper utilizes the knowledge of matrix to reduce greatly the computational burden of the Landweber method. Secondly, this paper imitates Landweber method to structure one kind regularization strategy which is suit in having the symmetric Kernel about the first kind of integral equation and has made the effective improvement to it, which improve computational efficiency obviously. Finally, this paper uses the regularization homotopy method to solve the first kind of integral equation. The most main merit of homotopy method is a wide range convergence method, but regarding to the ill-posed question, it can not overcome ill-posedness of the operator equation. In conquering ill-posed aspect of inverse problem, Tikhonov regularization method is without doubt most famous and the most effective method. The first kind of integral equation with the regularization homotopy method to solve is possible to unify both the merits.
引文
1 倪金福. 用卡尔曼滤波技术识别振系统参数. 振动与冲击. 1982, (2): 23~36
2 S. R. Ibrahim, E. C. Mikulcik. A Time Domain Modal Vibration Test Technique, The Shock and Vibration Bulletin. 1973, 43(4): 43~57
3 苏超伟. 偏微分方程逆问题的数值方法及其应用. 西北工业大学出版社, 19- 95:3~4
4 江泽坚, 吴智泉, 实变函数论. 高等教育出版社, 1994:144~161
5 A. N. Tikhonv, V. Y. Arenin. Solutions of Ill-Posed Problems. John Siley & Sons, 1977:136~163
6 Andreas. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996:1~233
7 潘军峰, 闵涛. 对流扩散方程逆过程反问题的稳定性及数值解. 武汉大学学报. 2005, 38(1):10~13
8 郭庆平, 王伟沧. 不适定问题研究的若干进展. 武汉理工大学学报. 2005, 25 (1):12~15
9 杨吉凡, 宋守根. 积分方程反演新解法. 中南工业大学学报. 1998, 29(5):416 ~417
10 李开泰, 黄艾香, 黄庆怀. 有限元方法及其应用, 2006:224~287
11 王德明, 刘家琦. 求解微分方程反问题的有限元技术. 应用数学与计算数学学报.1988, 2(2):26~34
12 韩波, 匡正, 刘家琦. 反演地层电阻率的单调同伦法. 地球物理学报. 1991, 34(4):97~112
13 张丽芹, 王家映, 严德天. 一维波动方程波阻抗反演的同伦方法. 地球物理学报. 2004, 47(6):1111~1117
14 G. F. Feng, B. Han, J. Q. Liu. A Widely Convergent Generalized Pulse Spec- trum Methods for 2-D Wave Equation Inversion. Chinese J. Geophys. 2003, 46(2):373~ 380
15 D. W. Vasco. Singularity and branching: a path-following formalism for geophysical inverse problems. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts. 1995, 32(6):273～274
16 D. W. Vasco. Regularization and Trade-Off Associated with Nonlinear Geophys-ical Inverse Problems: Penalty Homotopies. Inverse Problems. 1998, 14(4):103- 3~1052
17 M. D. Jegen, M. E. Everett, A. Schultz. Using Homotopy to Invert Geophysical Data. Geophysics, 2001, 66(6):1749~1760
18 匡启和, 刘建业. 基于小波变化的余度敏感器结构故障诊断方法. 南京航空航天大学学报. 2001, 33(4):329~333
19 D. E. Newland. Wavelet Analysis of Vibration. Journal of Vibration and Acousti- cs. 1994, 116(4):409~416
20 W. J. Staszewski, G. R. Tomlinson. Application of the Wavelet Transform to Fault Detection in Spur Gear. Mechanical systems and signal proceccing. 1994, 8(3): 289~307
21 赵纪元, 何正嘉. 小波包自回归谱分析及在振动诊断中的应用. 振动工程学报. 1994, 8(3):198~203
22 A. Rierer. A Wavelet Multilevel Method for Ill-Posed Problems Stabilized by Tikhonov Regularization. Numer Math. 1997, 75(4):501~522
23 D. L. Donoho. Nonlinear Solution of Linear Inverse Problems by Wavelets Vagu- elette decomposition. Applied Computational Analysis. 1995, 32(5):101~126
24 凌捷. 求解第一类积分方程的正则化-小波方法及其数值试验. 高等学校计算数学学报. 1998, (3):215~231
25 王德明, 刘家琦, 黄文虎. 适用于结构故障诊断中的微分方程反问题方法.振动与冲击. 1986, (3):19~26
26 D. M. Wang, B. Z. Gai. Way to Determine the Stiffness Function of Structure. Applied Mathematics and Mechanics. 2005, 26(12):1453~1458
27 戴嘉尊, 邱建贤. 微分方程数值解法. 东南大学出版社, 2002:42~56
28 徐长发, 李红. 实用偏微分方程数值解法. 华中科技大学出版社, 2000:9~16
29 Walter Rudin. Functional Analysis. 刘培德译. 机械工业出版社, 2004:137~154
30 肖庭延, 于慎根, 王彦飞. 反问题的数值解法. 科学出版社, 2003:93~126
31 H. W. Engl. Regularizations Methods for the Stable Solution of Inverse Problems. Surv. Math. Ind. 1993, 21(3):71~143
32 D. L. Phillips. A Technique for The Numerical Solution of Certain Integral Equa- utions of the First Kind. J. ACM. 1962, 9:84~97
33 M. Hanke. Accelerated Landweber Iterations for the Solution of Ill-Posed Equati- ons. Numer. Math. 1991, 60(2):341~373
34 A. Neubauer. On Landweber Iteration for Nonlinear Ill-Posed Problems in Hilbert Cales. Numer Math. 2000, 85(2):309~328.
35 H. W. Engl. Discrepancy Principles for Tikhonov Regularization of Ill-posed Pro- blems Leading to Optimal Convergence Rates. Optimal Theory Appl. 1987, 52(2): 209~215
36 H. W. Engl, A. Neubauer. Optimal Discrepancy Principle for the Tikhonov Regu- larization of Integral Equations of the First Kind. Brikhauser Verlag, 1985:120~1 41
37 H. J. Lai, G. Q. He. Simplified Iterative Tikhonov Regularization and Posteriori Parameter Choice Rules. Journal of Shanghai University. 2005, l9(4):314~319 ed Regularization of Ill-Posed Problems. Austria Math Science. 1994, 36:242~24 8
38 E. Schock. On the Asymptotic Order of Accuracy of Tikhonov Regularization, O- ptimal Theory Appl. 1984, 44(1):95~104
39 朱佑彬, 傅初黎, 邱春雨. 一类不适定问题具备停止规则的简化迭代技巧. 兰州大学学报. 2002, 38(2):1~6
40 Tikhonov, Aseninvy. Solutions of Ill - Posed Problems. New York: Winston Wil- ey, 1997:136~163.
41 罗兴钧, 陈仲英. 第一类算子方程的一种新的迭代正则化方法. 高校应用数学学报. 2006, 21(2):223~230
42 H. W. Engl, Hankem, A. Neubauer. Regularization of Inverse problems . Kluwer Academic Publishers, 1996:7~43
43 B. Blaschke. Some Newton Type Methods for the Regularization of Nonlinear il- l–Posed Problems. Inverse Problems. 1997, 13(3):729~753.
44 Ramlaur. A Steepest Descent Algorithm for the Global Minimization of the Tikh- onov-functional. Inverse Problems.2002, 18(2):381~405.
45 Doicua, Schreierf, Hesm. Iteratively Regularized Gauss–Newton Method for Bo- und-Constraint Problems in Atmospheric Remote Sensing. Computer Physics C- ommunications. 2003, 153(1):59~65.
46 魏培君, 章梓茂. 弹性动力学反问题的数值反演方法. 力学进展. 2001, 31(2): 172~180
47 尤承业. 基础拓扑学讲义. 北京大学出版社, 1997:103~115
2 S. R. Ibrahim, E. C. Mikulcik. A Time Domain Modal Vibration Test Technique, The Shock and Vibration Bulletin. 1973, 43(4): 43~57
3 苏超伟. 偏微分方程逆问题的数值方法及其应用. 西北工业大学出版社, 19- 95:3~4
4 江泽坚, 吴智泉, 实变函数论. 高等教育出版社, 1994:144~161
5 A. N. Tikhonv, V. Y. Arenin. Solutions of Ill-Posed Problems. John Siley & Sons, 1977:136~163
6 Andreas. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996:1~233
7 潘军峰, 闵涛. 对流扩散方程逆过程反问题的稳定性及数值解. 武汉大学学报. 2005, 38(1):10~13
8 郭庆平, 王伟沧. 不适定问题研究的若干进展. 武汉理工大学学报. 2005, 25 (1):12~15
9 杨吉凡, 宋守根. 积分方程反演新解法. 中南工业大学学报. 1998, 29(5):416 ~417
10 李开泰, 黄艾香, 黄庆怀. 有限元方法及其应用, 2006:224~287
11 王德明, 刘家琦. 求解微分方程反问题的有限元技术. 应用数学与计算数学学报.1988, 2(2):26~34
12 韩波, 匡正, 刘家琦. 反演地层电阻率的单调同伦法. 地球物理学报. 1991, 34(4):97~112
13 张丽芹, 王家映, 严德天. 一维波动方程波阻抗反演的同伦方法. 地球物理学报. 2004, 47(6):1111~1117
14 G. F. Feng, B. Han, J. Q. Liu. A Widely Convergent Generalized Pulse Spec- trum Methods for 2-D Wave Equation Inversion. Chinese J. Geophys. 2003, 46(2):373~ 380
15 D. W. Vasco. Singularity and branching: a path-following formalism for geophysical inverse problems. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts. 1995, 32(6):273～274
16 D. W. Vasco. Regularization and Trade-Off Associated with Nonlinear Geophys-ical Inverse Problems: Penalty Homotopies. Inverse Problems. 1998, 14(4):103- 3~1052
17 M. D. Jegen, M. E. Everett, A. Schultz. Using Homotopy to Invert Geophysical Data. Geophysics, 2001, 66(6):1749~1760
18 匡启和, 刘建业. 基于小波变化的余度敏感器结构故障诊断方法. 南京航空航天大学学报. 2001, 33(4):329~333
19 D. E. Newland. Wavelet Analysis of Vibration. Journal of Vibration and Acousti- cs. 1994, 116(4):409~416
20 W. J. Staszewski, G. R. Tomlinson. Application of the Wavelet Transform to Fault Detection in Spur Gear. Mechanical systems and signal proceccing. 1994, 8(3): 289~307
21 赵纪元, 何正嘉. 小波包自回归谱分析及在振动诊断中的应用. 振动工程学报. 1994, 8(3):198~203
22 A. Rierer. A Wavelet Multilevel Method for Ill-Posed Problems Stabilized by Tikhonov Regularization. Numer Math. 1997, 75(4):501~522
23 D. L. Donoho. Nonlinear Solution of Linear Inverse Problems by Wavelets Vagu- elette decomposition. Applied Computational Analysis. 1995, 32(5):101~126
24 凌捷. 求解第一类积分方程的正则化-小波方法及其数值试验. 高等学校计算数学学报. 1998, (3):215~231
25 王德明, 刘家琦, 黄文虎. 适用于结构故障诊断中的微分方程反问题方法.振动与冲击. 1986, (3):19~26
26 D. M. Wang, B. Z. Gai. Way to Determine the Stiffness Function of Structure. Applied Mathematics and Mechanics. 2005, 26(12):1453~1458
27 戴嘉尊, 邱建贤. 微分方程数值解法. 东南大学出版社, 2002:42~56
28 徐长发, 李红. 实用偏微分方程数值解法. 华中科技大学出版社, 2000:9~16
29 Walter Rudin. Functional Analysis. 刘培德译. 机械工业出版社, 2004:137~154
30 肖庭延, 于慎根, 王彦飞. 反问题的数值解法. 科学出版社, 2003:93~126
31 H. W. Engl. Regularizations Methods for the Stable Solution of Inverse Problems. Surv. Math. Ind. 1993, 21(3):71~143
32 D. L. Phillips. A Technique for The Numerical Solution of Certain Integral Equa- utions of the First Kind. J. ACM. 1962, 9:84~97
33 M. Hanke. Accelerated Landweber Iterations for the Solution of Ill-Posed Equati- ons. Numer. Math. 1991, 60(2):341~373
34 A. Neubauer. On Landweber Iteration for Nonlinear Ill-Posed Problems in Hilbert Cales. Numer Math. 2000, 85(2):309~328.
35 H. W. Engl. Discrepancy Principles for Tikhonov Regularization of Ill-posed Pro- blems Leading to Optimal Convergence Rates. Optimal Theory Appl. 1987, 52(2): 209~215
36 H. W. Engl, A. Neubauer. Optimal Discrepancy Principle for the Tikhonov Regu- larization of Integral Equations of the First Kind. Brikhauser Verlag, 1985:120~1 41
37 H. J. Lai, G. Q. He. Simplified Iterative Tikhonov Regularization and Posteriori Parameter Choice Rules. Journal of Shanghai University. 2005, l9(4):314~319 ed Regularization of Ill-Posed Problems. Austria Math Science. 1994, 36:242~24 8
38 E. Schock. On the Asymptotic Order of Accuracy of Tikhonov Regularization, O- ptimal Theory Appl. 1984, 44(1):95~104
39 朱佑彬, 傅初黎, 邱春雨. 一类不适定问题具备停止规则的简化迭代技巧. 兰州大学学报. 2002, 38(2):1~6
40 Tikhonov, Aseninvy. Solutions of Ill - Posed Problems. New York: Winston Wil- ey, 1997:136~163.
41 罗兴钧, 陈仲英. 第一类算子方程的一种新的迭代正则化方法. 高校应用数学学报. 2006, 21(2):223~230
42 H. W. Engl, Hankem, A. Neubauer. Regularization of Inverse problems . Kluwer Academic Publishers, 1996:7~43
43 B. Blaschke. Some Newton Type Methods for the Regularization of Nonlinear il- l–Posed Problems. Inverse Problems. 1997, 13(3):729~753.
44 Ramlaur. A Steepest Descent Algorithm for the Global Minimization of the Tikh- onov-functional. Inverse Problems.2002, 18(2):381~405.
45 Doicua, Schreierf, Hesm. Iteratively Regularized Gauss–Newton Method for Bo- und-Constraint Problems in Atmospheric Remote Sensing. Computer Physics C- ommunications. 2003, 153(1):59~65.
46 魏培君, 章梓茂. 弹性动力学反问题的数值反演方法. 力学进展. 2001, 31(2): 172~180
47 尤承业. 基础拓扑学讲义. 北京大学出版社, 1997:103~115
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