大跨度铁路上承式拱桥列车走行性研究
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摘要
大跨度上承式拱桥是一种应用范围较广的桥型,其优美的造型、强大的跨越能力使它在铁路桥梁中占有一席之地。本文以丽江至香格里拉新建铁路上的金沙江特大桥为工程背景对上承式拱桥进行了车桥耦合振动分析。全文主要研究内容如下：
1.综述了大跨度上承式拱桥的发展概况,回顾了国内外车桥振动研究的发展历程,阐述了上承式钢拱桥车—桥耦合振动系统分析的必要性。
2.基于弹性系统动力学总势能不变值原理及形成矩阵的“对号入座”法则建立了上承式钢拱桥的有限元模型,对该桥进行了自振特性分析。
3.基于以上分析模型,详细计算了不同工况下车桥系统的振动响应,对桥梁及列车的动力响应做出了评价,得出结论：金沙江特大桥具有良好的动力特性及列车走行性。
4.研究了桥梁各部分刚度对车桥系统动力特性的影响,计算结果表明：①在一定范围内提高桥面系刚度可以有效减少桥梁的振动动力响应。②拱上支撑墩刚度对列车桥梁振动响应的影响较小。③拱圈刚度对桥梁位移响应最大值的影响相对于其他构件刚度要大的多,提高拱圈刚度可以有效减小桥梁位移响应。④在拱圈中,拱肋刚度相对于联结系刚度对桥梁振动位移响应作用更大。提高拱肋刚度是一种减小桥梁振动位移的有效手段。
Span deck arch bridge has a wider range of applications; its graceful shape and strong ability make it beyond other railway bridges. With the Jinshajiang Bridge of the Lijiang-Shangri-La new railway as engeneering background, this paper makes a study of the vehicle-bridge coupling vibration of deck arch bridge. The works completed in this paper are as the following:
1. With the summary of the development of the span deck arch bridge, vehicle-bridge coupled vibration and vehicle-bridge coupled vibration at home and abroad, it describes the necessity of the study on vehicle-bridge coupled vibration systems.
2. Based on the principle of total potential energy with stationary value in elastic system dynamics and the "set in right position" rule for formulating system matrixes, the finite element model of deck steel arch bridge is established. And the natural vibration characteristics are analyzed.
3. Based on the above analysis models, the dynamic responses of the vehicle-bridge system under different conditions are analyzed. And we could come to the conclusion as follow:Jinsha River Bridge has good dynamic characteristics and train running property.
4. The dynamic responses of the vehicle-bridge system on different bridge system stiffness are analyzed.We could come to the conclusion as follow:①Increasing the stiffness of bridge deck system in a certain range can reduce the bridge dynamic response.②The stiffness of piers on arch has not a great impact on bridge and train vibration responses.③Compared with other parts of bridge, arch ring's stiffness has a great impact on the maximum of bridge displacement responses. Increasing the stiffness of arch ring can reduce the displacement response of the bridge In effect.④Compared with sway bracing, the arch ribs'stiffness has a great impact on bridge vibration displacements. Increasing the stiffness of arch ribs is an effective means of decreasing vibration displacement.
1.综述了大跨度上承式拱桥的发展概况,回顾了国内外车桥振动研究的发展历程,阐述了上承式钢拱桥车—桥耦合振动系统分析的必要性。
2.基于弹性系统动力学总势能不变值原理及形成矩阵的“对号入座”法则建立了上承式钢拱桥的有限元模型,对该桥进行了自振特性分析。
3.基于以上分析模型,详细计算了不同工况下车桥系统的振动响应,对桥梁及列车的动力响应做出了评价,得出结论：金沙江特大桥具有良好的动力特性及列车走行性。
4.研究了桥梁各部分刚度对车桥系统动力特性的影响,计算结果表明：①在一定范围内提高桥面系刚度可以有效减少桥梁的振动动力响应。②拱上支撑墩刚度对列车桥梁振动响应的影响较小。③拱圈刚度对桥梁位移响应最大值的影响相对于其他构件刚度要大的多,提高拱圈刚度可以有效减小桥梁位移响应。④在拱圈中,拱肋刚度相对于联结系刚度对桥梁振动位移响应作用更大。提高拱肋刚度是一种减小桥梁振动位移的有效手段。
Span deck arch bridge has a wider range of applications; its graceful shape and strong ability make it beyond other railway bridges. With the Jinshajiang Bridge of the Lijiang-Shangri-La new railway as engeneering background, this paper makes a study of the vehicle-bridge coupling vibration of deck arch bridge. The works completed in this paper are as the following:
1. With the summary of the development of the span deck arch bridge, vehicle-bridge coupled vibration and vehicle-bridge coupled vibration at home and abroad, it describes the necessity of the study on vehicle-bridge coupled vibration systems.
2. Based on the principle of total potential energy with stationary value in elastic system dynamics and the "set in right position" rule for formulating system matrixes, the finite element model of deck steel arch bridge is established. And the natural vibration characteristics are analyzed.
3. Based on the above analysis models, the dynamic responses of the vehicle-bridge system under different conditions are analyzed. And we could come to the conclusion as follow:Jinsha River Bridge has good dynamic characteristics and train running property.
4. The dynamic responses of the vehicle-bridge system on different bridge system stiffness are analyzed.We could come to the conclusion as follow:①Increasing the stiffness of bridge deck system in a certain range can reduce the bridge dynamic response.②The stiffness of piers on arch has not a great impact on bridge and train vibration responses.③Compared with other parts of bridge, arch ring's stiffness has a great impact on the maximum of bridge displacement responses. Increasing the stiffness of arch ring can reduce the displacement response of the bridge In effect.④Compared with sway bracing, the arch ribs'stiffness has a great impact on bridge vibration displacements. Increasing the stiffness of arch ribs is an effective means of decreasing vibration displacement.
引文
[1]吴冲.现代钢桥.人民交通出版社,2006
[2]小西一郎.钢桥(第三分册).北京：人民铁道出版社,1980
[3]周远棣,徐君兰.钢桥.北京：人民交通出版社,2003
[4]李富文,伏魁先,刘学信.钢桥.北京：中国铁道出版社,1996
[5]铁道专业设计院.钢桥.北京：中国铁道出版社,2003
[6]王承礼,徐名枢.铁路桥梁.北京：中国铁道出版社,1993
[7]陈宝春,高婧,吴庆雄.钢拱桥发展概况.北京交通大学学报,2006,30(增刊)：22～25
[8]颜毅.大跨度钢桁架拱桥受力特性分析：[硕士学位论文].重庆：重庆交通大学,2008
[9]Biggs I M. Introductiong to Structural Dynamics. NewYork:Mc-Graw-Hill Book Co Inc,1964
[10]Fryba L. Vibration of Solids and Structures under Moving loads. Netherlands:Noor-dhoff International Publishing Groningen,1972
[11]铁木辛柯等著,胡人礼译.工程中的振动问题.人民铁道出版社,1987
[12]陈英俊.车辆荷载下桥梁振动基本理论的演进.桥梁建设,1975(2)
[13]Krylov A N. Uber die Erzwungen Schwingungen Von Gleichformigen Elastischen Staben. Mathematische Analen,1905,4(3):6～11
[14]Timoshenko S P. On the forced vibration of Bridges. Philosoph Magazine Ser,1922,6(2):43～48
[15]C.E.Inglis. A Mathematical Treatise on Vibrations in Railway Bridges.Cambridge,1934:
[16]Mise K.,Kunii s.:A Theory for the Forced Vibration of Elastic Systems Subjected to Moving loads,Quarterly Journal of Mechanics &Applied Matrhematics,1956,9
[17]曾庆元,郭向荣.列车桥梁时变系统振动分析理论与应用.北京：中国铁道出版社,1999
[18]Fryba L. Dynamics of Railway Bridges Thomas Telford, London, England, 1996
[19]Fryba L. A Rough Assessment of Railway Bridges for High Speed Trains. Engineering Structures,2001,23:548～556
[20]松浦章夫.高速铁路车辆与桥梁的相互作用.铁道技术研究资料.1974.31 (5)14-17
[21]松浦章夫.高速铁路桥梁动力问题的研究.日本土木学会论文报告集.1976,12(256)：35-47.
[22]M H Bhatti. Vertical and lateral dynamic response of railway bridges due to nonlinear vehicle and track irregularities[D].Illinois Institute of Technology, Chicago,Illinois,1982
[23]Chu K H, Garg V K. Railway-bridge impact:simplified train and bridge model[J] Journal of the Structural Division, ASCE.1979,105, No. ST9
[24]Wiriyachai A. Impact and fatigue in open-deck railway truss bridge[D]. Illinois Institute of Technology, Chicago, Illinois,1980
[25]Ton-Lo Wang. Impact and fatigue in open-deck steel truss and ballasted prestressed concrete railway bridge[D]. Illinois Institute of Technology, Chicago, Illinois,1984
[26]Ton-Lo Wang. Ramp/Bridge interface in Railway Prestressed Concrete Railway Bridges. Journal of Structural Engineering,1990,116(6):1648-1659
[27]Diana G, et al. Dynamic Interaction between Railway Vehicles and Tracks. Report Dip. Di. Meccanica Politecnico di Milano,1986
[28]Diana G, et al. A Numerical Method to Define the Dynamic Behavior of a Train Running on a Deformable Structure. MECCANICA, Special Issue,1988: 27-42
[29]Olsson M. Finite Element Model Coordinate Analysis of Structures Subjected to Moving Loads. Journal of Sound&Vbration,1985,99(1):1～12
[30]Olsson M. On the Foundational Moving Load Problems. Journal of Sound & Vibration,1991,145 (2):299～307
[31]Tanabe M. Yamada Y. Model Method for Interaction of Train and Bridge. Journal of Computers & Structures,1987,27(1):119～127
[32]Bogaert Van. Dynamic Response of Trains Crossing Large Span Double-track Bridges. Journal of Constructional Steel Research,1993,24(1):57～74
[33]Green M F, Cebon D. Dynamic Response of Highway Bridges to Heavy Vehicles Loads:Theory and Experimental Validation. Journal of Sound & Vibration,1994,170(1):51～78
[34]Yang Y B, etal. Vehicle-Bridge Interaction Analysis by Dynamic Condensation Method. Journal of Structural Engineering. ASCE,1995,121(11):1636～1643
[35]Yang Y B, etal. Impact Formulas for Vehicle Moving over Simple and Continuous Beams. Journal of Structural Engineering, ASCE,1995, 121(11):1644～1650
[36]李国豪.桥梁结构稳定与振动.中国铁道出版社,2002.98-156
[37]李国豪.桁梁桥的扭转、稳定和振动.人民交通出版社,1975
[38]沈锐利.高速铁路简支梁桥竖向振动响应研究.中国铁道科学,1996,17(3)
[39]李小珍.高速铁路列车-桥梁系统耦合振动理论及应用研究：[博士学位论文].成都：西南交通大学,2000
[40]李小珍,强士中.京沪高速铁路南京越江钢斜拉桥车桥耦合振动分析.西南交通大学学报,1999,34(2)
[41]何发礼李乔.曲率和超高对曲线梁桥车桥耦合振动的影响.桥梁建设.1999(3)
[42]单德山,李乔.铁路曲线连续梁桥车桥耦合振动分析.中国铁道科学,2004,25(5)
[43]铁道部科学研究院铁道建筑研究所.高速铁路线桥结构与技术条件的研究研究报告.1996：12
[44]王福天.车辆系统动力学.北京：中国铁道出版社,1994
[45]程庆国,潘家英.大跨度铁路斜拉桥竖向刚度分析.全国桥梁结构学术大会论文集,武汉,1992：1163-1168
[46]许慰平.大跨度铁路桥梁车桥空间耦合振动研究：[博士学位论文].北京：铁道科学研究院,1988
[47]杨岳民.大跨度铁路桥桥梁车桥动力响应理论分析及试验研究：[博士学位论文].北京：铁道科学研究院,1995
[48]高芒芒.高速铁路列车-线路-桥梁耦合振动及列车走行性研究：[博士学位论文].北京：铁道科学研究院,2001
[49]夏禾.列车过桥时高桥墩的动力响应及其对车辆运行稳定性的影响：[硕士学位论文].北京：北方交通大学,1984
[50]夏禾,陈英俊.车—梁—墩体系动力相互作用分析.土木工程学报,1992,25(2)：3～12
[51]Xia He, Chen Yingjun. Dynamic Reliability of Train-Bridge System under Wind Action by Stochastic Extreme Analysis. Proc. EASEL-4, Seoul, Korea. 1993:2063～2068
[52]夏禾,陈英俊.风和列车荷载同时作用下车桥系统的动力可靠性.土木工程学报,1994,27(2)：14～21
[53]阎贵平,夏禾,陈英俊.列车与刚梁柔拱组合系桥系统的地震响应分析. 北方交通大学学报,1999,18(1)：9～16
[54]曾庆元、骆宁安、江锋.桥上列车横向摇摆力的初步研究.桥梁建设,1990(1)
[55]陈淮,曾庆元.特大钢桁梁桥振动分析.长沙铁道学院学报,1990,8
[56]郭文华,曾庆元.高速铁路多跨简支梁桥横向振动随机分析.长沙铁道学院学报,1997,15(1)
[57]郭文华,郭向荣,曾庆元.大跨度斜拉桥空间振动计算分析.振动与冲击,1998,17(1)
[58]曾庆元,郭向荣.列车桥梁时变系统振动分析理论及应用.北京：中国铁道出版社,1995
[59]王荣辉,郭向荣,曾庆元.高速列车—钢桁梁桥横向振动随机分析.铁道学报,1996,18(1)
[60]朱汉华.列车—桥梁时变系统振动能量随机分析理论：[博士学位论文].长沙：长沙铁道学院,1993
[61]陈淮.高架铁路桥横向刚度研究：[博士学位论文].长沙：长沙铁道学院1993
[62]郭向荣.大跨度铁路斜拉桥的车振分析：[硕士学位论文].长沙：长沙铁道学院,1993
[63]李德建.列车—轨道时变系统空间振动分析：[博士学位论文].长沙：长沙铁道学院,1996
[64]曾庆元,向俊,娄平.车桥及车轨时变系统横向振动计算中的根本问题与列车脱轨能量随机分析理论.中国铁道科学,2002,23(1)
[65]曾庆元.弹性系统动力学总势能不变值原理与列车桥梁时变系统振动分析.铁道建筑技术特稿.2001(1)
[66]曾庆元.弹性系统动力学总势能不变值原理.华中理工大学学报.2001,28(1)
[67]中华人民共和国铁道部.铁路桥涵设计基本规范TB10002.1-99.北京：人民铁道出版社,1978
[68]曹雪琴,陈晓.轮轨蛇行引起桥梁横向振动随机分析.铁道学报.1986,8(1)：89-97
[69]曹雪琴等.列车过桥箱型钢梁空间振动分析.上海铁道学院学报.1982,3(3)：17-35
[70]曹雪琴,钢桁梁桥横向振动.中国铁道出版社,1991.23-78
[71]曹雪琴,刘必胜,昊鹏贤.桥梁结构动力分析.中国铁道出版社,1987.
[72]王荣辉郭向荣曾庆元.车桥系统中的车辆分析模型.振动与冲击.第16 卷第3期
[73]曾庆元,郭向荣.列车桥梁时变系统振动分析理论及应用,北京：中国铁道出版社,1999.49-131
[74]娄平,曾庆元.车辆轨道桥梁系统竖向运动方程的建立.铁道学报.2004,10(5)：65～78
[75]曾庆元、杨平：形成矩阵的“对号入座”法则与桁段有限元法,铁道学报,1986(2)
[76]王福天.车辆系统动力学.北京：中国铁道出版社,1994
[77]翟婉明.车辆-轨道耦合动力学.北京：科学出版社,2007
[2]小西一郎.钢桥(第三分册).北京：人民铁道出版社,1980
[3]周远棣,徐君兰.钢桥.北京：人民交通出版社,2003
[4]李富文,伏魁先,刘学信.钢桥.北京：中国铁道出版社,1996
[5]铁道专业设计院.钢桥.北京：中国铁道出版社,2003
[6]王承礼,徐名枢.铁路桥梁.北京：中国铁道出版社,1993
[7]陈宝春,高婧,吴庆雄.钢拱桥发展概况.北京交通大学学报,2006,30(增刊)：22～25
[8]颜毅.大跨度钢桁架拱桥受力特性分析：[硕士学位论文].重庆：重庆交通大学,2008
[9]Biggs I M. Introductiong to Structural Dynamics. NewYork:Mc-Graw-Hill Book Co Inc,1964
[10]Fryba L. Vibration of Solids and Structures under Moving loads. Netherlands:Noor-dhoff International Publishing Groningen,1972
[11]铁木辛柯等著,胡人礼译.工程中的振动问题.人民铁道出版社,1987
[12]陈英俊.车辆荷载下桥梁振动基本理论的演进.桥梁建设,1975(2)
[13]Krylov A N. Uber die Erzwungen Schwingungen Von Gleichformigen Elastischen Staben. Mathematische Analen,1905,4(3):6～11
[14]Timoshenko S P. On the forced vibration of Bridges. Philosoph Magazine Ser,1922,6(2):43～48
[15]C.E.Inglis. A Mathematical Treatise on Vibrations in Railway Bridges.Cambridge,1934:
[16]Mise K.,Kunii s.:A Theory for the Forced Vibration of Elastic Systems Subjected to Moving loads,Quarterly Journal of Mechanics &Applied Matrhematics,1956,9
[17]曾庆元,郭向荣.列车桥梁时变系统振动分析理论与应用.北京：中国铁道出版社,1999
[18]Fryba L. Dynamics of Railway Bridges Thomas Telford, London, England, 1996
[19]Fryba L. A Rough Assessment of Railway Bridges for High Speed Trains. Engineering Structures,2001,23:548～556
[20]松浦章夫.高速铁路车辆与桥梁的相互作用.铁道技术研究资料.1974.31 (5)14-17
[21]松浦章夫.高速铁路桥梁动力问题的研究.日本土木学会论文报告集.1976,12(256)：35-47.
[22]M H Bhatti. Vertical and lateral dynamic response of railway bridges due to nonlinear vehicle and track irregularities[D].Illinois Institute of Technology, Chicago,Illinois,1982
[23]Chu K H, Garg V K. Railway-bridge impact:simplified train and bridge model[J] Journal of the Structural Division, ASCE.1979,105, No. ST9
[24]Wiriyachai A. Impact and fatigue in open-deck railway truss bridge[D]. Illinois Institute of Technology, Chicago, Illinois,1980
[25]Ton-Lo Wang. Impact and fatigue in open-deck steel truss and ballasted prestressed concrete railway bridge[D]. Illinois Institute of Technology, Chicago, Illinois,1984
[26]Ton-Lo Wang. Ramp/Bridge interface in Railway Prestressed Concrete Railway Bridges. Journal of Structural Engineering,1990,116(6):1648-1659
[27]Diana G, et al. Dynamic Interaction between Railway Vehicles and Tracks. Report Dip. Di. Meccanica Politecnico di Milano,1986
[28]Diana G, et al. A Numerical Method to Define the Dynamic Behavior of a Train Running on a Deformable Structure. MECCANICA, Special Issue,1988: 27-42
[29]Olsson M. Finite Element Model Coordinate Analysis of Structures Subjected to Moving Loads. Journal of Sound&Vbration,1985,99(1):1～12
[30]Olsson M. On the Foundational Moving Load Problems. Journal of Sound & Vibration,1991,145 (2):299～307
[31]Tanabe M. Yamada Y. Model Method for Interaction of Train and Bridge. Journal of Computers & Structures,1987,27(1):119～127
[32]Bogaert Van. Dynamic Response of Trains Crossing Large Span Double-track Bridges. Journal of Constructional Steel Research,1993,24(1):57～74
[33]Green M F, Cebon D. Dynamic Response of Highway Bridges to Heavy Vehicles Loads:Theory and Experimental Validation. Journal of Sound & Vibration,1994,170(1):51～78
[34]Yang Y B, etal. Vehicle-Bridge Interaction Analysis by Dynamic Condensation Method. Journal of Structural Engineering. ASCE,1995,121(11):1636～1643
[35]Yang Y B, etal. Impact Formulas for Vehicle Moving over Simple and Continuous Beams. Journal of Structural Engineering, ASCE,1995, 121(11):1644～1650
[36]李国豪.桥梁结构稳定与振动.中国铁道出版社,2002.98-156
[37]李国豪.桁梁桥的扭转、稳定和振动.人民交通出版社,1975
[38]沈锐利.高速铁路简支梁桥竖向振动响应研究.中国铁道科学,1996,17(3)
[39]李小珍.高速铁路列车-桥梁系统耦合振动理论及应用研究：[博士学位论文].成都：西南交通大学,2000
[40]李小珍,强士中.京沪高速铁路南京越江钢斜拉桥车桥耦合振动分析.西南交通大学学报,1999,34(2)
[41]何发礼李乔.曲率和超高对曲线梁桥车桥耦合振动的影响.桥梁建设.1999(3)
[42]单德山,李乔.铁路曲线连续梁桥车桥耦合振动分析.中国铁道科学,2004,25(5)
[43]铁道部科学研究院铁道建筑研究所.高速铁路线桥结构与技术条件的研究研究报告.1996：12
[44]王福天.车辆系统动力学.北京：中国铁道出版社,1994
[45]程庆国,潘家英.大跨度铁路斜拉桥竖向刚度分析.全国桥梁结构学术大会论文集,武汉,1992：1163-1168
[46]许慰平.大跨度铁路桥梁车桥空间耦合振动研究：[博士学位论文].北京：铁道科学研究院,1988
[47]杨岳民.大跨度铁路桥桥梁车桥动力响应理论分析及试验研究：[博士学位论文].北京：铁道科学研究院,1995
[48]高芒芒.高速铁路列车-线路-桥梁耦合振动及列车走行性研究：[博士学位论文].北京：铁道科学研究院,2001
[49]夏禾.列车过桥时高桥墩的动力响应及其对车辆运行稳定性的影响：[硕士学位论文].北京：北方交通大学,1984
[50]夏禾,陈英俊.车—梁—墩体系动力相互作用分析.土木工程学报,1992,25(2)：3～12
[51]Xia He, Chen Yingjun. Dynamic Reliability of Train-Bridge System under Wind Action by Stochastic Extreme Analysis. Proc. EASEL-4, Seoul, Korea. 1993:2063～2068
[52]夏禾,陈英俊.风和列车荷载同时作用下车桥系统的动力可靠性.土木工程学报,1994,27(2)：14～21
[53]阎贵平,夏禾,陈英俊.列车与刚梁柔拱组合系桥系统的地震响应分析. 北方交通大学学报,1999,18(1)：9～16
[54]曾庆元、骆宁安、江锋.桥上列车横向摇摆力的初步研究.桥梁建设,1990(1)
[55]陈淮,曾庆元.特大钢桁梁桥振动分析.长沙铁道学院学报,1990,8
[56]郭文华,曾庆元.高速铁路多跨简支梁桥横向振动随机分析.长沙铁道学院学报,1997,15(1)
[57]郭文华,郭向荣,曾庆元.大跨度斜拉桥空间振动计算分析.振动与冲击,1998,17(1)
[58]曾庆元,郭向荣.列车桥梁时变系统振动分析理论及应用.北京：中国铁道出版社,1995
[59]王荣辉,郭向荣,曾庆元.高速列车—钢桁梁桥横向振动随机分析.铁道学报,1996,18(1)
[60]朱汉华.列车—桥梁时变系统振动能量随机分析理论：[博士学位论文].长沙：长沙铁道学院,1993
[61]陈淮.高架铁路桥横向刚度研究：[博士学位论文].长沙：长沙铁道学院1993
[62]郭向荣.大跨度铁路斜拉桥的车振分析：[硕士学位论文].长沙：长沙铁道学院,1993
[63]李德建.列车—轨道时变系统空间振动分析：[博士学位论文].长沙：长沙铁道学院,1996
[64]曾庆元,向俊,娄平.车桥及车轨时变系统横向振动计算中的根本问题与列车脱轨能量随机分析理论.中国铁道科学,2002,23(1)
[65]曾庆元.弹性系统动力学总势能不变值原理与列车桥梁时变系统振动分析.铁道建筑技术特稿.2001(1)
[66]曾庆元.弹性系统动力学总势能不变值原理.华中理工大学学报.2001,28(1)
[67]中华人民共和国铁道部.铁路桥涵设计基本规范TB10002.1-99.北京：人民铁道出版社,1978
[68]曹雪琴,陈晓.轮轨蛇行引起桥梁横向振动随机分析.铁道学报.1986,8(1)：89-97
[69]曹雪琴等.列车过桥箱型钢梁空间振动分析.上海铁道学院学报.1982,3(3)：17-35
[70]曹雪琴,钢桁梁桥横向振动.中国铁道出版社,1991.23-78
[71]曹雪琴,刘必胜,昊鹏贤.桥梁结构动力分析.中国铁道出版社,1987.
[72]王荣辉郭向荣曾庆元.车桥系统中的车辆分析模型.振动与冲击.第16 卷第3期
[73]曾庆元,郭向荣.列车桥梁时变系统振动分析理论及应用,北京：中国铁道出版社,1999.49-131
[74]娄平,曾庆元.车辆轨道桥梁系统竖向运动方程的建立.铁道学报.2004,10(5)：65～78
[75]曾庆元、杨平：形成矩阵的“对号入座”法则与桁段有限元法,铁道学报,1986(2)
[76]王福天.车辆系统动力学.北京：中国铁道出版社,1994
[77]翟婉明.车辆-轨道耦合动力学.北京：科学出版社,2007