高速列车—轨道—路基耦合系统垂向振动特性分析
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摘要
高速铁路以运行速度快、运输能力大、能源消耗低、环境影响轻、土地占用少、安全性好、正点率高等综合优势，在世界各国得到了迅速发展。然而，列车运行速度的提高和载重的不断增加，使得车辆、轨道与路基之间的动力相互作用越来越大，行车的安全性和舒适性、轨道结构的安全性问题也越来越突出。因此，对车辆轨道路基耦合系统动力学问题的研究具有重要意义。本文主要工作和研究内容如下：
（1）建立了车辆轨道路基耦合动力分析模型。将车辆轨道路基耦合系统分解成三个子系统：车辆子系统、轨道路基子系统和轮轨子系统。根据对车辆结构和轨道结构的假定，分别建立了车辆和轨道路基两个子系统模型，并由达朗贝尔原理建立了车辆动力学方程和轨道路基系统动力学方程。针对轮轨子系统，垂向处理采用了Hertz非线性弹性接触理论。
（2）介绍了轨道不平顺的种类和常见的数值积分方法。本文以谐波不平顺作为系统激励，采用新型快速显式积分法对耦合系统动力学的方程进行逐步积分。并列出了对系统进行数值求解的程序流程图。
（3）采用新型快速显式积分法对车辆轨道路基耦合系统的垂向振动响应进行数值计算，分析了不同轨道结构、列车速度、轨枕间距、轨道不平顺、车体质量以及道床厚度等参数对耦合系统结构动力响应的影响规律。
With its comprehensive advantages such as the high operating speed, largetransport capacity, low energy consumption, light pollution of the environment, lessoccupation of the land aera, high safety and high punctuality, the high-speed railwayhas been developed rapidly all the world. However, with the improving of train speedand the increasing of load, the dynamic interaction of Vehicle-Track-Subgrade isbecoming more and more serious, the safety and comfort of the running train, thereliability of tracks and subgrade structure are even more conspicuous. Therefore, theresearch of Vehicle-Track-Subgrade coupled dynamics problem is of great significantIn this paper, the main work are as follows,
(1) The dynamic analysis model for Vehicle-Track-Subgrade coupled system isestablished. The coupled system is divided into three subsystems: the vehiclesubsystem, the track-subgrade subsystem and wheel-track subsystem. Based on theassumptions of vehicle structure and track-subgrade structure, vehicle model andtrack-subgrade model are built respectively, and the dynamic equations of vehicle andtrack-subgrade system are derived by using the D’Alembert principle. For thewheel/rail subsystem, the vertical interaction is handled as the nonlinear interactionelastic Hertz contact.
(2) The track irregularity and the kinds of common numerical integration methodare introduced. Based on the harmonic irregularity as the excitation system, thedynamic equation of the coupling system is integrated step by step by using the newdisplay integral method. And the program flow chart of integral solution for system isgiven.
(3) With the Vehicle-Track-Subgrade vertical vibrating response calculatingprocedure is programmed by using the new display integral method on the platformMatlab software, The coupled system dynamic response are analyzed with the effectsof different track structures, velocity, sleeper spacing, track irregularity, body mass, thickness of ballast.
（1）建立了车辆轨道路基耦合动力分析模型。将车辆轨道路基耦合系统分解成三个子系统：车辆子系统、轨道路基子系统和轮轨子系统。根据对车辆结构和轨道结构的假定，分别建立了车辆和轨道路基两个子系统模型，并由达朗贝尔原理建立了车辆动力学方程和轨道路基系统动力学方程。针对轮轨子系统，垂向处理采用了Hertz非线性弹性接触理论。
（2）介绍了轨道不平顺的种类和常见的数值积分方法。本文以谐波不平顺作为系统激励，采用新型快速显式积分法对耦合系统动力学的方程进行逐步积分。并列出了对系统进行数值求解的程序流程图。
（3）采用新型快速显式积分法对车辆轨道路基耦合系统的垂向振动响应进行数值计算，分析了不同轨道结构、列车速度、轨枕间距、轨道不平顺、车体质量以及道床厚度等参数对耦合系统结构动力响应的影响规律。
With its comprehensive advantages such as the high operating speed, largetransport capacity, low energy consumption, light pollution of the environment, lessoccupation of the land aera, high safety and high punctuality, the high-speed railwayhas been developed rapidly all the world. However, with the improving of train speedand the increasing of load, the dynamic interaction of Vehicle-Track-Subgrade isbecoming more and more serious, the safety and comfort of the running train, thereliability of tracks and subgrade structure are even more conspicuous. Therefore, theresearch of Vehicle-Track-Subgrade coupled dynamics problem is of great significantIn this paper, the main work are as follows,
(1) The dynamic analysis model for Vehicle-Track-Subgrade coupled system isestablished. The coupled system is divided into three subsystems: the vehiclesubsystem, the track-subgrade subsystem and wheel-track subsystem. Based on theassumptions of vehicle structure and track-subgrade structure, vehicle model andtrack-subgrade model are built respectively, and the dynamic equations of vehicle andtrack-subgrade system are derived by using the D’Alembert principle. For thewheel/rail subsystem, the vertical interaction is handled as the nonlinear interactionelastic Hertz contact.
(2) The track irregularity and the kinds of common numerical integration methodare introduced. Based on the harmonic irregularity as the excitation system, thedynamic equation of the coupling system is integrated step by step by using the newdisplay integral method. And the program flow chart of integral solution for system isgiven.
(3) With the Vehicle-Track-Subgrade vertical vibrating response calculatingprocedure is programmed by using the new display integral method on the platformMatlab software, The coupled system dynamic response are analyzed with the effectsof different track structures, velocity, sleeper spacing, track irregularity, body mass, thickness of ballast.
引文
[1]金学松，温泽峰，张卫华等.世界铁路发展状况及其关键力学问题全国结构工程学术会议特邀报告[J].工程力学，2004，1.
[2]钱立新.世界高速铁路技术[M].北京：中国铁道出版社，2003.
[3]翟婉明.车辆-轨道耦合动力学(第三版)[M].北京:科学出版社,2007.
[4] Garg V K. Dynamics of railway vehicle systems[J].1984.
[5]詹斐生.机车动力学[M].北京：中国铁道出版社，1990.
[6]王福天.车辆系统动力学[M].北京：中国铁道出版社，1994.
[7] Knothe K. Gleisdynamik[M]. Berlin：Ernst&Sohn，2001.
[8]左藤裕，卢肇英译.轨道力学[M].北京：中国铁道出版社，1981.
[9]练松良.轨道动力学[M].上海：同济大学出版社，2003.
[10] FRYBA，L. History of Winkler foundation[J]. Vehicle System Dynamics，1995，24(sup1)：7-12.
[11] Prud’Homme M A，Janin M G. The stability of tracks laid with long welded rails,partsⅠ andⅡ[C]. Bulletin of the International Railway Congress Association，1969，88：460-480.
[12] Lyon D. The calculation of track forces due to dipped rail joints，wheel flats andrail welds[C]. The Second ORE Colloquium on Technical Computer Programs，1972.
[13] Jenkins H H, Stephenson J E, Clayton G A，etal. The effect of track and vehicleparameters on wheel/rail vertical dynamic forces[J]. Railway EngineeringJournal，1974,3(1)：2-16.
[14] Sato Y. Abnormal wheel load of test train[J]. Permanent Way,1973,14：1-8
[15] Ahlbeck D R, Meacham H C, Prause R H. The development of analyticalmodels for railroad track dynamics[C]. Railroad Track Mechanics&Technology,1975:197-220.
[16]Clark R.A, Dean P A., Elkins J A, Newton S G. An investigation into the dynamiceffects of railway vehicle running on corrugated rails[J]. Journal of MechanicalEngineering Science,1982,24(2):65-76.
[17] Grassie S L, Gregory R W, Johnson K L. The dynamic response of railway trackto high frequency lateral excitation[J]. Journal of Mechanical EngineeringScience,1982,24(2):91-95.
[18] Grassie S L, Gregory R W, Harrison D, et al. The dynamic response of railwaytrack to high frequency vertical excitation[J]. Journal of Mechanical EngineeringScience,1982,24(2):77-90
[19] Grassie S L, Gregory R W, Johnson K L. The behaviour of railway wheelsetsand track at high frequencies of excitation[J]. Journal of Mechanical EngineeringScience,1982,24(2):103-111.
[20] Grassie S L, Gregory R W, Johnson K L. The dynamic loading of rails atcorrugation frequencies[J]. Contact Mechanics and Wear of Rail/Wheel Systems,1982:209-228.
[21] Cai Z, Raymond G P. Theoretical model for dynamic wheel/rail and trackinteraction[C]. Proceedings of10th Interactional Wheelset Congress, Australia,1992,127-131.
[22] Fermer M. Wheel/rail contact forces for flexible versus solid wheels due to treadirregularities[J].Vehicle System Dynamics,1994,23(Suppl.):142-157.
[23] Nielsen J. Train/track Interaction Coupling of Moving and Stationary DynamicSystems[D]. Ph. D. Dissertation, Chalmers University of Technology, Gotebory,Sweden,1993.
[24] RIPKE B, KNOTHE K. Simulation of High Frequency Vehicle-trackInteractions[J]. Vehicle System Dynamics,1995,24(Suppl.):72-85.
[25] Oscarsson J， Dahlberg T. Dynamic train/track/ballast interaction-computermodels and full-scale experiments[J]. Vehicle System Dynamics，1998，29(Suppl.)：73-84.
[26] Frohling R D. Low frequency dynamic vehicle-track interaction: modeling andsimulation[J]. Vehicle System Dynamics,1998,28(Suppl.):30-46.
[27] AUERSCHL L. Vehicle-track interaction and soil dynamics[J]. Vehicle SystemDynamics,1998,28(Suppl.):553-558.
[28] Popp K, Kruse H, Kaiser I. Vehicle-track dynamics in the mid-frequencyrange[J]. Vehicle System Dynamics,1999,31(5-6):423-464.
[29] Andersson C, Oscarsson J. Dynamic train/track interaction includingstate-dependent track properties and flexible vehicle components[J]. VehicleSystem Dynamics,1999,33(Suppl.)：47-58.
[30] Drozdziel J. The effect of railway vehicle-track system geometric deviation onits dynamics in the turnout zone[J]. Vehicle System Dynamics,1999,33(Suppl.)：641-652.
[31] Gurule S，Wilson N. Simulation of wheel/rail interaction in turnouts and specialtrack work[J]. Vehicle System Dynamics，1999，33(Suppl.)：143-154.
[32] Oscarsson J. Dynamic train-track-ballast interaction with unevenly distributedtrack properties[J]. Vehicle System Dynamics,2003,37:385-396.
[33] Dietz S, Hippmann G, Schupp G. Interaction of vehicle and flexible tracks byco-simulation of multibody vehicle system and finite element track models[J].Vehicle System Dynamics,2003,37:372-385.
[34] Szole T. Simulation of vehicle-track interaction in Medium frequency range withapplication to analysis of mechanical and thermal loading in contact[C].17thIAVSD Symposium on Dynamics of Vehicles on Roads and Tracks，Lyngby，Denmark，2001
[35]吴章江,王浦强,李湘敏等.车辆通过轨道抵扣接头的垂向轮轨作用力计算[J].铁道机车车辆,1982,1：24-30.
[36]李定清.轮轨垂直相互动力作用及其动力响应[J].铁道学报，1987，9(1)：1~8
[37]许实儒,徐维杰,仲延禧.钢轨接头处轮轨冲击力的模拟分析[J].铁道学报,1989,1.
[38]王澜,姚明初.轨道结构随机振动理论及其在轨道结构减振研究中的应用[J].中国铁道科学,1989,10(2):41-59.
[39]翟婉明.车辆-轨道垂向耦合动力学[D].成都:西南交通大学,1991.
[40]翟婉明.车辆-轨道垂向系统的统一模型及其耦合动力学原理[J].铁道学报，1992,14(3):10-21.
[41] Zai W M, Sun X. A detailed model for investigating vertical interaction betweenrailway vehicle and track[J]. Vehicle System Dynamics,1994,23(Suppl.):603-615.
[42]翟婉明，机车-轨道耦合动力学理论及其应用[J].中国铁道科学,1996,17(2):58-73.
[43] Zhai W M, Cai C B, Guo S Z. Coupling model of vertical and lateralvehicle/track interactions[J]. Vehicle System Dynamics,1996,26(1):61~79.
[44]李德建,曾庆元.列车-直线轨道空间时耦合时变系统振动分析[J].铁道学报,1997,19(1):101-107.
[45]翟婉明，蔡成标，王开云.轨道刚度对列车走形性能的影响.铁道机车车辆，1997，16(3)
[46]翟婉明，蔡成标，王其昌.高速铁路轨道刚度与胶垫的应用.铁道学报，2000，22(6)
[47]陈果.车辆-轨道耦合系统随机振动分析[D].成都:西南交通大学,2000.
[48]刘学毅,王平.轮轨空间耦合振动分析模型及其应用[J].铁道学报,1998,20(3):102-108.
[49]李成辉.轮轨结构振动理论及应用研究[D].成都:西南交通大学,1996.
[50]梁波.高速铁路路基的动力特性及土工合成材料的应用研究[D].成都:西南交通大学,1998.
[51]梁波,蔡英.不平顺条件下高速铁路路基的动力分析[J].铁道学报,1999，21(2):84-88.
[52]梁波,蔡英,朱东生.车-路垂向耦合系统的动力分析[J].铁道学报,2000,22(5):65-71.
[53]雷晓燕,陈水生.高速铁路轨道结构空间动力分析[J].铁道学报,2000,22(5):76-80.
[54]毛利军,雷晓燕.车辆-轨道耦合系统随机振动分析[J].华东交通大学学报,2001,18(2):6~12.
[55]王平.道岔区轮轨系统动力学研究[D].成都:西南交通大学,1997.
[56]任尊松.车辆-道岔系统动力学研究[D].成都:西南交通大学,2000.
[57]苏谦,蔡英.高速铁路路基结构空间时变系统耦合振动分析[J].西南交通大学学报,2001,36(5):509-513.
[58]苏谦.高速铁路路基空间时变耦合系统动力分析模型及其应用研究[D].成都：西南交通大学,2001.
[59]雷震宇，周顺华，许凯.列车动荷载对下立交结构的影响分析[J].岩石力学与工程学报，2004,23（20）:3536-3540.
[60]雷震宇.高速铁路路基基床动力特性的研究[D].上海：同济大学，2003.
[61]夏禾,张楠.车辆与结构动力相互作用[M].北京：科学出版社,2005.
[62]夏禾等著.交通环境振动工程[M].北京：科学出版社,2010.
[63] Ahlbeck D R, Meacham H C, Prause R H. The development of analytical models forrailroad track dynamics[C]. Railroad Track Mechanics&Technology,1978.
[64] Rucker W. Dynamic interaction of railroad-bed with the subsoil[C]. SoilDynamic&Earthquake Engineering Conference, Southampton,1982,1:435-448.
[65]翟婉明,林建辉,王开云.铁路道床振动的理论模拟与试验研究[J].振动工程学报,2003,16(4):404-408.
[66] Zhai Wan-ming. Two simple fast integration methods for large-scale dynamicproblems in engineering. International Journal for Numerical Methods inEngineering,1996,39(24):4199~4214.
[67]卢乃宽.世界高速铁路建设发展趋势[J].中国铁路，2000，3：19-24.
[68]刘学毅,王平.车辆轨道路基系统动力学[M].成都：西南交通大学出版社,2010.
[69] Singiresu S. Rao著,李欣业,张明路译.机械振动(第四版)[M].北京:清华大学出版社,2009.
[2]钱立新.世界高速铁路技术[M].北京：中国铁道出版社，2003.
[3]翟婉明.车辆-轨道耦合动力学(第三版)[M].北京:科学出版社,2007.
[4] Garg V K. Dynamics of railway vehicle systems[J].1984.
[5]詹斐生.机车动力学[M].北京：中国铁道出版社，1990.
[6]王福天.车辆系统动力学[M].北京：中国铁道出版社，1994.
[7] Knothe K. Gleisdynamik[M]. Berlin：Ernst&Sohn，2001.
[8]左藤裕，卢肇英译.轨道力学[M].北京：中国铁道出版社，1981.
[9]练松良.轨道动力学[M].上海：同济大学出版社，2003.
[10] FRYBA，L. History of Winkler foundation[J]. Vehicle System Dynamics，1995，24(sup1)：7-12.
[11] Prud’Homme M A，Janin M G. The stability of tracks laid with long welded rails,partsⅠ andⅡ[C]. Bulletin of the International Railway Congress Association，1969，88：460-480.
[12] Lyon D. The calculation of track forces due to dipped rail joints，wheel flats andrail welds[C]. The Second ORE Colloquium on Technical Computer Programs，1972.
[13] Jenkins H H, Stephenson J E, Clayton G A，etal. The effect of track and vehicleparameters on wheel/rail vertical dynamic forces[J]. Railway EngineeringJournal，1974,3(1)：2-16.
[14] Sato Y. Abnormal wheel load of test train[J]. Permanent Way,1973,14：1-8
[15] Ahlbeck D R, Meacham H C, Prause R H. The development of analyticalmodels for railroad track dynamics[C]. Railroad Track Mechanics&Technology,1975:197-220.
[16]Clark R.A, Dean P A., Elkins J A, Newton S G. An investigation into the dynamiceffects of railway vehicle running on corrugated rails[J]. Journal of MechanicalEngineering Science,1982,24(2):65-76.
[17] Grassie S L, Gregory R W, Johnson K L. The dynamic response of railway trackto high frequency lateral excitation[J]. Journal of Mechanical EngineeringScience,1982,24(2):91-95.
[18] Grassie S L, Gregory R W, Harrison D, et al. The dynamic response of railwaytrack to high frequency vertical excitation[J]. Journal of Mechanical EngineeringScience,1982,24(2):77-90
[19] Grassie S L, Gregory R W, Johnson K L. The behaviour of railway wheelsetsand track at high frequencies of excitation[J]. Journal of Mechanical EngineeringScience,1982,24(2):103-111.
[20] Grassie S L, Gregory R W, Johnson K L. The dynamic loading of rails atcorrugation frequencies[J]. Contact Mechanics and Wear of Rail/Wheel Systems,1982:209-228.
[21] Cai Z, Raymond G P. Theoretical model for dynamic wheel/rail and trackinteraction[C]. Proceedings of10th Interactional Wheelset Congress, Australia,1992,127-131.
[22] Fermer M. Wheel/rail contact forces for flexible versus solid wheels due to treadirregularities[J].Vehicle System Dynamics,1994,23(Suppl.):142-157.
[23] Nielsen J. Train/track Interaction Coupling of Moving and Stationary DynamicSystems[D]. Ph. D. Dissertation, Chalmers University of Technology, Gotebory,Sweden,1993.
[24] RIPKE B, KNOTHE K. Simulation of High Frequency Vehicle-trackInteractions[J]. Vehicle System Dynamics,1995,24(Suppl.):72-85.
[25] Oscarsson J， Dahlberg T. Dynamic train/track/ballast interaction-computermodels and full-scale experiments[J]. Vehicle System Dynamics，1998，29(Suppl.)：73-84.
[26] Frohling R D. Low frequency dynamic vehicle-track interaction: modeling andsimulation[J]. Vehicle System Dynamics,1998,28(Suppl.):30-46.
[27] AUERSCHL L. Vehicle-track interaction and soil dynamics[J]. Vehicle SystemDynamics,1998,28(Suppl.):553-558.
[28] Popp K, Kruse H, Kaiser I. Vehicle-track dynamics in the mid-frequencyrange[J]. Vehicle System Dynamics,1999,31(5-6):423-464.
[29] Andersson C, Oscarsson J. Dynamic train/track interaction includingstate-dependent track properties and flexible vehicle components[J]. VehicleSystem Dynamics,1999,33(Suppl.)：47-58.
[30] Drozdziel J. The effect of railway vehicle-track system geometric deviation onits dynamics in the turnout zone[J]. Vehicle System Dynamics,1999,33(Suppl.)：641-652.
[31] Gurule S，Wilson N. Simulation of wheel/rail interaction in turnouts and specialtrack work[J]. Vehicle System Dynamics，1999，33(Suppl.)：143-154.
[32] Oscarsson J. Dynamic train-track-ballast interaction with unevenly distributedtrack properties[J]. Vehicle System Dynamics,2003,37:385-396.
[33] Dietz S, Hippmann G, Schupp G. Interaction of vehicle and flexible tracks byco-simulation of multibody vehicle system and finite element track models[J].Vehicle System Dynamics,2003,37:372-385.
[34] Szole T. Simulation of vehicle-track interaction in Medium frequency range withapplication to analysis of mechanical and thermal loading in contact[C].17thIAVSD Symposium on Dynamics of Vehicles on Roads and Tracks，Lyngby，Denmark，2001
[35]吴章江,王浦强,李湘敏等.车辆通过轨道抵扣接头的垂向轮轨作用力计算[J].铁道机车车辆,1982,1：24-30.
[36]李定清.轮轨垂直相互动力作用及其动力响应[J].铁道学报，1987，9(1)：1~8
[37]许实儒,徐维杰,仲延禧.钢轨接头处轮轨冲击力的模拟分析[J].铁道学报,1989,1.
[38]王澜,姚明初.轨道结构随机振动理论及其在轨道结构减振研究中的应用[J].中国铁道科学,1989,10(2):41-59.
[39]翟婉明.车辆-轨道垂向耦合动力学[D].成都:西南交通大学,1991.
[40]翟婉明.车辆-轨道垂向系统的统一模型及其耦合动力学原理[J].铁道学报，1992,14(3):10-21.
[41] Zai W M, Sun X. A detailed model for investigating vertical interaction betweenrailway vehicle and track[J]. Vehicle System Dynamics,1994,23(Suppl.):603-615.
[42]翟婉明，机车-轨道耦合动力学理论及其应用[J].中国铁道科学,1996,17(2):58-73.
[43] Zhai W M, Cai C B, Guo S Z. Coupling model of vertical and lateralvehicle/track interactions[J]. Vehicle System Dynamics,1996,26(1):61~79.
[44]李德建,曾庆元.列车-直线轨道空间时耦合时变系统振动分析[J].铁道学报,1997,19(1):101-107.
[45]翟婉明，蔡成标，王开云.轨道刚度对列车走形性能的影响.铁道机车车辆，1997，16(3)
[46]翟婉明，蔡成标，王其昌.高速铁路轨道刚度与胶垫的应用.铁道学报，2000，22(6)
[47]陈果.车辆-轨道耦合系统随机振动分析[D].成都:西南交通大学,2000.
[48]刘学毅,王平.轮轨空间耦合振动分析模型及其应用[J].铁道学报,1998,20(3):102-108.
[49]李成辉.轮轨结构振动理论及应用研究[D].成都:西南交通大学,1996.
[50]梁波.高速铁路路基的动力特性及土工合成材料的应用研究[D].成都:西南交通大学,1998.
[51]梁波,蔡英.不平顺条件下高速铁路路基的动力分析[J].铁道学报,1999，21(2):84-88.
[52]梁波,蔡英,朱东生.车-路垂向耦合系统的动力分析[J].铁道学报,2000,22(5):65-71.
[53]雷晓燕,陈水生.高速铁路轨道结构空间动力分析[J].铁道学报,2000,22(5):76-80.
[54]毛利军,雷晓燕.车辆-轨道耦合系统随机振动分析[J].华东交通大学学报,2001,18(2):6~12.
[55]王平.道岔区轮轨系统动力学研究[D].成都:西南交通大学,1997.
[56]任尊松.车辆-道岔系统动力学研究[D].成都:西南交通大学,2000.
[57]苏谦,蔡英.高速铁路路基结构空间时变系统耦合振动分析[J].西南交通大学学报,2001,36(5):509-513.
[58]苏谦.高速铁路路基空间时变耦合系统动力分析模型及其应用研究[D].成都：西南交通大学,2001.
[59]雷震宇，周顺华，许凯.列车动荷载对下立交结构的影响分析[J].岩石力学与工程学报，2004,23（20）:3536-3540.
[60]雷震宇.高速铁路路基基床动力特性的研究[D].上海：同济大学，2003.
[61]夏禾,张楠.车辆与结构动力相互作用[M].北京：科学出版社,2005.
[62]夏禾等著.交通环境振动工程[M].北京：科学出版社,2010.
[63] Ahlbeck D R, Meacham H C, Prause R H. The development of analytical models forrailroad track dynamics[C]. Railroad Track Mechanics&Technology,1978.
[64] Rucker W. Dynamic interaction of railroad-bed with the subsoil[C]. SoilDynamic&Earthquake Engineering Conference, Southampton,1982,1:435-448.
[65]翟婉明,林建辉,王开云.铁路道床振动的理论模拟与试验研究[J].振动工程学报,2003,16(4):404-408.
[66] Zhai Wan-ming. Two simple fast integration methods for large-scale dynamicproblems in engineering. International Journal for Numerical Methods inEngineering,1996,39(24):4199~4214.
[67]卢乃宽.世界高速铁路建设发展趋势[J].中国铁路，2000，3：19-24.
[68]刘学毅,王平.车辆轨道路基系统动力学[M].成都：西南交通大学出版社,2010.
[69] Singiresu S. Rao著,李欣业,张明路译.机械振动(第四版)[M].北京:清华大学出版社,2009.
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