时域精细算法求解移动荷载下梁的正、反问题
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摘要
许多工程结构承受移动载荷的作用，如桥梁，飞机跑道、停车场和防波堤等。对移动荷载下结构的动力分析研究，对这类结构的设计、控制和安全稳定性分析有着重要意义。
由于实际问题的复杂性，移动荷载动力问题的解析求解一般较为困难，发展行之有效的数值求解技术十分必要。本文发展了一种求解梁移动荷载动力问题的时域自适应精细算法，通过离散时段内的时间相关变量的展开，将时空耦合移动荷载问题转化为一系列递推形式的空间问题，并采用自适应技术以保证计算精度，避免步长选择不当可能造成的计算误差。分析了梁在匀速移动定常力、简谐力，变速移动力，移动质量，移动振动等载荷条件下梁的动力响应，与解析解及Runge-Kutta法的计算结果进行比较有很好吻合。
在移动荷载动力正问题建模和求解的基础上，本文提出了移动荷载动力反问题的数值求解模型，考虑了待识别物性参数的空间分布特性，并利用Levenberg-Marquardt方法，对匀速移动载荷下梁的未知刚度系数进行了识别，得到了令人满意的结果。计算结果表明：载荷的移动，可使结构各个部分的物性参数对附加信息的敏感度更加均衡，因此，对有空间分布特性的动力物性参数反问题，如采用移动载荷激励可能会使求解更为效。
本文的研究工作为数值求解移动荷载正／反问题提供了新的途径，经进一步完善改进，有望在实际问题中得到初步应用。
Lots of engineering structures are under moving loads, such as bridges, airstrips, parkinglots groynes and so on. Dynamic analysis of structures under moving loads has greatsignificance.
Due to complexity of practical engineering, analytical solutions of moving loads aregenerally difficult, and therefore effective numerical methods are in great need. Aself-adaptive precise algorithm in time domain is developed for moving load problems. Byexpanding variables in discrete time intervals, problems with time and space coupled can betransformed into a series space problems which can be solved recursively. Responses ofbeams under moving force, moving mass and moving oscillator models are studied, numericalresults are satisfactory compared with analytical solutions and those from Runge-Kuttamethod.
Based on the forward model, inverse model for moving load problems are established,taking distributing of unknown variables into account, and Levenberg-Marquardt method isapplied. Numerical results show that moving load excitation are effective in identification ofdistributing variables.
This paper provide a new numerical approach to dynamic analysis and inverse problems,its application in practice is looked forward to.
由于实际问题的复杂性，移动荷载动力问题的解析求解一般较为困难，发展行之有效的数值求解技术十分必要。本文发展了一种求解梁移动荷载动力问题的时域自适应精细算法，通过离散时段内的时间相关变量的展开，将时空耦合移动荷载问题转化为一系列递推形式的空间问题，并采用自适应技术以保证计算精度，避免步长选择不当可能造成的计算误差。分析了梁在匀速移动定常力、简谐力，变速移动力，移动质量，移动振动等载荷条件下梁的动力响应，与解析解及Runge-Kutta法的计算结果进行比较有很好吻合。
在移动荷载动力正问题建模和求解的基础上，本文提出了移动荷载动力反问题的数值求解模型，考虑了待识别物性参数的空间分布特性，并利用Levenberg-Marquardt方法，对匀速移动载荷下梁的未知刚度系数进行了识别，得到了令人满意的结果。计算结果表明：载荷的移动，可使结构各个部分的物性参数对附加信息的敏感度更加均衡，因此，对有空间分布特性的动力物性参数反问题，如采用移动载荷激励可能会使求解更为效。
本文的研究工作为数值求解移动荷载正／反问题提供了新的途径，经进一步完善改进，有望在实际问题中得到初步应用。
Lots of engineering structures are under moving loads, such as bridges, airstrips, parkinglots groynes and so on. Dynamic analysis of structures under moving loads has greatsignificance.
Due to complexity of practical engineering, analytical solutions of moving loads aregenerally difficult, and therefore effective numerical methods are in great need. Aself-adaptive precise algorithm in time domain is developed for moving load problems. Byexpanding variables in discrete time intervals, problems with time and space coupled can betransformed into a series space problems which can be solved recursively. Responses ofbeams under moving force, moving mass and moving oscillator models are studied, numericalresults are satisfactory compared with analytical solutions and those from Runge-Kuttamethod.
Based on the forward model, inverse model for moving load problems are established,taking distributing of unknown variables into account, and Levenberg-Marquardt method isapplied. Numerical results show that moving load excitation are effective in identification ofdistributing variables.
This paper provide a new numerical approach to dynamic analysis and inverse problems,its application in practice is looked forward to.
引文
[1] A. V. Pesterev, L.A.B., C. A. Tan, T.-C. Tsao, and B. Yang. SOME RECENT RESULTS IN MOVING LOAD PROBLEMS WITH APPLICATION TO HIGHWAY BRIDGES. in Sixth International Conference on Motion and Vibration Control (MoViC), August 20-24.2002. Saitama, Japan.
[2] Fryba, L., Vibration of solid and structure under moving loads. 1972, Groningen, The Netherland: Noordhoff International Publishing.
[3] Y.H.LIN, M.W.T., Finite element analysis of elastic beams subjected to moving dynamic loads. Journal of Sound and Vibration, 1990. 136(2): p. 323-342.
[4] ZHUGE, D.T.A.Y., DYNAMIC ANALYSIS OF BEAMS ON AN ELASTIC FOUNDATION SUBJECTED TO MOVING LOADS. Journal of Sound and Vibration, 1996. 198(2): p. 149-169.
[5] L. Andersen, S.R.K.N., R. Iwankiewicz, Vehicle Moving Along an Infinite Beam With Random Surface Irregularities on a Kelvin Foundation. Journal of Applied Mechanics, 2002. 69: p. 69-75.
[6] C. G. Koh, J.S.Y.O., D. K. H. Chua and J. Feng, Moving element method for train-track dynamics. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2003. 56: p. 1549-1567.
[7] Yeong-Bin Yang, J.-D.Y., vehicle-bridge interaction element for dynamic analysis. Journal of Structural Engineering, 1997. 123(11): p. 1512-1518.
[8] YEONG-BIN YANG, C.-H.C.AJ.-D.Y., An element for analysis vehicle-bridge systems considering vehicle's pitching effect. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 1999. 46: p. 1031-1047.
[9] Pesterev A.V., B.L.A., Vibration of Elastic Continuum Carrying Moving Linear Oscillator. ASCE Journal of Engineering Mechanics, 1997. 123: p. 878-884.
[10] Pesterev A.V., B.L.A., Vibration of Elastic Continuum Carrying Accelerating Oscillator. ASCE Journal of Engineering Mechanics, 1997. 123: p. 886-889.
[11] Pesterev A.V., B.L.A., Response of a Nonconservative Continuous System to a Moving Concentrated Load. ASME Journal of Applied Mechanics, 1998. 65: p. 436-444.
[12] Rao, G.V., Linear Dynamics of an Elastic Beam Under Moving Loads. Journal of Vibration and Acoustics, 2000. 122: p. 281-289.
[13] U, L., Revisiting the moving mass problem: Onset of separation between the mass and beam. Journal of vibration and acoustics, 1996. 118(3): p. 516-521.
[14] JAMAL AL-DUAIJ, M.A.-R., REENA JAMES P., Maximum Acceleration for the Vibration of Single Span Bridges with Even and Uneven Decks. Dynamics and Control, 1999. 9: p. 247-254.
[15] A. S. J. Suiker, R.d.B., C. Esveld, Critical behaviour of a Timoshenko beam-half plane system under a moving load. Archive of Applied Mechanics, 1998, 68: p. 158-168.
[16]Shen-Haw Ju, H.-T.L., Chung-Cheng Hsueh and Shin-Lin Wang, A simple finite element model for vibration analyses induced by moving vehicles. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2006: p. 1748-1772.
[17]Y.S. Cheng, F.T.K.A., Y.K. Cheung, Vibration of railway bridges under a moving train by using bridgetrack-vehicle element. Engineering Structures, 2001. 23: p. 1597-1606.
[18] Alawneh, K.A.B.-H.a.M.R., Prestressed active post-tensioned tendons control for bridges under moving loads. STRUCTURAL CONTROL AND HEALTH MONITORING, 2007. 14: p. 357-383.
[19] 武秀丽，弹性点支连续等跨梁的横向振动(Ⅰ)——自由振动．石家庄铁道学院学报，1995．8(3)：p．14-20．
[20] 武秀丽，弹性点支连续等跨梁的横向振动(Ⅱ)——匀速移动载荷作用下的动态响应．石家庄铁道学院学报，1995．8(4)：p．57-63．
[21] 施丽娟，林铸明，崔维成，浮桥结构在移动荷载作用下的振动响应．海洋工程，2003．21(3)：p．6-17．
[22] 洪芳鹏，卜昌富，黄醒春，车载作用下箱形截面梁振动特性的瞬态动力学分析．地震工程与工程振动，2004．24(6)：p．53-57．
[23] 李红兵，谢．于．，高速移动荷载下轨道系统振动模拟的数值算法．武汉理工大学学报，2001．23(12)：p．57-60．
[24] 姚春桥，谢．于．，高速移动荷载下轨道系统的振动研究．地震工程与工程振动，2002．22(2)：p．85-90．
[25] MANJRIKER GUNARATNE, O.S., Ill', RESPONSE OF A LAYERED ELASTIC MEDIUM TO A MOVING STRIP LOAD. INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, 1996.20: p. 191-208.
[26] Li, H.O.J.E.M.W., A Moving-Load Model for Disc-Brake Stability Analysis. Journal of Vibration and Acoustics, 2003. 125: p. 53-58.
[27] Wu, J.-J., Vibration analyses of a portal frame under the action of a moving distributed mass using moving mass element. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2005.62: p. 2028-2052.
[28] He Xia1, (?), Yan Han2, Nan Zhang1 and Weiwei Guo, Dynamic analysis of train-bridge system subjected to non-uniform seismic excitations. EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, 2006.
[29] S. Park, W.K.C., Y. Yourn and J.W. Lee, Motion Analysis of a Moving Elastic Beam with a Moving Mass. International Conference on Advanced Intelligent Mechatronics, 1999: p. 167-172.
[30] GOLNARAGHI, S.A.Q.S.a.M.F., Dynamics of a Flexible Cantilever Beam Carrying a Moving Mass. Nonlinear Dynamics, 1998.15: p. 137-154.
[31] Hung, C.S.H.Y.P.T.a.C.L., An accurate solution for the responses of circular curved beams subjected to a moving load. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2000.48: p. 1723-1740.
[32] CHAN, S.S.L.A.T.H.T., Moving Force Identificationa time domain method. Journal of Sound and Vibration, 1997. 201(1): p. 1-22.
[33] YUNG, T.H.T.C.S.S.L.A.T.H., An interpretive method for moving force identification. Journal of Sound and Vibration, 1999.219(3): p. 503-524.
[34] S. S. LAW, X.Q.Z., Study on different beam models in moving force identification. Journal of Sound and [35] S. S. Law, T.H.T.C., 2 Q. X. Zhu,3 and Q. H. Zeng4, Regularization in Moving Force Identification. JOURNAL OF ENGINEERING MECHANICS, 2001: p. 136-148.
[36] Jaureguiz, C.R.F.a.D.A., Comparative study of damage identification algorithms applied to a bridge: Ⅰ Experiment. Smart Mater. Struct., 1998.7: p. 704-719.
[37] Scott W. Doebling, C.R.F., Michael B. Prime, Daniel W. Shevitz, Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review. 1996, Los Alamos National Lab., NM (United States).
[38] Haitian, Y., A new approach of time stepping for solving transfer problem. Communications in numerical methods in engineering, 1999. 15: p. 325-334.
[39] Haitian Yang, Z. H., Solving non-linear viscoelastic problems via a self-adaptive precise algorithm in time domain. International Journal of Solids and Structures, 2004.41: p. 5483-5498.
[40] Yang Haitian, G.Q., A Precise time stepping scheme to solve hyperbolic and parabolic heat transfer problems with radiative boundary condition. Heat and Mass Transfer, 2003.39: p. 571-577.
[41] Yang Haitian, G. Q., Guo Xinglin and Wu Chengwei, A new algorithm of time stepping in the non-linear dynamic analysis. Communications in numerical methods in engineering, 2001.17: p. 597-611.
[42] 唐驾时，自激振动系统的参数识别．振动与冲击，1991．1：p．49-54．
[43] 张亚辉，张守云，赵岩，宋刚，林家浩，桥梁受移动荷载动力响应的一种精细积分法．计算力学学报，2006．23(3)：p．290-294．
[44] G. R. Liu, X.H., COMPUTATIONAL INVERSE TECHNIQUES in NONDESTRUCTIVE EVALUATION. 2003, Boca Raton London New York Washington D.C. 592.
[2] Fryba, L., Vibration of solid and structure under moving loads. 1972, Groningen, The Netherland: Noordhoff International Publishing.
[3] Y.H.LIN, M.W.T., Finite element analysis of elastic beams subjected to moving dynamic loads. Journal of Sound and Vibration, 1990. 136(2): p. 323-342.
[4] ZHUGE, D.T.A.Y., DYNAMIC ANALYSIS OF BEAMS ON AN ELASTIC FOUNDATION SUBJECTED TO MOVING LOADS. Journal of Sound and Vibration, 1996. 198(2): p. 149-169.
[5] L. Andersen, S.R.K.N., R. Iwankiewicz, Vehicle Moving Along an Infinite Beam With Random Surface Irregularities on a Kelvin Foundation. Journal of Applied Mechanics, 2002. 69: p. 69-75.
[6] C. G. Koh, J.S.Y.O., D. K. H. Chua and J. Feng, Moving element method for train-track dynamics. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2003. 56: p. 1549-1567.
[7] Yeong-Bin Yang, J.-D.Y., vehicle-bridge interaction element for dynamic analysis. Journal of Structural Engineering, 1997. 123(11): p. 1512-1518.
[8] YEONG-BIN YANG, C.-H.C.AJ.-D.Y., An element for analysis vehicle-bridge systems considering vehicle's pitching effect. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 1999. 46: p. 1031-1047.
[9] Pesterev A.V., B.L.A., Vibration of Elastic Continuum Carrying Moving Linear Oscillator. ASCE Journal of Engineering Mechanics, 1997. 123: p. 878-884.
[10] Pesterev A.V., B.L.A., Vibration of Elastic Continuum Carrying Accelerating Oscillator. ASCE Journal of Engineering Mechanics, 1997. 123: p. 886-889.
[11] Pesterev A.V., B.L.A., Response of a Nonconservative Continuous System to a Moving Concentrated Load. ASME Journal of Applied Mechanics, 1998. 65: p. 436-444.
[12] Rao, G.V., Linear Dynamics of an Elastic Beam Under Moving Loads. Journal of Vibration and Acoustics, 2000. 122: p. 281-289.
[13] U, L., Revisiting the moving mass problem: Onset of separation between the mass and beam. Journal of vibration and acoustics, 1996. 118(3): p. 516-521.
[14] JAMAL AL-DUAIJ, M.A.-R., REENA JAMES P., Maximum Acceleration for the Vibration of Single Span Bridges with Even and Uneven Decks. Dynamics and Control, 1999. 9: p. 247-254.
[15] A. S. J. Suiker, R.d.B., C. Esveld, Critical behaviour of a Timoshenko beam-half plane system under a moving load. Archive of Applied Mechanics, 1998, 68: p. 158-168.
[16]Shen-Haw Ju, H.-T.L., Chung-Cheng Hsueh and Shin-Lin Wang, A simple finite element model for vibration analyses induced by moving vehicles. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2006: p. 1748-1772.
[17]Y.S. Cheng, F.T.K.A., Y.K. Cheung, Vibration of railway bridges under a moving train by using bridgetrack-vehicle element. Engineering Structures, 2001. 23: p. 1597-1606.
[18] Alawneh, K.A.B.-H.a.M.R., Prestressed active post-tensioned tendons control for bridges under moving loads. STRUCTURAL CONTROL AND HEALTH MONITORING, 2007. 14: p. 357-383.
[19] 武秀丽，弹性点支连续等跨梁的横向振动(Ⅰ)——自由振动．石家庄铁道学院学报，1995．8(3)：p．14-20．
[20] 武秀丽，弹性点支连续等跨梁的横向振动(Ⅱ)——匀速移动载荷作用下的动态响应．石家庄铁道学院学报，1995．8(4)：p．57-63．
[21] 施丽娟，林铸明，崔维成，浮桥结构在移动荷载作用下的振动响应．海洋工程，2003．21(3)：p．6-17．
[22] 洪芳鹏，卜昌富，黄醒春，车载作用下箱形截面梁振动特性的瞬态动力学分析．地震工程与工程振动，2004．24(6)：p．53-57．
[23] 李红兵，谢．于．，高速移动荷载下轨道系统振动模拟的数值算法．武汉理工大学学报，2001．23(12)：p．57-60．
[24] 姚春桥，谢．于．，高速移动荷载下轨道系统的振动研究．地震工程与工程振动，2002．22(2)：p．85-90．
[25] MANJRIKER GUNARATNE, O.S., Ill', RESPONSE OF A LAYERED ELASTIC MEDIUM TO A MOVING STRIP LOAD. INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, 1996.20: p. 191-208.
[26] Li, H.O.J.E.M.W., A Moving-Load Model for Disc-Brake Stability Analysis. Journal of Vibration and Acoustics, 2003. 125: p. 53-58.
[27] Wu, J.-J., Vibration analyses of a portal frame under the action of a moving distributed mass using moving mass element. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2005.62: p. 2028-2052.
[28] He Xia1, (?), Yan Han2, Nan Zhang1 and Weiwei Guo, Dynamic analysis of train-bridge system subjected to non-uniform seismic excitations. EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, 2006.
[29] S. Park, W.K.C., Y. Yourn and J.W. Lee, Motion Analysis of a Moving Elastic Beam with a Moving Mass. International Conference on Advanced Intelligent Mechatronics, 1999: p. 167-172.
[30] GOLNARAGHI, S.A.Q.S.a.M.F., Dynamics of a Flexible Cantilever Beam Carrying a Moving Mass. Nonlinear Dynamics, 1998.15: p. 137-154.
[31] Hung, C.S.H.Y.P.T.a.C.L., An accurate solution for the responses of circular curved beams subjected to a moving load. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2000.48: p. 1723-1740.
[32] CHAN, S.S.L.A.T.H.T., Moving Force Identificationa time domain method. Journal of Sound and Vibration, 1997. 201(1): p. 1-22.
[33] YUNG, T.H.T.C.S.S.L.A.T.H., An interpretive method for moving force identification. Journal of Sound and Vibration, 1999.219(3): p. 503-524.
[34] S. S. LAW, X.Q.Z., Study on different beam models in moving force identification. Journal of Sound and
[36] Jaureguiz, C.R.F.a.D.A., Comparative study of damage identification algorithms applied to a bridge: Ⅰ Experiment. Smart Mater. Struct., 1998.7: p. 704-719.
[37] Scott W. Doebling, C.R.F., Michael B. Prime, Daniel W. Shevitz, Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review. 1996, Los Alamos National Lab., NM (United States).
[38] Haitian, Y., A new approach of time stepping for solving transfer problem. Communications in numerical methods in engineering, 1999. 15: p. 325-334.
[39] Haitian Yang, Z. H., Solving non-linear viscoelastic problems via a self-adaptive precise algorithm in time domain. International Journal of Solids and Structures, 2004.41: p. 5483-5498.
[40] Yang Haitian, G.Q., A Precise time stepping scheme to solve hyperbolic and parabolic heat transfer problems with radiative boundary condition. Heat and Mass Transfer, 2003.39: p. 571-577.
[41] Yang Haitian, G. Q., Guo Xinglin and Wu Chengwei, A new algorithm of time stepping in the non-linear dynamic analysis. Communications in numerical methods in engineering, 2001.17: p. 597-611.
[42] 唐驾时，自激振动系统的参数识别．振动与冲击，1991．1：p．49-54．
[43] 张亚辉，张守云，赵岩，宋刚，林家浩，桥梁受移动荷载动力响应的一种精细积分法．计算力学学报，2006．23(3)：p．290-294．
[44] G. R. Liu, X.H., COMPUTATIONAL INVERSE TECHNIQUES in NONDESTRUCTIVE EVALUATION. 2003, Boca Raton London New York Washington D.C. 592.
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