基于模态分析的轨道系统参数识别
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摘要
随着我国轨道交通的不断发展，列车运行速度不断提高，轨道系统振动的强度也随之增加，从而加速了轨道系统的变形和破坏。轨道振动是导致轨道失效的主要原因，高速铁路对轨道的性能提出了更高的要求，因此需要对轨道系统的动力特性有更深入更全面的了解。轨道结构是一个非常复杂的结构，一般由铁轨、扣件、枕木、碎石垫层组成，利用模态分析技术建立轨道结构动力学模型并辨识其参数是一条有效的途径为此，本文在模态分析理论的基础上主要研究了以下几个方面的内容：
1、研究振动系统的物理参数模型、模态参数模型和非参数模型的关系，即三种模型的理论建模问题。对粘性比例阻尼与一般粘性阻尼两种情况分别进行了多自由度系统的实模态与复模态分析。该部分是全文的理论基础。
2、以第二章的模态分析理论为基础，用ANSYS模拟了几种轨道模型，对建立的模型分别施加冲击荷载，简谐荷载和移动荷载，然后对得到的时域响应进行FFT变换，采用半功率法结合MATLAB编程，识别出几种模型的频率、阻尼和振型，并和理论计算结果进行了比较，得到了较满意的识别结果。这些结果对采用精确的频域法识别轨道模态参数具有很现实的意义。
3、首次将特征系统实现算法(ERA)引入轨道结构参数识别的研究中，详细推导了该方法的基本理论过程，用ANSYS建立了二维和三维轨道模型，采用MIMO的方法得到轨道模型的多点脉冲响应，构造hankel矩阵，结合MATLAB编程，识别出频率和阻尼比，并和理论计算值进行了对比，得到一些有用的结论。从结论中可以看出，该方法对轨道结构模态耦合密集的情况识别效果好，从而证明了运用该方法识别轨道模态参数的可行性。
最后，在全面总结论文工作的基础上，提出本课题尚待深入研究的若干问题。
With the development of track traffic, the deformation and destruction of track system is speeded up consequently. The major reason of track invalidation is the track vibration. Due to the higher demand of track capability, it is needed to search the dynamic characteristic of track system. In general the track structure consists of track, rail pad, sleeper and ballast and it is a very complex structure. It will be a valid method to constitute track structure dynamic modal and identify the modal parameters by modal analysis technology. Therefore based on modal analysis theory, the main ideas of this paper can be summarized as follows:1 The relations of physical parameter model, modal parameter model and non-parameter-model in vibration system have been researched. That s the problem of theoretic modeling of the three models. The real modal analysis and complex modal analysis are respectively applied to the multi-degree-of -freedom system according to the different damp. This chapter is the theory basis of the paper.2 Based on modal analysis theory of chapter two, several track models are simulated by ANSYS and the impact load, harmonic load and moving load are respectively put on the models. Then the time domain responses are transferred to the frequency domain by the FFT. Final, frequency, damp and mode shape of the several models are identified by half power method and MATLAB procedure. The identification results are compared with theory results, and the satisfied results have been got.3 The Eigensystem Realization Algorithm (ERA) is first applied to identify modal parameters of track structure in this paper. Then the basis theory process of this method has been deduced. The two-dimension track model and three-dimension track model are set up using ANSYS. After the several impulse responses are got by MIMO method. The frequency and damp ratio are estimated by constructing Hankel matrix and MATLAB procedure. Finally the identification results are compared with theory results, some useful conclusions have been got. From the results, we find out that this method is applicable to estimate the coupled-modes of railway system, therefore it is feasible to identify modal parameters of track by this method.
In the end, the content of this paper has been summarized and some problems, which need further study, have been proposed.
1、研究振动系统的物理参数模型、模态参数模型和非参数模型的关系，即三种模型的理论建模问题。对粘性比例阻尼与一般粘性阻尼两种情况分别进行了多自由度系统的实模态与复模态分析。该部分是全文的理论基础。
2、以第二章的模态分析理论为基础，用ANSYS模拟了几种轨道模型，对建立的模型分别施加冲击荷载，简谐荷载和移动荷载，然后对得到的时域响应进行FFT变换，采用半功率法结合MATLAB编程，识别出几种模型的频率、阻尼和振型，并和理论计算结果进行了比较，得到了较满意的识别结果。这些结果对采用精确的频域法识别轨道模态参数具有很现实的意义。
3、首次将特征系统实现算法(ERA)引入轨道结构参数识别的研究中，详细推导了该方法的基本理论过程，用ANSYS建立了二维和三维轨道模型，采用MIMO的方法得到轨道模型的多点脉冲响应，构造hankel矩阵，结合MATLAB编程，识别出频率和阻尼比，并和理论计算值进行了对比，得到一些有用的结论。从结论中可以看出，该方法对轨道结构模态耦合密集的情况识别效果好，从而证明了运用该方法识别轨道模态参数的可行性。
最后，在全面总结论文工作的基础上，提出本课题尚待深入研究的若干问题。
With the development of track traffic, the deformation and destruction of track system is speeded up consequently. The major reason of track invalidation is the track vibration. Due to the higher demand of track capability, it is needed to search the dynamic characteristic of track system. In general the track structure consists of track, rail pad, sleeper and ballast and it is a very complex structure. It will be a valid method to constitute track structure dynamic modal and identify the modal parameters by modal analysis technology. Therefore based on modal analysis theory, the main ideas of this paper can be summarized as follows:1 The relations of physical parameter model, modal parameter model and non-parameter-model in vibration system have been researched. That s the problem of theoretic modeling of the three models. The real modal analysis and complex modal analysis are respectively applied to the multi-degree-of -freedom system according to the different damp. This chapter is the theory basis of the paper.2 Based on modal analysis theory of chapter two, several track models are simulated by ANSYS and the impact load, harmonic load and moving load are respectively put on the models. Then the time domain responses are transferred to the frequency domain by the FFT. Final, frequency, damp and mode shape of the several models are identified by half power method and MATLAB procedure. The identification results are compared with theory results, and the satisfied results have been got.3 The Eigensystem Realization Algorithm (ERA) is first applied to identify modal parameters of track structure in this paper. Then the basis theory process of this method has been deduced. The two-dimension track model and three-dimension track model are set up using ANSYS. After the several impulse responses are got by MIMO method. The frequency and damp ratio are estimated by constructing Hankel matrix and MATLAB procedure. Finally the identification results are compared with theory results, some useful conclusions have been got. From the results, we find out that this method is applicable to estimate the coupled-modes of railway system, therefore it is feasible to identify modal parameters of track by this method.
In the end, the content of this paper has been summarized and some problems, which need further study, have been proposed.
引文
[13] Ibrahim S. R. Random decrement technique for model identification of structures, Journal of Spacecraft and Rockets, 1977, 14(11), 696- 700.
[14] S . R. Ibrahim. Application of Randon Time Domain Analysis to Dynamic Flight Measurements, Shock and Vibration Bulletin, 1979, 49(2), 165-170.
[15] S. R. Ibrahim, and S. R. Pappa. Large Modal Survey Testing using the Ibrahim Time Domain Identify Technique, AIAA J. Spacecraft Rockets, 1982, 19(5), 459-465.
[16] C .S. Huang, I. C. Tsai and C. H. Yeh. Application of random decrement technique to identify the characteristics of a building in time domain from ambient vibration measurement. J. Chinese Inst. Civil Hydraulic. Engng. 1998, 10(3), 537-547.
[17] Spitznogle, F. R. and Quazi, A. H. Representation and analysis of time-limited signals using a complex exponential algorithm. Journal of the Acoustical Society of America, 1970,47(5), 1150-1155.
[18] Brown, D. L., Allemang, R. J., Zimmerman, R., and Mergeay, M. Parameter estimation techniques for modal analysis. SAE Technical Paper series No.790221, 1979, 88(1), 828-846.
[19] Void, H., Kundrat, J., Rocklin, G.T., and Russel, R. A multi-input modal estimation algorithm for mini-computers. SAE Technical report series, No.820194, 1982, 91(1), 815-821.
[20] Deblauwe, F. and Allemang, R.J. The Polyreference Time-Domain Technique. Proceedings of the 10th International seminar on modal Analysis, 1985.
[21] Maia, N. M. M. and Silva, J. M. M., Theoretical and experimental modal analysis. John Wiley, Inc. 1997.
[22] Juang, J. N. and Pappa, R. S . An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, J. Guidance, Control and Dynamics. 1985, Vol.8, No.5,620-627.
[23] Gilbert E. G. Controllability and observability in Multivariable Control Systems, SIAM J. on Control, 1963,1(2):128-151
[24] Juang, J. N., Cooper, J. E. and Wright, J.R.. An Eigensystem Realization Algorithm Using Data Correlations (ERA/DC) for Modal Parameter Identification, Control Theory and Advanced Technology, 1988, VO1.4, No.l, 5-14.
[25] Ljung, L. and Soderstrom, T, Theory and Practice of Recursive Identification. MIT Press, Cambridge, MA. 1983.
[26] Young, P. C. Recursive Estimation and Time-Series Analysis. Springer-Verlag, New York. 1984.
[13] Ibrahim S. R. Random decrement technique for model identification of structures, Journal of Spacecraft and Rockets, 1977, 14(11), 696- 700.[14] S . R. Ibrahim. Application of Randon Time Domain Analysis to Dynamic Flight Measurements, Shock and Vibration Bulletin, 1979, 49(2), 165-170.[15] S. R. Ibrahim, and S. R. Pappa. Large Modal Survey Testing using the Ibrahim Time Domain Identify Technique, AIAA J. Spacecraft Rockets, 1982, 19(5), 459-465.[16] C .S. Huang, I. C. Tsai and C. H. Yeh. Application of random decrement technique to identify the characteristics of a building in time domain from ambient vibration measurement. J. Chinese Inst. Civil Hydraulic. Engng. 1998, 10(3), 537-547.[17] Spitznogle, F. R. and Quazi, A. H. Representation and analysis of time-limited signals using a complex exponential algorithm. Journal of the Acoustical Society of America, 1970,47(5), 1150-1155.[18] Brown, D. L., Allemang, R. J., Zimmerman, R., and Mergeay, M. Parameter estimation techniques for modal analysis. SAE Technical Paper series No.790221, 1979, 88(1), 828-846.[19] Void, H., Kundrat, J., Rocklin, G.T., and Russel, R. A multi-input modal estimation algorithm for mini-computers. SAE Technical report series, No.820194, 1982, 91(1), 815-821.[20] Deblauwe, F. and Allemang, R.J. The Polyreference Time-Domain Technique. Proceedings of the 10th International seminar on modal Analysis, 1985.[21] Maia, N. M. M. and Silva, J. M. M., Theoretical and experimental modal analysis. John Wiley, Inc. 1997.[22] Juang, J. N. and Pappa, R. S . An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, J. Guidance, Control and Dynamics. 1985, Vol.8, No.5,620-627.[23] Gilbert E. G. Controllability and observability in Multivariable Control Systems, SIAM J. on Control, 1963,1(2):128-151[24] Juang, J. N., Cooper, J. E. and Wright, J.R.. An Eigensystem Realization Algorithm Using Data Correlations (ERA/DC) for Modal Parameter Identification, Control Theory and Advanced Technology, 1988, VO1.4, No.l, 5-14.[25] Ljung, L. and Soderstrom, T, Theory and Practice of Recursive Identification. MIT Press, Cambridge, MA. 1983.[26] Young, P. C. Recursive Estimation and Time-Series Analysis. Springer-Verlag, New York. 1984.
[27] 王成瑞．ITD法辨识模态参数的有效性．机械设计．1995，9，5～7．
[28] 朱东生，朱唏，田琪．识别桥梁模态参数时域法的仿真研究．兰州铁道学院学报．1995，14(3)，1～9．
[29] Brian Schwarz, Mark Richardson. Modal Parameter Estimation from Ambient Response Data. IMAC 2001, 2001, February:5-8.
[30] Brian Schwarz, Mark Richardson. Modal Parameter Estimation from Operating Data. Sound and Vibration Magazine. 2003, January.
[31] 谢伟平，王国波，于艳丽．移动荷载作用下双层Euler梁模型土动力响应分析．地震工程与工程振动，2004，24(1)：82-86．
[32] 李锁全．弹性交叉梁系的分析计算及应用．石家庄铁道学院学报，2001，14(3)，55～58．
[33] 雷晓燕．高速铁路轨道结构力学模型参数研究．华东交通大学学报，1999，16(2)：1～6．
[34] 童大坝．铁路轨道．北京：中国铁路出版社，1988．36～37．
[35] Gilbert EG. Controllability and Observability in Multivariable Control Systems,SIAM J. on Control, 1963, 1(2):128～151.
[36] Kalman, R.E., Mathematical Description of Linear Dynamical Systems, SIAM J. on control, Col.1, No.2, 1963, 152～192.
[37] Ho B.L. and Kalman R.E., Effective Construction of Linear State Variable Models from Input/Output Data, Proceeding of the 3rd Annual Allerton Conference on Circuit and System Theory,1965, 449～459.
[38] Rossen, R. H., and Lapidus, L., Minimum Realizations and System Modeling:1.Fundamental Theory and Algorithm, AIChe Journal, 18(4), 1972, 673～684.
[39] Zeiger, H.P., and McEwen, A.J., Approximate Linear Realizations of Given Dimension via Ho's Algorithm, IEEE Transactions Automatic Control, 1974, 19(2), 153.
[40] Juang, J.N. and Pappa. R.S., An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, J. Guidance, Control and Dynamics, 1985,18(5), 620～627.
[14] S . R. Ibrahim. Application of Randon Time Domain Analysis to Dynamic Flight Measurements, Shock and Vibration Bulletin, 1979, 49(2), 165-170.
[15] S. R. Ibrahim, and S. R. Pappa. Large Modal Survey Testing using the Ibrahim Time Domain Identify Technique, AIAA J. Spacecraft Rockets, 1982, 19(5), 459-465.
[16] C .S. Huang, I. C. Tsai and C. H. Yeh. Application of random decrement technique to identify the characteristics of a building in time domain from ambient vibration measurement. J. Chinese Inst. Civil Hydraulic. Engng. 1998, 10(3), 537-547.
[17] Spitznogle, F. R. and Quazi, A. H. Representation and analysis of time-limited signals using a complex exponential algorithm. Journal of the Acoustical Society of America, 1970,47(5), 1150-1155.
[18] Brown, D. L., Allemang, R. J., Zimmerman, R., and Mergeay, M. Parameter estimation techniques for modal analysis. SAE Technical Paper series No.790221, 1979, 88(1), 828-846.
[19] Void, H., Kundrat, J., Rocklin, G.T., and Russel, R. A multi-input modal estimation algorithm for mini-computers. SAE Technical report series, No.820194, 1982, 91(1), 815-821.
[20] Deblauwe, F. and Allemang, R.J. The Polyreference Time-Domain Technique. Proceedings of the 10th International seminar on modal Analysis, 1985.
[21] Maia, N. M. M. and Silva, J. M. M., Theoretical and experimental modal analysis. John Wiley, Inc. 1997.
[22] Juang, J. N. and Pappa, R. S . An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, J. Guidance, Control and Dynamics. 1985, Vol.8, No.5,620-627.
[23] Gilbert E. G. Controllability and observability in Multivariable Control Systems, SIAM J. on Control, 1963,1(2):128-151
[24] Juang, J. N., Cooper, J. E. and Wright, J.R.. An Eigensystem Realization Algorithm Using Data Correlations (ERA/DC) for Modal Parameter Identification, Control Theory and Advanced Technology, 1988, VO1.4, No.l, 5-14.
[25] Ljung, L. and Soderstrom, T, Theory and Practice of Recursive Identification. MIT Press, Cambridge, MA. 1983.
[26] Young, P. C. Recursive Estimation and Time-Series Analysis. Springer-Verlag, New York. 1984.
[13] Ibrahim S. R. Random decrement technique for model identification of structures, Journal of Spacecraft and Rockets, 1977, 14(11), 696- 700.[14] S . R. Ibrahim. Application of Randon Time Domain Analysis to Dynamic Flight Measurements, Shock and Vibration Bulletin, 1979, 49(2), 165-170.[15] S. R. Ibrahim, and S. R. Pappa. Large Modal Survey Testing using the Ibrahim Time Domain Identify Technique, AIAA J. Spacecraft Rockets, 1982, 19(5), 459-465.[16] C .S. Huang, I. C. Tsai and C. H. Yeh. Application of random decrement technique to identify the characteristics of a building in time domain from ambient vibration measurement. J. Chinese Inst. Civil Hydraulic. Engng. 1998, 10(3), 537-547.[17] Spitznogle, F. R. and Quazi, A. H. Representation and analysis of time-limited signals using a complex exponential algorithm. Journal of the Acoustical Society of America, 1970,47(5), 1150-1155.[18] Brown, D. L., Allemang, R. J., Zimmerman, R., and Mergeay, M. Parameter estimation techniques for modal analysis. SAE Technical Paper series No.790221, 1979, 88(1), 828-846.[19] Void, H., Kundrat, J., Rocklin, G.T., and Russel, R. A multi-input modal estimation algorithm for mini-computers. SAE Technical report series, No.820194, 1982, 91(1), 815-821.[20] Deblauwe, F. and Allemang, R.J. The Polyreference Time-Domain Technique. Proceedings of the 10th International seminar on modal Analysis, 1985.[21] Maia, N. M. M. and Silva, J. M. M., Theoretical and experimental modal analysis. John Wiley, Inc. 1997.[22] Juang, J. N. and Pappa, R. S . An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, J. Guidance, Control and Dynamics. 1985, Vol.8, No.5,620-627.[23] Gilbert E. G. Controllability and observability in Multivariable Control Systems, SIAM J. on Control, 1963,1(2):128-151[24] Juang, J. N., Cooper, J. E. and Wright, J.R.. An Eigensystem Realization Algorithm Using Data Correlations (ERA/DC) for Modal Parameter Identification, Control Theory and Advanced Technology, 1988, VO1.4, No.l, 5-14.[25] Ljung, L. and Soderstrom, T, Theory and Practice of Recursive Identification. MIT Press, Cambridge, MA. 1983.[26] Young, P. C. Recursive Estimation and Time-Series Analysis. Springer-Verlag, New York. 1984.
[27] 王成瑞．ITD法辨识模态参数的有效性．机械设计．1995，9，5～7．
[28] 朱东生，朱唏，田琪．识别桥梁模态参数时域法的仿真研究．兰州铁道学院学报．1995，14(3)，1～9．
[29] Brian Schwarz, Mark Richardson. Modal Parameter Estimation from Ambient Response Data. IMAC 2001, 2001, February:5-8.
[30] Brian Schwarz, Mark Richardson. Modal Parameter Estimation from Operating Data. Sound and Vibration Magazine. 2003, January.
[31] 谢伟平，王国波，于艳丽．移动荷载作用下双层Euler梁模型土动力响应分析．地震工程与工程振动，2004，24(1)：82-86．
[32] 李锁全．弹性交叉梁系的分析计算及应用．石家庄铁道学院学报，2001，14(3)，55～58．
[33] 雷晓燕．高速铁路轨道结构力学模型参数研究．华东交通大学学报，1999，16(2)：1～6．
[34] 童大坝．铁路轨道．北京：中国铁路出版社，1988．36～37．
[35] Gilbert EG. Controllability and Observability in Multivariable Control Systems,SIAM J. on Control, 1963, 1(2):128～151.
[36] Kalman, R.E., Mathematical Description of Linear Dynamical Systems, SIAM J. on control, Col.1, No.2, 1963, 152～192.
[37] Ho B.L. and Kalman R.E., Effective Construction of Linear State Variable Models from Input/Output Data, Proceeding of the 3rd Annual Allerton Conference on Circuit and System Theory,1965, 449～459.
[38] Rossen, R. H., and Lapidus, L., Minimum Realizations and System Modeling:1.Fundamental Theory and Algorithm, AIChe Journal, 18(4), 1972, 673～684.
[39] Zeiger, H.P., and McEwen, A.J., Approximate Linear Realizations of Given Dimension via Ho's Algorithm, IEEE Transactions Automatic Control, 1974, 19(2), 153.
[40] Juang, J.N. and Pappa. R.S., An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, J. Guidance, Control and Dynamics, 1985,18(5), 620～627.