摘要
我国列车不断提速。随着列车速度的不断提高,轨道、桥梁结构受到的动力冲击作用越来越大,列车自身的动力响应也在增大。国内外对列车—轨道(桥梁)竖向振动的研究很多,但在下列方面仍存在一些问题:例如:钢轨截面剪力和弯矩的计算公式,高速列车—板式轨道—桥梁系统动力响应的有限元分析,车轮跳离钢轨的安全性,桥梁的挠度和应力冲击系数等。本文针对上述问题,开展了一系列的工作,取得了如下成果:
1.首次基于单元动力的平衡条件,提出了计算移动列车作用下连续或离散粘弹性基础钢轨任意截面剪力和弯矩的有限元公式。所提出的公式不仅能退化为计算移动列车作用下简支或连续Bernoulli-Euler梁任意截面剪力和弯矩的有限元公式,也能退化为计算静止荷载作用下Winkler基础Bernoulli-Euler梁任意截面剪力和弯矩的有限元公式。用几个算例验证了所提出公式的正确性,并与使用直接对位移求导方法的计算结果进行了对比。数值结果表明:与直接对位移求导的方法相比,运用本文所提出的有限元公式可以显著提高计算结果的精度及计算效率。
2.视列车、桥梁两端有碴轨道、桥上板式轨道以及多座桥梁为一个整体系统,将轨道和桥梁用有限元离散,利用曾庆元院士提出的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,首次建立了有限元形式的系统振动矩阵方程。分析了两种确定性轨道不平顺(钢轨扣件失效不平顺、钢轨表面存在单一谐波型不平顺)对系统动力响应的影响;研究了轨道高低随机不平顺激扰下高速列车与桥上板式轨道系统的动力响应。数值结果表明:(1)为了高速列车安全运行及轨道处于良好的受力状态,应保证扣件处于正常的工作状态,严格限制短波长不平顺的波深;(2)简支梁桥设计宜用应力冲击系数,而不宜用挠度冲击系数;(3)秦沈客运专线24m跨度桥梁以及轨道参数能保证高速列车正常运行,故能用于高速铁路设计之中。
3.基于轮轨刚性接触模型,首次导出了考虑轮对跳离钢轨的2轴车辆—桥梁振动矩阵方程,数值模拟了车轮从正常接触钢轨、跳离钢轨到再次接触钢轨的动态过程。数值结果显示:当车轮跳离钢轨时,对于同样的不平顺波长l_a,不平顺波深(?)越大、车速越高,车轮跳离得越高,轮轨冲击力越大,第2次轮轨分离的持续时间越长,轮轨两次分离之间的时间间隔越短。
4.将下承式简支桁架桥、桥上的双层悬挂机车及其牵引的多辆具有一系中央悬挂的货车视为一个系统,首次导出了该系统的竖向振动矩阵方程。利用我国三大干线轨道高低随机不平顺功率谱密度函数,计算了提速货物列车和旅客列车通过64m跨度下承式简支钢桁架桥时下弦杆节点的挠度冲击系数和每根杆件的应力冲击系数。数值结果显示:(1)提速旅客列车以速度160、180、200 km/h运行时的桥梁挠度冲击系数和杆件应力冲击系数均小于货物列车以速度60、70、80、90 km/h运行时相应的冲击系数;(2)在桁梁设计中宜用应力冲击系数,而不宜用挠度冲击系数。
5.分别导出了有限元法和模态分析法的双层轨道结构模型的列车—有碴轨道—桥梁系统竖向振动的矩阵方程。对比了两种方法的计算结果和计算效率,数值结果表明:(1)用有限元法计算钢轨响应和桥梁响应,均能获得较高精度的计算结果;(2)用模态分析法计算桥梁响应时,可以获得较高精度的计算结果;(3)当采用相同的自由度计算钢轨响应时,与有限元法相比,模态分析法的计算精度低(特别是弯矩的计算精度低),并且计算效率低。
Our country keeps on raising the speed of the trains.With the increase of the train speed,the dynamic impacts on the track structure or the bridge structure increase as well as the dynamic responses of the trains.Though the vertical vibration of the train-track/bridge system has been investigated in the extensive literature,the following issues remain such as the formulae calculating the shear force and the bending moment at any cross-section of a rail,the finite element analysis of the dynamic responses of the high-speed train-slab track-bridge system,the safety for the wheel-rail separation,and the stress/deflection impact factor of the bridge.Aiming at these issues,the following series of studies have been conducted and some conclusions drawn.
1.Based on the equilibrium condition of force of beam element,some finite-element formulae are presented to calculate the shear force and the bending moment at any cross-section of a rail on either continuously or discretely viscoelastic foundation subjected to a moving train.Furthermore,the proposed formulae can be easily degenerated into either formulae calculating the shear force and the bending moment of a simply supported or a continuous Bernoulli-Euler beam subjected to a moving train,or formulae calculating the shear force and the bending moment of a Bernoulli-Euler beam on Winlder foundation under static loads.The correctness of the proposed formulae has been verified by the analytical solutions to some numerical examples.The solutions obtained by the proposed formulae have been compared with those calculated directly from the derivatives of the element displacements,and the numerical results have shown that the proposed formulae not only raise the efficiency but also improve the accuracy of calculation.
2.A train,the ballast track on each approach embankment of bridges,the slab track on bridges,and a series of bridges have been considered as an entire system,in which the rail,the slabs and the bridges have been divided into a finite number of beam elements,respectively.The equation of motion for this system has been formulated by means of the principle of total potential energy with stationary value in elastic system dynamics and the "set-in-right-position" rule for formulating matrices, with both the principle and the rule being presented by academician Qingyuan Zeng. The effects of the two types of definite track irregularities,that is,the irregularity caused by the invalid rail fastenings and the single harmonic irregularity on track surface,on the dynamic responses of this system have been investigated.The dynamic responses of the trains,the slab tracks and the bridges caused by a high-speed train passing through the track with profile random irregularities have also been studied.The numerical results have shown that(ⅰ)in order to guarantee the safe running of high-speed trains or that the rails being acted on by sustainable forces, the fastenings must be kept in normal working condition,and the maximum depth of track unevenness has to be kept within limits;(ⅱ)the stress impact factors,not the deflection impact factors should be applied in the design of simply-supported beam bridges;(ⅲ)the parameters for the 24m-span bridge beams and for the tracks on the Qinhuangdao-Shenyang passenger transport line can be applied in the high-speed railway design to guarantee the normal running of high-speed trains.
3.Based on the rigid wheel-rail contact model,the equation of motion of a two-axle vehicle-bridge system with one or two wheelsets of vehicle separating from the bridge has been derived.The whole process of the wheel-rail contact,separation, and re-contact has been simulated.The numerical results have shown that:when the wheel separates from the rail,on the condition of equal wavelength l_αof the track irregularity,the bigger the maximum depth of track unevenness is and the higher the train speed becomes,the higher the wheel will separate from the rail,the bigger the wheel-track impact force,the longer the duration of the second wheel-rail separation, and the shorter the duration between the two separations.
4.A matrix equation is derived for the vertical motion of a system that is composed of a simply supported through truss bridge subjected to a moving train consisting of a locomotive with two-stage suspension and a set of freight cars with central single-stage suspension.Based on the spectral density power function of the random profile irregularity on the three main railway lines in China,both the deflection impact factors at the nodes of the down chord member and the stress impact factors of each member of a simply supported 64m-span through steel truss bridge subjected to a speed-raising freight train or passenger train are calculated, respectively.The numerical results have shown that:(ⅰ)for a bridge under a speed-raising passenger train at a constant speed of 160,180 or 200 km/h,its deflection impact factors and stress impact factors are smaller than those corresponding factors for a bridge under a speed-raising freight train at a constant speed of 60,70,80 or 90 km/h;(ⅱ)in the truss bridge design,the stress impact factor,not the deflection impact factor,should be used.
5.Two types of equations of motion,one based on the finite element method and the other on the modal analysis method,for the vertical vibration of the train,track and bridge interaction system are derived with two-layer track model.The numerical results obtained by the finite element method are compared with those by the modal analysis method.The numerical results have shown:(ⅰ)the finite element method can yield results of high accuracy for the responses of either the rail or the bridge;(ⅱ)the modal analysis method can yield results of high accuracy for the responses of the bridge;(ⅲ)when the same degrees of freedom of rail have been adopted by the two methods,compared with the finite element method,the modal analysis method yields the results for the responses of the rail not only with lower efficiency but with lower accuracy,in particular and the results for the bending moment of the rail are not accurate enough.
引文
[1]李国豪.桥梁结构稳定与振动.北京:人民铁道出版社,1965
[2]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
[3]Timoshenko S.Young D.H.and Weaver W.J.R.Vibration Problems in Engineering(4~(th)edition).New York:John Wiley & Sons,1974
[4]Timoshenko S.Method of analysis of statical and dynamical stresses in rail.Proceedings of the Second International Congress for Applied Mechanics,Zurich,Swiss,1926,pp.407-418.
[5]Kenney J.T.Steady-state vibrations of beam on elastic foundation for moving load.Journal of Applied Mechanics,ASME,1954,21(4):359-364
[6]Florence A.L.Traveling force on a Timoshenko beam.Journal of Applied Mechanics,ASME,1965,32(2):351-358
[7]Sato Y.Study on high frequency vibration in track operated with high-speed trains.Quarterly Report of Railway Technical Research Institute,1977,24(2):109-114
[8]Grassie S.L.Gregory R.W.Harrison D.and Johnson K.L.The dynamic response of railway track to high frequency vertical excitation.Journal of Mechanical Engineering Science,1982,24(2):77-90
[9]Patil S.P.Natural frequencies of an infinite beam on a simple inertial foundation model.International Journal of Solids and Structures,1987,23(12):1615-1623
[10]Patil S.P.Response of infinite railway track to vibrating mass.Journal of Engineering Mechanics ASCE,1988,114(4):68-703
[11]Duffy D.G.The response of an infinite railroad track to a moving vibrating mass.Journal of Applied Mechanics ASME,1990,57(1):66-73
[12]Fryba L.Vibration of Solids and Structures under Moving Loads(3rd Edition).Prague:Academia,1999
[13]Thambiratnam D.and Zhuge Y Dynamic analysis of beams on an elastic foundation subjected to moving loads.Journal of Sound and Vibration,1996,198(2):149-169
[14]Wu J.S.and Shih P.Y.Dynamic responses of railway and carriage under the high-speed moving loads.Journal of Sound and Vibration,2000,236(1):61-87
[15]Wu J.S.and Shih P.Y.The dynamic behaviour of a finite railway under the high-speed multiple moving forces by using finite element method.Communications in Numerical Methods in Engineering.2000.16(12):851-866
[16]Lei X.and Noda N.A.Analyses of dynamic response of vehicle and track coupling system with random irregularity of track vertical profile.Journal of Sound and Vibration,2002,258(1):147-165
[17]Hou K.,Kalousek J.and Dong R.A dynamic model for an asymmetrical vehicle/track system.Journal of Sound and Vibration,2003,267(3):591-604
[18]Young T.H.and Li C.Y Vertical vibration analysis of vehicle/imperfect track systems.Vehicle System Dynamics,2003,40(5):329-349
[19]Zhai W.and Cai Z.Dynamic interaction between a lumped mass vehicle and a discretely supported continuous rail track.Computers and Structures,1997,63(5):987-997
[20]翟婉明.车辆-轨道耦合动力学(第二版).北京:中国铁道出版社,2002
[21]Sun Y.Q.and Dhanasekar M.A dynamic model for the vertical interaction of the rail track and wagon system.International Journal of Solids and Structures,2002,39(5):1337-1359
[22]Shah V.N.Cook R.D.and Huang T.C.Loads moving on beam supported by layered elastic foundation.Journal of Mechanical Design,ASME,1980,102(2):295-302
[23]Clark R.A.Dean P.A.Elkins J.A.and Newton S.G.An investigation into the dynamic effects of railway vehicles running on corrugated rails.Journal of Mechanical Engineering Science,1982,24(2):65-76
[24]李定清.轮轨垂直相互动力作用及其动力响应.铁道学报,1987,9(1):1-8
[25]王澜.轨道结构随机振动理论及其在轨道结构减振中的应用:[博士学位论文].北京:铁道部科学研究院,1988
[26].Nielsen J.C.O.and Abrahamsson T.J.S.Coupling of physical and modal components for analysis of moving non-linear dynamic systems on general beam structures.International Journal for Numerical Methods in Engineering 1992,33(9):1843-1859
[27]翟婉明.车辆—轨道垂向系统的统—模型及其耦合动力学原理.铁道学报,1992,14(3):10-21
[28]Cai Z.and Raymond G P.Theoretical model for dynamic wheel/rail and track interaction.Proceedings ofthe 10~(th)International Wheelset Congress,Sydney,Australia,1992,pp.127-131
[29]Fryba L.Nakagiri S.and Yoshikawa N.Stochastic finite elements for a beam on a random foundation with uncertain damping under a moving force.Journal of Sound and Vibration,1993,163(1):31-45
[30]翟婉明,孙翔,詹斐生.机车车辆与轨道垂向相互作用的计算机仿真研究.中国铁道科学,1993,14(1):42-50
[31].Cai Z.arid Raymond G.P.Modelling the dynamic response of railway track to wheel/rail impact loading.Structural Engineering and Mechanics,1994,2(1):95-112
[32]翟婉明,王其昌.轮轨动力分析模型研究.铁道学报,1994,16(1):64-72
[33]Dong R.G.Sankar S.and Dukkipati R.V.A finite element model of railway track and its application to wheel flat problem.Proceedings of the Institution of Mechanical Engineers Part F Journal of Rail and Rapid Transit,1994,208(1):61-72
[34]Zhai W.and Sun X.A detailed model for investigating vertical interaction between railway vehicle and track.Vehicle System Dynamics,1994,23(supplement):603-615
[35]Nielsen J.C.O.and Igeland A.Vertical dynamic interaction between train and track—influence of wheel and track imperfections.Journal of Sound and Vibration,1995,187(5):825-839
[36]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
[37]李德建,曾庆元.列车—直线轨道空间耦合时变系统振动分析.铁道学报,1997,19(1):101-107
[38]雷晓燕.铁路轨道结构数值分析方法.北京:中国铁道出版社,1998
[39]Esveld C.and Kok A.W.M.Interaction between moving vehicles and railway track at high speed.Rail Engineering International,1998,27(3):14-16
[40]Oscarsson J.and Dahlberg T.Dynamic train/track/ballast interaction—computer models and full-scale experiments.Vehicle System Dynamics,1998,28(supplement):73-84
[41]Dukkipatti R.V.and Dong R.Idealized steady state interaction between railway vehicle and track.Proceedings of the Institution of Mechanical Engineers Part F Journal of Rail and Rapid Transit,1999,213(1):15-29
[42]雷晓燕,陈水生.高速铁路轨道结构空间动力分析.铁道学报,2000,22(5):76-80
[43]翟婉明,蔡成标,王其昌,卢祖文,吴细水.列车提速对线路的动力影响研究与对策.中国铁道科学,2000,21(3):11-20
[44]Lei X.and Mao L.Dynamic responses analyses of vehicle and track coupling system on track transition of conventional high speed railway.Journal of Sound and Vibration,2004,271(3-4):1133-1146
[45]练松良.轨道动力学.上海:同济大学出版社,2003
[46]Koh C.GOng J.S.Y.Chua D.K.H.and Feng,J.Moving element method for train-track dynamics.International Journal for Numerical Methods in Engineering,2003,56(11):1549-1567
[47]Nielsen J.C.O.and Oscarsson J.Simulation of dynamic train-track interaction with state-dependent track properties.Journal of Sound and Vibration,2004,275(3-5):515-532
[48]Baeza L.Roda A.and Nielsen J.C.O.Railway vehicle/track interaction analysis using a modal substructuring approach.Journal of Sound and Vibration,2006,293(1-2):112-124
[49]Lou P.and Zeng Q.Y.Vertical vehicle-track coupling element.Proceedings of the Institution of Mechanical Engineers Part F Journal of Rail and Rapid Transit,2006,220(3):293-304
[50]Cook R.D.Concepts and Applications of Finite Element Analysis(2nd Edition).New York: John Wiley & Sons,1981
[51]陈淮,殷学纲.具有不变轴向力的梁式结构动力分析的动态有限元法.铁道学报,1989,工务工程专辑,9-15
[52]Inglis C.E.A Mathematical Treatise on Vibration in Railway Bridges.Cambridge,1934
[53]Lou P.and Zeng Q.Y.Finite element based formulations for vehicle-bridge interaction system considering vertical motion.Journal of Structural Engineering,2005,32(4):261-273
[54]Xia H.and Zhang N.Dynamic analysis of railway bridge under high-speed trains.Computers and Structures,2005,83(23-24):1891-1901
[55]Lou P.Vertical dynamic responses of a simply supported bridge subjected to a moving train with two-wbeelset vehicles using modal analysis method.International Journal for Numerical Methods in Engineering,2005,64(9):1207-1235
[56]Lou P.Dai G.L.and Zeng Q.Y.Modal coordinate formulations for a simply supported bridge subjected to a moving train modelled as two-stage suspension vehicles.Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science,2005,219(10):1027-1040
[57]Lou P.and Zeng Q.Y.Formulation of equations of motion for a simply supported bridge under a moving railway freight vehicle.Shock and Vibration(in press)
[58]Humar J.L.and Kashif A.H.Dynamic response analysis of dab-type bridges.Journal of Structural Engineefing,ASCE,1995,121(1):48-62
[59]Fryba L.Dynamics of Railway Bridges.London:Thomas Telford,1996
[60]Matsuura A.Dynamic interaction between vehicles and girders in high speed railways.Quarterly Report of Railway Technical Research Institute,1974,15(3):133-136
[61]Matsuttra A.Dynamic behavior of bridge girder for high speed railway bridge.Quarterly Report of Railway Technical Research Institute,1979,20(2):70-76
[62]Chu K.H.Garg V.K.and Wiriyachai A.Dynamic interaction of railway train and bridges.Vehicle System Dynamics,1980,9(4):207-236
[63]Wiriyachai A.Chu K.H.and Garg V.K.Bridge impact due to wheel and track irregularities.Journal of Engineering Mechanics,ASCE,1982,108(4):648-666
[64]Dahlberg T.Vehicle-bridge interaction.Vehicle System Dynamics,1984,13(1):187-206
[65]Chu K.H.Garg V.K.and Wang T.L.Impact in railway prestressed concrete bridges.Journal of Structural Engineering,ASCE,1986,112(5):1036-1051
[66]Dianna G.and Cheli F.Dynamic interaction of railway systems with large bridges.Vehicle System Dynamics,1989,18(1):71-106
[67]Zaman M.Taheri M.R.and Khanna A.Dynamic response of cable-stayed bridges to moving vehicles using the structural impedance method.Applied Mathematical Modelling,1996,20(12):875-889
[68]Wang T.L.Garg V.K.and Chu K.H.Railway bridge/vehicle interaction studies with new model vehicle model.Journal of Structural Engineering,ASCE,1991,117(7):2099-2116
[69]Au F.T.K.,Wang J.J.and Cheung Y.K.Impact study of cable-stayed bridge under railway traffic using various models.Journal of Sound and Vibration,2001,240(3):467-481
[70]Zhang Q.L.Vrouwenvelder A.and Wardenier J.Numerical simulation of train-bridge interactive dynamics.Computers and Structures,2001,79(10):1059-1075
[71]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
[72]夏禾 陈英俊.车-桥-墩体系动力相互作用分析.土木工程学报,1992,25(2):3-12
[73]杨岳民.大跨度铁路桥梁车桥动力响应理论分析及实验研究:[博士学位论文].北京:铁道科学研究院,1995.
[74]王荣辉,郭向荣,曾庆元.高速列车—钢桁梁桥系统横向振动随机分析.铁道学报,1996,18(1):90-95
[75]夏禾,陈英俊,张煅,柯在田.列车提速情况下铁路双线简支钢桁梁动力响应分析.铁道学报 1996,18(5):75-82
[76]陈淮,郭向荣.大跨度斜拉桥动力特性分析.计算力学学报,1997,14(1):57-63
[77]郭文华,郭向荣,曾庆元.京沪高速铁路南京长江大桥斜拉桥方案车桥系统振动分析.土木工程学报,1999,32(3):23-27
[78]郭向荣,曾庆元.高速铁路多∏形预应力混凝土梁桥动力特性及列车走行性分析.铁道学报,2000,22(1):72-78
[79]郭向荣,曾庆元.高速铁路结合梁桥与列车系统振动分析模型.华中理工大学学报,2000,28(3):60-62
[80]王刚,曹雪琴.高速铁路大跨度斜拉桥车桥动力分析.上海铁道大学学报,2000,21(8):7-11
[81]高芒芒,李永强,许兆军,潘家英.高速列车作用下的芜湖长江大桥车桥耦合振动分析.中国铁道科学,2001,22(5):34-40
[82]陈淮,杨磊,郭向荣.列车通过连续钢桁梁桥时的动力响应研究.郑州大学学报(理学版),2004,36(4):80-84
[83]张楠,夏禾.铁路桥梁在高速列车作用下的动力响应分析.工程力学,2005,22(3):144-151
[84]Wu Y.S.Yang Y.B.and Yau J.D.Three-Dimensional analysis of train-rail-bridge interaction problems.Vehicle System Dynamics,2001,36(1):1-35
[85]Cheng Y.S.Au F.T.K.and Cheung Y.K.Vibration of railway bridges under a moving train by using bridge-track-vehicle element.Engineering Structures,2001,23(12):1597-1606
[86]Wu Y.S.and Yang Y.B.Steady-state response and riding comfort of trains moving over a series of simply supported bridges.Engineering Structures,2003,25(2):251-265
[87]Song M.K.Nob H.C.and Choi C.K.A new three-dimensional finite element analysis model of high-speed train-bridge interactions.Engineering Structures,2003,25(13):1611-1626
[88]Lou P.A vehicle-track-bridge interaction element considering vehicle's pitching effect.Finite Elements in Analysis and Design,2005,41(4):397-427
[89]Lou P.and Zeng Q.Y.Formulation of equations of motion of finite element form for vehicle-track-bridge interaction system with two types of vehicle model.International Journal for Numerical Methods in Engineering 2005,62(3):435-474
[90]Lou P.Finite element analysis for train-track-bridge interaction system.Archive of Applied Mechanics,2007,77(10):707-728
[91]蔡成标,翟婉明.机车-轨道-桥梁垂向耦合动力学分析.西南交通大学学报,1997,32(6):628-632
[92]蔡成标,翟婉明,王其昌.高速列车与高架桥上无碴轨道相互作用研究.铁道工程学报,2000,(3):29-32
[93]蔡成标,翟婉明,王开云.高速列车与桥上板式轨道动力学仿真分析.中国铁道科学,2004,25(5):57-60
[94]翟婉明,蔡成标,王开云.高速列车-轨道-桥梁动态相互作用原理及模型.土木工程学报,2005,38(11):132-137
[95]Le R.Ripke B.and Zaeher M.Ballast mats on high speed bridges.In:Proceedings of the Fourth European Conference on Structural Dynamics,EURODYN'99,Prague(Czech Republic),1999,pp.699-703.
[96]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,2004,26(5):71-80.
[97]Lou P.Comparison of two types of deflection functions for analyzing the responses of the rail and the bridge under static or moving vehicles.Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics,2006,220(2):105-123
[98]曾庆元,杨毅,骆宁安,江锋,张麒.列车—桥梁时变系统横向振动分析.铁道学报,1991,13(2):38-46
[99]郭向荣,曾庆元.高速铁路简支钢桁梁桥横向刚度限值研究.长沙铁道学院学报,1998,16(2):1-6
[100]郭向荣,陈淮,曾庆元.铁路连续钢桁梁桥横向刚度限值分析.桥梁建设,2000,(1):13-16
[101]马坤全,曹雪琴,吴定俊.列车准高速通过半穿式钢桁梁桥横向振动分析.上海铁道大学学报,1996,17(3):1-10
[102]葛玉梅.大跨度铁路钢桁梁桥的车—桥耦合横向振动分析.铁道学报,1998,20(4):26-29
[103]郭文华.中小跨度铁路桥梁横向刚度分析:[博士学位论文].长沙:长沙铁道学院,1999
[104]郭文华,曾庆元.半穿式钢桁梁桥与列车系统空间振动分析.长沙铁道学院学报,2002,20(1):36-40
[105]曾庆元.弹性系统总势能不变值原理.华中理工大学学报,2000,28(1):1-3
[106]曾庆元,杨平.形成矩阵的“对号入座”法则与桁梁空间分析的桁段有限元法.铁道学报,1986,8(2):48-59
[107]向俊,周智辉,曾庆元.南京长江大桥128 m钢桁梁上列车走行安全性分析.铁道科学与工程学报,2006,3(2):26-29
[108]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):1-10
[109]曾庆元,向俊,娄平.突破列车脱轨难题的能量随机分析道路.中国工程科学,2002,4(12):9-20
[110]曾庆元,向俊,娄平,周智辉.列车脱轨的力学机理与防止脱轨理论.铁道科学与工程学报,2004,1(1):19-31
[111]曾庆元,向俊,周智辉,娄平.列车脱轨分析理论与应用.长沙:中南大学出版社,2006
[112]Olsson M.Finite element,modal co-ordinate analysis of structures subjected to moving loads.Journal of Sound and Vibration,1985,99(1):1-12
[113]Lin Y.H.and Trethewey M.W.Finite element analysis of elastic beams subjected to moving dynamic loads.Journal of Sound and Vibration,1990,136(2):323-342
[114]Yang Y.B.and Lin B.H.Vehicle-bridge interaction analysis by dynamic condensation method.Journal of Structural Engineering,ASCE,1995,121(11):1636-1643
[115]Yang Y.B.and Yau J.D.Vehicle-bridge interaction element for dynamic analysis.Journal of Structural Engineering,ASCE,1997,123(11):1512-1518
[116]Yang Y.B.Chang C.H.and Yau J.D.An element for analysing vehicle-bridge systems considering vehicle's pitching effect.International Journal for Numerical Methods in Engineering,1999,46(7):1031-1047
[117]Au F.T.K.Wang J.J.and Cheung Y.K.Impact study of cable-stayed bridge under railway traffic using various models.Journal of Sound and Vibration,2001,240(3):447-465
[118]Au F.T.K.Wang J.J.and Cheung Y.K.Impact study of cable-stayed railway bridges with random rail irregularities.Engineering Structures,2002,24(5):529-541
[119]Chu K.H.Garg V.K.and Dhar C.L.Railway-bridge impact:simplified train and bridge model.Journal of the Structural Division,ASCE,1979,105(9):1823-1844
[120]曾京,胡松.轮轨摩擦碰撞及脱轨的研究.振动工程学报,2001,14(1):1-6
[121]Lee U.Revisiting the moving mass problem:onset of separation between the mass and beam.Journal of Vibration and Acoustics,ASME,1996,118(3):516-521
[122]Lee U.Separation between the flexible structure and the moving mass sliding on it.Journal of Sound and Vibration,1998,209(5):867-877
[123]Cheng Y.S.Au F.T.K.Cheung Y.K.and Zheng D.Y.On the separation between moving vehicles and bridge.Journal of Sound and Vibration,1999,225(5):781-801
[124]Fryba L.Impacts of two-axle system traversing a beam.International Journal of Solids and Structures,1968,4(11):1107-1123
[125]Sack R.L.Structural Analysis.New York:McGraw-Hill Book Company,1984
[126]Clough R.W.and Penzien J.Dynamics of Structures(2~(nd)edition).New York:McGraw-Hill,Inc.,1993
[127]Desai C.S.Elementary Finite Element Method.New Jersey:Prentice-Hall,Inc.,Englewood Cliffs,1979.181-185
[128]Feng Z.and Cook R.D.Beam element on two-parameter elastic foundations.Journal of Engineerin8 Mechanics ASCE,1983,109(6):1390-1402
[129]Timoshenko S.Strength of Materials Part Ⅱ Advance Theory and Problems.London:VanNostrand Reinhold Company Ltd,1956.
[13O]Profillidis V.A.Railway Engineering.Aldershot:Ashgate Publishing Limited,2000
[131]童大埙.铁路轨道.北京:中国铁道出版社,1988
[132]陈秀方.轨道工程.北京:中国建筑工业出版社,2005
[133]Dong R.G.Vertical Dynamics of Railway Vehicle-Track System.Ph.D.Thesis,Concordia University,1994
[134]Lou P.Zhong X.G.Tang J.F.and Zeng Q.Y.Finite-element analysis of discretely supported rail subjected to multiple moving concentrated forces.Proceedings of the Institution of Mechanical Engineers Part F Journal of Rail and Rapid Transit,2006,220(3):305-315
[135]Yoshida D.M.and Weaver W.Finite element analysis of beams and plates with moving loads.Publication of International Association for Bridge and Structural Engineering,1971,31(1):179-195
[136]Filho F.V.Finite element analysis of structures under moving loads.Shock and Vibration Digest,1978,10(8):27-35
[137]Hino J.Yoshimura T.and Ananthanarayana N.Vibration analysis of non-linear beams subjected to a moving load using the finite element method.Journal of Sound and Vibration,1985,100(4):477-491
[138]Hino J.Yoshimura T.Konishi K.and Ananthanarayana N.A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load.Journal of Sound and Vibration,1984,96(1):45-53
[139]Olsson,M.On the fundamental moving load problem.Journal of Sound and Vibration,1991,145(2):299-307
[140]Yener M.and Chompooming K.Numerical method of lines for analysis of vehicle-bridge dynamic interaction.Computers and Structures,1994,53(3):709-726
[141]Yang Y.B.and Wu Y.S.A versatile element for analysing vehicle-bridge interaction response.Engineering Structures,2001,23(5):452-469
[142]Zheng D.Y.Cheung Y.K.Au F.T.K.and Cheng Y.S.Vibration of multi-span non-uniform beams under moving loads by using modified beam vibration functions.Journal of Sound and Vibration,1998,212(3):455-467
[143]Cheung Y.K.Au F.T.K.Zheng D.Y.and Cheng Y.S.Vibration of multi-span non-uniform bridges under moving vehicles and trains by using modified beam vibration functions.Journal of Sound and Vibration,1999,228(3):611-628
[144]Esmailzadeh E.and Jalili N.Vehicle-passenger-structure interaction of uniform bridges traversed by moving vehicles.Journal of Sound and Vibration,2003,260(4):611-635
[145]Pesterev A.V.Yang B.Bergman L.A.and Tan C.A.Revisiting the moving force problem.Journal of Sound and Vibration,2003,261(1):75-91
[146]Yang Y.B.and Lin B.H.Vehicle-bridge interaction dynamics and potential applications.Journal of Sound and Vibration,2005,284(1-2):205-226
[147]Biggs J.M.Introduction to Structural Dynamics.New York:McCwaw-Hill,Inc.,1964
[148]Newmark N.M.A method of computation for structural dynamics.Journal of the Engineering Mechanics Division,Proceedings of the American Society of Civil Engineers,1959,85(3):67-94
[149]Bathe K.J.and Wilson E.L.Numerical Methods in Finite Element Analysis.New Jersey:Prentice-Hall,Englewood Cliffs,1976
[150]王福天.车辆系统动力学.北京:中国铁道出版社,1994
[151]Yang C.Y.Random Vibration of Structures.New York:John Wiley & Sons,1986
[152]李富文,伏魁先,刘学信.钢桥.北京:中国铁道出版社,1992
[153]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
[154]铁道部科学研究院铁道建筑研究所.我国干线轨道不平顺功率谱的研究.TY-1215,北京:铁道部科学研究院,1999