Research Article Dynamic Analysis of the High Speed Train and … · 2019. 7. 30. · Bernoulli-Euler beam model which includes the rail, slab, and roadbed. Linear wheel-rail contact

Research ArticleDynamic Analysis of the High Speed Train and Slab TrackNonlinear Coupling System with the Cross Iteration Algorithm

Xiaoyan Lei Shenhua Wu and Bin Zhang

Engineering Research Center of Railway Environment Vibration and Noise Ministry of Education East China Jiaotong UniversityNanchang 330013 China

Correspondence should be addressed to Xiaoyan Lei xiaoyanlei2013163com

Received 28 October 2015 Revised 13 January 2016 Accepted 28 January 2016

Academic Editor Usama El Shamy

Copyright copy 2016 Xiaoyan Lei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Amodel for dynamic analysis of the vehicle-track nonlinear coupling system is established by the finite element methodThewholesystem is divided into two subsystems the vehicle subsystem and the track subsystemCoupling of the two subsystems is achieved byequilibrium conditions for wheel-to-rail nonlinear contact forces and geometrical compatibility conditions To solve the nonlineardynamics equations for the vehicle-track coupling system a cross iteration algorithm and a relaxation technique are presentedExamples of vibration analysis of the vehicle and slab track coupling system induced by Chinarsquos high speed train CRH3 are givenIn the computation the influences of linear and nonlinear wheel-to-rail contact models and different train speeds are consideredIt is found that the cross iteration algorithm and the relaxation technique have the following advantages simple programming fastconvergence shorter computation time and greater accuracyThe analyzed dynamic responses for the vehicle and the trackwith thewheel-to-rail linear contactmodel are greater than thosewith thewheel-to-rail nonlinear contactmodel where the increasing rangeof the displacement and the acceleration is about 10 and the increasing range of the wheel-to-rail contact force is less than 5

1 Introduction

Since the opening of the worldrsquos first high speed railwayin 1964 (with its advantages of high speed and conve-nience safety and comfort environmental-friendliness andlow energy consumption larger carrying capacity and theavailability of all-day transportation) high speed railwayhas shown strong competitiveness amongst other modes oftransport According to the statistics by the InternationalUnion of Railways as of November 1 2013 there are a totalof 11605 km of high speed railways in operation 4883 km inconstruction and another 12570 km of high speed railwaysplanned to be built in other countries of the world In Chinathere are 11028 km of high speed railways in operation andanother 12000 km under construction making up half thetotal amount of the worldrsquos total high speed railways with thecountry owning the longest and the largest scale of high speedrailways both in operation and construction With the rapiddevelopment of high speed railways ballastless slab track hasbeen widely used throughout the world [1 2] In China over80 high speed railways are ballastless slab track Compared

with the traditional ballast track it has the advantages of highstability long service life high geometric regularity uniformtrack stiffness low maintenance and good durability

It is well known that the transportation function ofa railway system is achieved by wheel-to-rail interactionand the design and the manufacture of locomotives androlling stock as well as the design and the construction ofthe line itself are required for understanding the dynamiccharacteristics of the vehicle and track coupling system Toensure the train moves safely and smoothly a good dynamicperformance of the railway line is requiredThe developmentof a mathematical model and the simulation technique foranalysis of the dynamic behavior of ballastless slab trackwill be helpful to achieve improved component design andmaintenance schedulesThesemodels are used to understandthe interactions of the track and vehicle components and tomeasure the vibration characteristics of the track structureSome literature has focused on recent developments in slabtrack and slab track design [3 4] However only a fewreferences have been made to the dynamic behavior of aballastless slab track under moving vehicles Zhai et al

Hindawi Publishing CorporationJournal of Nonlinear DynamicsVolume 2016 Article ID 8356160 17 pageshttpdxdoiorg10115520168356160

2 Journal of Nonlinear Dynamics

developed a vehicleslab track interactionmodel based on thetheory of vehicle-track coupling dynamics and analyzed thedynamic properties of slab tracks used in high speed railwaysThe effects of the elasticity and damping of the CA-layerunder the slab on system dynamics were also investigated[5] Xiang et al established a dynamic analysis model ofthe lateral finite strip and slab segment element based onthe structural characteristics of the ballastless track such asthe Bogl slab track [6] Both the motor car and trailers ofthe high speed train were modeled as a multibody systemwith two suspensions The vertical displacements of the railand the slab were obtained by the traditional static modeland by the dynamic analysis model of the lateral finite stripand slab segment element respectivelyThe calculated resultsdemonstrate that the vertical static and dynamic maximumdisplacements of the rail and the slab are close and the valuesare in general range Next Xiang et al put forward a newspatial vibration model of track segment element of the slabtrack according to structural characteristic of a slab track[7] The spatial vibration equation set of the high speedtrain and slab track system was then established on thebasis of the rule of ldquoset-in-right-positionrdquo for formulatingsystem matrices The equation set was solved by the Wilson-120579 direct integration method The theory was verified bythe high speed running experiment carried out on the slabtrack of the Qinghuang dao-Shenyang passenger transportline Dong et al carried out experimental validation of anumerical model for prediction of the dynamic response ofballastless subgrade of high speed railways [8] 3D consistentviscous-spring artificial boundary elements are introduced inthe ABAQUS software the effects of the distance betweenmultisource loading points and the viscous-spring artificialboundary on the simulation results under train loadingconditions are discussed and 3D finite element ballastlesstracksubgrade models have been developed Comparisonbetween the field measured data of CRH2 at the NationalRailway Test Site and the ABAQUS simulation results showsgood agreement The results indicate that the dynamic stressof the subgrade increases with the speed Lei and Zhangpresented a model for dynamic analysis of vehicle-track-subgrade coupling system to the CRTS II (China RailwayTrack System) ballastless slab track system [9] Based onthe model a new type of ballastless slab track element isdeveloped and the associated stiffness matrix mass matrixand dampingmatrix for the element is deducedThis elementincludes rail rail pad and fastener prefab slab cement-asphalt mortar hydraulically bonded layer and subgrade Bymeans of the Lagrange equation the finite element equationfor analysis of dynamic behavior of the ballastless slab trackis formulated As application examples parameter studieson the track vibration of the ballastless slab track structuresuch as stiffness and damping resulting from the rail padand fastener CA mortar and subgrade are investigated Inorder to investigate the dynamic behavior of the train and slabtrack coupling system Lei andWang studied a new approachwith finite elements in a moving frame of reference basedon conceptions of vehicle element and track element [10]By discretizing the slab track subsystem into track elementsthat flow with the moving vehicle the proposed method

eliminates the need for keeping track of the vehicle positionwith respect to the track model The governing equationsare formulated in a coordinate system moving at a constantvelocity and the associated stiffness matrix mass matrix anddamping matrix for the track element in a moving frameof reference are derived The vehicle element is introducedto model a car with primary and secondary suspensionsystems which has 26 degrees of freedom where 10 degreesof freedom are used to describe the vertical movement ofthe car and 16 degrees of freedom are associated with raildisplacementsThemethod is shown towork for varying trainspeed and track parameters and has several advantages overthe conventional finite element method in a fixed system ofreference (It is known that track irregularity is the principalexciting source inducing vibrations of the train and the track)Kang et al proposed a calculation method fitting functionand fitting spectrum of power spectral density (PSD) oftrack irregularity for high speed railways [11] To improvethe calculation accuracy of PSD of track irregularity a linearinterpolation and a wavelet analysis are given and used toeliminate the outliers and trends in the track irregularitiesBased on the method and the data from the high speedcomprehensive inspection train the track irregularity PSDfitting equation was obtained as well as the frequencymultiplication energy table to reflect the impact of trackperiodical structure of high speed railways PSD of theballastless track irregularities of high speed railway includesfitting spectrum and frequency multiplication energy tablewhich provides the basis for maintenance and optimizationdesign for high speed railways By means of a parallelalgorithm based on the pseudo excitation method (PEM)Zhang et al investigated the nonstationary random responseof a vertically coupled vehicle-slab track system subjectedto random excitation induced by track irregularity [12] Thevehicle is simplified as a multibody system with 10 degreesof freedom and the slab track is represented by a three-layerBernoulli-Euler beam model which includes the rail slaband roadbed Linear wheel-rail contact provides interactionbetween the vehicle and slab track models In the abovecontributions studies are mainly focused on the vehicle-track linear coupling dynamic problems and have their owncharacteristics whereas the analysis of dynamic responsefor the vehicle-track nonlinear coupling system is rarelygiven attention as is the corresponding algorithm Varandaset al presented a methodology to predict the settlementof railway track in transition zones due to train loading[13] The methodology is based on dynamic calculationsusing a (non-linear) train-track interaction model and anincremental settlement model Nguyen et al carried outcomparison of dynamic effects of high speed traffic loadon ballasted track using a simplified two-dimensional andfull three-dimensional model [14] In these models thevehicle and track are coupled via a nonlinear Hertz contactmechanism The method of Lagrange multipliers is usedfor the contact constraint enforcement between the wheeland rail Due to the contact nonlinearities the numericalsimulations are performed in the time domain using a directintegration method for the transient problem Yang andFonder presented an iterative scheme to analyze the dynamic

Journal of Nonlinear Dynamics 3

response of a bridge-vehicle system [15]Themethod consistsin dividing the whole system into two subsystems at theinterface of the bridge and vehicles and these two subsystemsare solved separately Their compatibility at the interfaceis achieved by an iterative procedure with underrelaxationor with Aitken acceleration The proposed method is moreefficient in nonlinear dynamic responses because in thiscase the iterations are necessary whether the system issolved as a whole or not In each iteration step judgmentmust be made to satisfy equilibrium conditions for wheel-rail nonlinear contact forces and geometrical compatibilityconditions and the select of the time stepwill be limitedwhendealing with problem of the larger amplitude of the trackrandom irregularity A fundamental model is establishedfor analyzing the train-track-bridge dynamic interactions byZhai and Xia in which the vehicle subsystem is coupled withthe track subsystem through a spatially interacted wheel-rail model [16 17] and the track subsystem is coupled withthe bridge subsystem by a track-bridge dynamic interactionmodel An explicit-implicit integration scheme is adoptedto numerically solve the equations of motion of the largenonlinear dynamic system in the time domain Computersimulation software that is the train-track-bridge interactionsimulation software (TTBSIM) is developed to predict thevertical and lateral dynamic responses of the train-track-bridge coupled system and the effectiveness of the TTB-SIM simulation for dynamic evaluation of complex bridgestructures in high speed railways is demonstrated [16 17]Neves et al pointed out that the equations of motion ofthe structure and vehicles are complemented with additionalcompatibility equations that relate nodal displacements of thevehicles to the displacements of the corresponding points onthe surface of the structure with no sliding or separation[18 19] In order to avoid the system matrix being timedependent at each time step and deal with the nonlinearcontact problem a model for dynamic analysis of the vehicleand slab track nonlinear coupling system is proposed by finiteelement method in this paper The whole system is dividedinto two subsystems the vehicle subsystem (considered asa complete locomotive or rolling stock unit with a primaryand secondary suspension system) and the track subsystem(regarded as three-layer elastic beammodel) Coupling of thetwo systems is achieved by equilibrium conditions for wheel-to-rail nonlinear contact forces and geometrical compatibilityconditions A cross iteration algorithm is presented to solvethe dynamics equations of the vehicle-track nonlinear cou-pling system In order to accelerate the iteration convergencerate a relaxation technique is introduced to modify thewheel-to-rail contact forces By contrasting with a referenceexample the correctness of the algorithm is verified Theexample of vehicle and track vibration induced by Chinahigh speed train CRH3 moving on the slab track is given inwhich the influences of the linear and nonlinear wheel-to-railcontact model and the different train speeds are consideredThe results demonstrate that the cross iteration algorithm hasthe advantages of simple programming fast convergence rateless computation time and high accuracy

Mc Jc120593c

c

Bogie 2 Bogie 1Mt

Mw4 Mw3

Ful4 Ful3

Mw2 Mw1

Ful2 Ful1

kw

Ks1 Cs1

Jt

Ks2 Cs2t1

w1

120593t1

Figure 1 Vehicle model as upper subsystem

2 Fundamental Assumptions

The following assumptions are made in establishing themodel to analyze the dynamic behavior of the ballastless slabtrack of high speed railways

(1) Only vertical dynamic loads are considered in themodel

(2) Since the vehicle and the slab track are symmetricalabout the center-line of the track only half of thecoupling system is used for ease of calculation

(3) The upper structure in the vehicle and slab trackcoupling system is a complete locomotive or rollingstock unit with a primary and secondary suspensionsystem in which vertical and pitch motion for bothvehicle and bogie are considered

(4) The lower structure in the coupling system is a CRTSII slab track where rails are considered as beams withfinite length resting on a discrete pad The elastic anddamping behavior for the rail pad and the fastener ofthe track structure are represented with stiffness anddamping coefficients 1198961199101 and 1198881199101

(5) The concrete slab is simplified as a beam and onlyvertical dynamic responses are consideredThe elasticand damping behavior resulting from cement-asphaltmortar (CAmortar) of the track structure are 1198961199102 and1198881199102

(6) The hydraulically bonded layer (HBL) is simplifiedas a beam and only vertical dynamic responsesare considered The elastic and damping behaviorresulting from subgrade of the track structure are 1198961199103and 1198881199103

(7) A nonlinear relationship between two elastic contactcylinders perpendicular to each other is used incoupling the vehicle and the track

3 Vehicle Subsystem

The vehicle subsystem is a complete locomotive or rollingstock unit with a primary and secondary suspension systemas shown in Figure 1 In the model 2119872119888 and 2119869119888 are mass


and pitch inertia for the car body 2119872119905 and 2119869119905 are massand pitch inertia for the bogie 1198961199041 1198961199042 and 1198881199041 1198881199042 standfor stiffness and damping coefficients for the primary andsecondary suspension systems of the vehicle respectively V119888and 120593119888 are vertical displacement and angular displacement ofpitch motion for the car body V119905119894 (119894 = 1 2) and 120593119905119894 (119894 = 1 2)

are the vertical displacement and angular displacement ofpitchmotion for the 119894th bogie V119908119894 (119894 = 1 2 3 4) is the verticaldisplacement for the 119894th wheel119872119908119894 (119894 = 1 2 3 4) is the massof the 119894th wheel and 119865119906119897119894 (119894 = 1 2 3 4) is the wheel-to-railcontact force for the 119894th wheelThe nodal displacement vectorfor this element can be defined as

119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (1)

By means of the Lagrange equation the dynamic equation ofvehicle for the coupling system can be obtained as

119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906 (2)

where 119872119906 119862119906 and 119870

119906represent the mass damping and

stiffness matrixes for the upper structure and the explicitexpressions can be described as

119872119906

= diag (119872119888 119869119888 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084)

(3)

119862119906and119870

119906can be found in the literature [9]

119876119906is the load vector for the upper structure

119876119906

= minus119872119888119892 0 minus119872119905119892 minus119872119905119892 0 0 1198751 1198752 1198753 1198754119879

(4)

119875119894 = minus119872119908119894119892 + 119865119906119897119894 and 119865119906119897119894 stands for the wheel-to-railcontact force for the 119894th wheel which can be calculated bythe nonlinear Hertz formula

119865119906119897119894

=

1

11986632

1003816100381610038161003816V119908119894 minus (V119897119888119894 + 120578119894)100381610038161003816100381632 V119908119894 minus (V119897119888119894 + 120578119894) lt 0

0 V119908119894 minus (V119897119888119894 + 120578119894) ge 0

(5)

where V119908119894 and V119897119888119894 are the vertical displacement for the 119894thwheel and the 119894th rail on the 119894th wheelrail contact pointrespectively 120578119894 stands for irregularity of the track verticalprofile at the rail contact point with 119894th wheel and 119866 is thedeflection coefficient of the contact between the wheel andthe rail which can be referenced in the literature [20]

4 CRTS II Slab Track Subsystem

A slab track element model with three layer beams isdeveloped in Figure 2 and the whole slab track subsystemmodel is described in Figure 3 In the model V1 V4 and 1205791 1205794are vertical rail displacements and rail slopes for node 1 andnode 4 V2 V5 and 1205792 1205795 are vertical displacements and slopesof the concrete slab for node 2 and node 5 and V3 V6 and 1205793

1

2

3

4

5

6

1

2

3

4

5

6

1205791

1205792

1205793

1205794

1205795

1205796

cy1ky1

cy2ky2

cy3ky3

Figure 2 Slab track element model

cy1ky1

cy2ky2

cy3ky3

Ful4 Ful3 Ful2 Ful1

Figure 3 Ballastless slab track model as lower subsystem

1205796 are vertical displacements and slopes of the hydraulicallybonded layer for node 3 and node 6

The nodal displacement vector for the slab track elementcan be defined as

119886119890

119897= V1 1205791 V2 1205792 V3 1205793 V4 1205794 V5 1205795 V6 1205796

119879 (6)

Using finite element method the dynamic equation of slabtrack in the coupling system can be described as

119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876119897 (7)

where 119872119897 119862119897 and 119870

119897stand for global mass matrix

global damping matrix and global stiffness matrix of theballastless slab track subsystem respectively and the explicitexpressions can be described as

119872119897= sum

119890

119898119890

119897

119862119897= sum

119890

119888119890

119897

119870119897= sum

119890

119896119890

119897

(8)

where 119898119890

119897 119888119890119897 and 119896

119890

119897are the mass matrix damping matrix

and stiffness matrix of the slab track element respectivelyand the expressions can be represented as

119898119890

119897= 119898119890

119903+ 119898119890

119904+ 119898119890

ℎ

119896119890

119897= 119896119890

119903+ 119896119890

119904+ 119896119890

ℎ+ 119896119890

1119888+ 119896119890

2119888+ 119896119890

3119888

119888119890

119897= 119888119890

119903+ 119888119890

1119888+ 119888119890

2119888+ 119888119890

3119888

(9)


where 119898119890119903 119898119890119904 and 119898

119890

ℎare mass matrixes associated with the

rail the concrete slab and the hydraulically bonded layer 119896119890119903

119896119890

119904 and 119896

119890

ℎare modified versions of stiffness matrixes for the

standard beam elements due to strain energies of the rail theconcrete slab and the hydraulically bonded layer 119896119890

1119888 1198961198902119888 and

119896119890

3119888are stiffness matrixes resulting from elastic elements of

the rail pad and fastener the cement-asphalt mortar and thesubgrade 119888119890

119903is a dampingmatrix for the rail the concrete slab

and the hydraulically bonded layer 1198881198901119888 1198881198902119888 and 119888

119890

3119888are damp-

ing matrixes resulting from damping elements of the rail padand fastener the cement-asphalt mortar and the subgradeAll the expressions can be found in the literature [9]

5 Cross Iteration Algorithm

In order to solve the dynamic equations of the vehicle-tracknonlinear coupling system theNewmark integrationmethodis used and the detailed procedure of cross iteration algorithmis given as follows For the vehicle subsystem (2) can berewritten as

119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906119892

+ 119865119906119897 (10)

where 119876119906119892

is the gravity vector for the vehicle and 119865119906119897is the

vector of wheel-to-rail contact force which can be calculatedby nonlinear Hertz formula (5)

Taking the Hertz nonlinear stiffness into linearizationformula (5) can be rewritten as

119865119906119897

=

minus119870119908(119886119906minus 119886119897119888minus 120578) (119886

119906minus 119886119897119888minus 120578)119894lt 0

0 (119886119906minus 119886119897119888minus 120578)119894ge 0

(11)

where 119870119908is the linearized stiffness matrix of wheel-to-rail

contact 120578 is the vector of track irregularity at the wheel-to-rail contact points

119876119906119892

= minus119892 119872119888 0 119872119905 119872119905 0 0 1198721199081 1198721199082 1198721199083 1198721199084119879

(12)

119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (13)

119886119897119888= 0 0 0 0 0 0 V1198971198881 V1198971198882 V1198971198883 V1198971198884

119879 (14)

119870119908= diag 0 0 0 0 0 0 119896119908 119896119908 119896119908 119896119908 (15)

119896119908 =3

2119866119901013

(Ncm) (16)

where 119896119908 is the linearized contact stiffness between the wheeland the rail and 1199010 is the static load of the wheel

Substituting (11) into (10) (10) can be expressed as

119872119906119906+ 119862119906119906+ (119870119906+ 119870119908) 119886119906= 119876119906119892

+ 119870119908(119886119897119888+ 120578) (17)

For the subsystem of the slab track (7) can be rewritten as

119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876lg minus 119865

119906119897 (18)

where 119876lg is the nodal load vector induced by gravity of thetrack structure and 119865

119906119897is the wheel-to-rail force vector

The Newmark integration method is used to solve thedynamic responses of the vehicle subsystem and the tracksubsystem which is widely used in engineering practiceParameters and procedures of the Newmark method can befound in the literature [21]

The fundamental computation steps are summarized asfollows

Stage 1 (initial computation) (1) In the first time step andfirst iteration assume the initial displacement 119886

0

119897of the

track structure (usually taking 1198860

119897= 0) Based on the

value of 1198860

119897 the initial displacement of the rail V0119897119888119894 (119894 =

1 2 3 4) can be derived at the 119894th wheel-to-rail contact pointNext substituting V0119897119888119894 (119894 = 1 2 3 4) into (14) the initialdisplacement vector 1198860

119897119888can be obtained

(2) Substituting 1198860119897119888into (17) the displacement 1198860

119906 velocity

0

119906 and acceleration

0

119906of the vehicle can be evaluated by

solving for the motion of vehicle equation (17)

Stage 2 (step by step procedure in time domain) Assume thatin time step (119905) the (119896 minus 1)th iteration has been done andvectors of the displacement velocity and acceleration at timestep (119905) for the vehicle and the track structure are knownNowlet us consider the (119896)th iteration

(1) By substituting 119905119886119896minus1119906

and 119905119886119896minus1119897119888

into (5) calculate theinteraction force 119905119865

119896

119906119897119894(119894 = 1 2 3 4) by flowing formula

119905119865119896

119906119897119894=

1

11986632

100381610038161003816100381610038161003816

119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905120578119896minus1

119894)100381610038161003816100381610038161003816

32119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) lt 0

0119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) ge 0

(19)

(2)The relaxationmethod is introduced tomodify the wheel-to-rail contact force 119905119865

119896

119906119897119894(119894 = 1 2 3 4) that is

119905119865119896

119906119897119894=119905119865119896minus1

119906119897119894+ 120583 (119905119865119896

119906119897119894minus119905119865119896minus1

119906119897119894) (20)

where 120583 is relaxation coefficient which has to be in a certainrange (0 lt 120583 lt 1) to ensure the convergence If 120583 = 1it is difficult to converge especially in case of dealing withproblem by consideration of the track random irregularitywith larger amplitude It shows that it is optimal when the


Table 1 Parameters for China high speed train CRH3

Parameters Value Parameters Value

Mass of car body 2119872119888(kg) 40000 Damping of primary suspension system

21198881199041(KNsdotsm) 100

Mass of bogie 2119872119905(kg) 3200 Damping of secondary suspension system

21198881199042(kNsdotsm) 120

Mass of wheel119872119908119894(kg) 1200 Wheelbase 2119897

1(m) 250

Pitch inertia of car body 2119869119888(kgsdotm2) 547 times 105 Distance between center of front bogie

and center of rear bogie 21198972(m) 17375

Pitch inertia of bogie 2119869119905(kgsdotm2) 6800 Stiffness of wheelrail contact 119896

119888(MNm) 1325 times 103

Stiffness of primary suspension system 21198961199041

(MNm) 208 Axle load (kN) 140

Stiffness of secondary suspension system 21198961199042

(MNm) 08 Length of vehicle (m) 25675

relaxation coefficient meets 03 lt 120583 lt 05 and sufficientaccuracy can be obtained with less computation time

(3) Assembling 119905119865119896

119906119897119894into the wheel-to-rail force vector

119905119865119896

119906119897 which is applied to the lower part of the track structure

as shown in Figure 3 the displacement 119905119886119896119897 velocity 119905119896

119897 and

acceleration 119905119896119897for the track structure can be evaluated by

solving (18)(4) Based on 119905119886119896

119897 the displacement 119905V119896

119897119888119894(119894 = 1 2 3 4) of

the rail can be obtained at the 119894th wheel-to-rail contact pointand the updated wheel-to-rail force 119905119865

119896

119906119897119894acting on the vehicle

can be calculated with formula (5)(5) Assembling the updated 119905119865

119896

119906119897119894into the wheel-to-rail

force vector 119905119865119896119906119897 which is applied to the upper structure

as external forces the displacement 119905119886119896119906 velocity 119905119896

119906 and

acceleration 119905119896119906for the vehicle subsystem can be evaluated

by solving (10)(6) Calculate track displacement difference and its norm

Δ119905119886119896

119897=119905119886119896

119897minus119905119886119896minus1

119897 (21)

where 119905119886119896119897and 119905119886119896minus1

119897are displacement vectors of the lower

structure at current iteration and previous iteration respec-tively

Now define the convergence criterion

Norm Δ119905119886119896

119897

Norm (119905119886119896

119897)

le 120576 (22)

where

Norm Δ119905119886119896

119897=

119899

sum

119894=1

Δ1199051198862

(119894)119896

119897

Norm (119905119886119896

119897) =

119899

sum

119894=1

119905119886119896

119897(119894)2

(23)

and 120576 is a specified tolerance and assumed between 10times10minus5

and 10 times 10minus8 usually sufficient to obtain the solution with

reasonable accuracy

(7) Check convergence criterion (22)If convergence criterion (22) is not satisfied go to Stage

2 and enter the next iteration step (119896 + 1) If it is satisfiedproceed to the next instant (119905 + Δ119905) and define

119905+Δ1199051198860

119906=119905119886119896

119906

119905+Δ1199050

119906=119905

119896

119906

119905+Δ1199050

119906=119905119896

119906

119905+Δ1199051198860

119897=119905119886119896

119897

119905+Δ1199050

119897=119905119896

119897

119905+Δ1199050

119897=119905119896

119897

(24)

Hence return to Stage 2 enter the next Newmark time stepand continue the calculation until the last time step

6 Verification

In order to verify the correctness of the cross iteration algo-rithm for the model and the method proposed in this paperto calculate the example in literature [6] analysis of dynamicresponse for vehicle and track is carried out where Chinarsquoshigh speed train CRH3 with the speed of 200 kmh and theCRTS II ballastless slab track are considered Parameters ofthe high speed train CRH3 and the CRTS II slab track aregiven in Tables 1 and 2 respectively As excitation sourceof the track profile irregularity periodic sine function withamplitude 3mm and wave length 125m is adopted in thecomputation The calculated vertical rail displacements andthe wheel-rail contact forces derived from the two differentcomputational methods are shown and compared in Figures4 and 5 respectively Good agreements between the twocalculations are observed which confirms the validity andpracticability of the algorithm

Figures 6 and 7 show the iteration convergence process ofthe wheel-rail contact force and the rail displacement of thewheel-rail contact point at the time 119905 It can be observed that


Table 2 Parameters for CRTS II slab track

Parameters Value

Rail padDistance of ties 119897 (m) 0625

Stiffness of clips and pad 1198961199101(MNsdotmminus1) 60

Damping of clips and pad 1198881199101(kNsdotssdotmminus1) 50

Concrete slab

Length (m) 645Width (m) 255Height (m) 020

Density 120588119904(kgsdotmminus3) 2500

Youngrsquos modulus 119864119904(MPa) 39 times 104

Cement-asphalt mortar Stiffness 1198961199102(MNsdotmminus1) 09 times 103

Damping 1198881199102(kNsdotssdotmminus1) 80

Hydraulically bonded layer

Upper bottom width (m) 295Lower bottom width (m) 325

Height (m) 030Density 120588

ℎ(kgsdotmminus3) 2500

Damping 119864ℎ(MPa) 33 times 104

Subgrade Stiffness 1198961199103(MNsdotmminus1) 65


08

06

04

02

0

Rail

disp

lace

men

t (m

m)

1 2 3 40

Running time of the train (s)

(a) The results in this paper

Rail

disp

lace

men

t (m

m)

09

06

03

00

1 2 3 40


(b) The results in the literature [6]

Figure 4 Time history of the vertical rail displacements

starting from the previous equilibrium state at the time (119905 minusΔ119905) the solution quickly approaches convergence after severaliterations

7 Dynamic Analysis of the High Speed Trainand Slab Track Nonlinear Coupling System

As an application example dynamic analysis of the vehicleand track nonlinear coupling system induced by China highspeed train CRH3 moving on the ballastless slab track iscarried out Parameters for high speed train CRH3 and theballastless slab track are given in Tables 1 and 2 respectivelyIn order to reduce the boundary effect of the track structurethe total track length for computation is 370m Random

irregularity of the track vertical profile with line grade 6 PSDof the United State is regarded as the external excitationInfluences of the linear and nonlinear wheel-to-rail contactmodels and different train speeds (119881 = 200 kmh 250 kmh300 kmh and 350 kmh) on the dynamic responses of thetrain and the track are investigated The time step of numeri-cal integration for the Newmark scheme adopts 00005 s

The results computed by varying train speeds and consid-eration of linear and nonlinear wheel-to-rail contact modelupon the dynamic response of the track and the train havebeen given in Tables 3ndash15 and shown in Figures 8 and 9where the outputs are the maximum and total amplitudeof displacements and accelerations for the rail the concreteslab the hydraulically bonded layer the wheel the car bodyand the bogie and the maximum wheel-rail contact forces


Table 3 The maximum and the total amplitude of the rail displacements

Speed (kmh) Wheel-to-rail linear contact Wheel-to-rail nonlinear contactMax (mm) Min (mm) Total amplitude (mm) Max (mm) Min (mm) Total amplitude (mm)

350 028 minus190 218 028 minus170 198300 022 minus170 192 028 minus160 188250 020 minus160 180 019 minus160 179200 015 minus140 155 016 minus130 146

21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)


Figure 5 Time history of the wheel-rail contact force

minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number

Figure 6 Iteration process for the wheel-rail contact force at time 119905

Figures 10 and 11 are the time history of the wheel-to-railcontact forces with the linear and nonlinear contact modelfor different train speeds respectively where the values of thewheel-to-rail contact forces with the linear and the nonlinearcontact model without irregularity excitation are shown inTable 16

It is indicated that the train speed has a great influence onthe displacements of the slab track structure shown in Tables3ndash5 and Figures 8(a)ndash8(c) With the increase of the train

minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number

Figure 7 Iteration process for the rail displacement of wheel-railcontact point at time 119905

speed from 200 kmh to 350 kmh the total amplitudes ofdisplacement for rail concrete slab and hydraulically bondedlayer in case of thewheel-to-rail linear contactmodel increasefrom 155mm 083mm and 08mm to 218mm 124mmand 123mm respectively and the increased amplitude ratiosare 403 49 and 53 respectively For the wheel-to-railnonlinear contact model the increased amplitude ratios are357 468 and 455


Table 4 The maximum and the total amplitude of the concrete slab displacements



Table 5 The maximum and the total amplitude of the hydraulically bonded layer displacements



Table 6 The maximum and the total amplitude of the rail accelerations

Speed (kmh) Wheel-to-rail linear contact Wheel-to-rail nonlinear contactMax (ms2) Min (ms2) Total amplitude (ms2) Max (ms2) Min (ms2) Total amplitude (ms2)


Table 7 The maximum and the total amplitude of the concrete slab accelerations



Table 8 The maximum and the total amplitude of the hydraulically bonded layer accelerations



Table 9 The maximum and the total amplitude of the car body displacements


350 minus02991 minus03013 00022 minus02989 minus03014 00025300 minus03001 minus03011 00010 minus03001 minus03013 00012250 minus03000 minus03011 00011 minus03001 minus03012 00011200 minus02998 minus03013 00015 minus02998 minus03014 00016


Table 10 The maximum and the total amplitude of the bogie displacements



Table 11 The maximum and the total amplitude of the wheel displacements



Table 12 The maximum and the total amplitude of the car body accelerations



Table 13 The maximum and the total amplitude of the bogie accelerations



Table 14 The maximum and the total amplitude of the wheel accelerations



The calculated maximum and the total amplitude ofdisplacements for the rail concrete slab and hydraulicallybonded layer with the linear contact model are larger thanthose with the nonlinear contact model and the increasingrange is within 10

The maximum and the total amplitudes of accelerationsfor the rail concrete slab and hydraulically bonded layerincrease with the increase of the train speeds as shown in

Tables 6ndash8 and Figures 8(d)ndash8(f) With the increase of thetrain speed from 200 kmh to 350 kmh the total amplitudesof the rail concrete slab and hydraulically bonded layeracceleration in case of the wheel-to-rail linear contact modelincrease from 42676ms2 4456ms2 and 4456ms2 to23994ms2 8232ms2 and 729ms2 respectively and theincreased amplitude ratios are 4622 847 and 389for the wheel-to-rail nonlinear contact model the increased


The maximum value in wheel-to-rail linear contactThe minimum value in wheel-to-rail linear contactThe total amplitude in wheel-to-rail linear contactThe maximum value in wheel-to-rail nonlinear contactThe minimum value in wheel-to-rail nonlinear contactThe total amplitude in wheel-to-rail nonlinear contact

The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m

s2 )The maximum value in wheel-to-rail linear contactThe minimum value in wheel-to-rail linear contactThe total amplitude in wheel-to-rail linear contactThe maximum value in wheel-to-rail nonlinear contactThe minimum value in wheel-to-rail nonlinear contactThe total amplitude in wheel-to-rail nonlinear contact

(f)

Figure 8 Dynamic response of the ballastless slab track subsystem (a) the rail displacements (b) the concrete slab displacements (c) thehydraulically bonded layer displacements (d) the rail accelerations (e) the concrete slab accelerations and (f) the hydraulically bonded layeraccelerations

Table 15 The maximum wheel-to-rail contact forces

Speed (kmh) Wheel-to-raillinear contact (kN)

Wheel-to-railnonlinear contact (kN)

350 17238 16631300 16005 15216250 14241 14195200 12253 12580

Table 16 The maximum wheel-to-rail contact forces withoutirregularities excitation



350 6958 6954300 6968 6963250 6982 6979200 7017 7009

amplitude ratios are 224 802 and 392 The higher thetrain speed the more significant the difference of the railacceleration

The calculated total amplitudes of the rail accelerationwith the wheel-to-rail linear contact model are larger than

those with the wheel-to-rail nonlinear contact model Thehigher the train speed is the more obvious the differencewill be whereas the differences of the total amplitude for theconcrete slab and the hydraulically bonded layer accelerationwith both contact models are within 5

As indicated in Tables 9ndash11 and Figures 9(a)ndash9(c) themaximum displacements of the car body the bogie and thewheel vary little with the train speed

The calculated maximum absolute values of displace-ments for the bogie and the wheel with the wheel-to-raillinear contact model are larger than those with the nonlinearcontact model the increased ranges being within 5 and25 respectively However the maximum absolute value ofdisplacement for the car body with the wheel-to-rail linearcontact model is nearly equal to that with the nonlinearcontact model at the same train speed

As indicated in Tables 12ndash14 and Figures 9(d)ndash9(f) themaximum and total amplitudes of acceleration for the carbody the bogie and the wheel increase with the increase oftrain speed from 200 kmh to 350 kmh Yet the maximumand the total amplitudes of the bogie acceleration at a speedof 200 kmh are larger than those at a speed of 250 kmhThecause may be track irregularity of an individual point beingtoo large The total amplitudes of acceleration for the carbody the bogie and the wheel with the wheel-to-rail linear


0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms

2 )The maximum value in wheel-to-rail linear contactThe minimum value in wheel-to-rail linear contactThe total amplitude in wheel-to-rail linear contactThe maximum value in wheel-to-rail nonlinear contactThe minimum value in wheel-to-rail nonlinear contactThe total amplitude in wheel-to-rail nonlinear contact

(f)

Figure 9 Dynamic response of the vehicle subsystem (a) the car body displacements (b) the bogie displacements (c) the wheeldisplacements (d) the car body accelerations (e) the bogie accelerations and (f) the wheel accelerations

contact model are nearly equal to those with the nonlinearcontact model where the differences are within 5

It is indicated that the train speed has a great influenceon the wheel-to-rail contact forces as shown in Figures 10and 11 and Table 15 With the increase of the train speedfrom 200 kmh to 350 kmh the calculated maximumwheel-rail contact force increases from 12253 kN to 17238 kN withthe wheel-to-rail linear contact model and the increasedamplitude ratio is 4068 For the wheel-to-rail nonlinearcontact model the calculated maximum wheel-rail contactforce increases from 1258 kN to 16631 kN and the increasedamplitude ratio is 322

The maximum wheel-rail contact force with the linearcontact model is slightly larger than that with the nonlinearcontact model where the increasing range is within 5However at a train speed of 200 kmh the maximum wheel-rail contact force with the linear contact model is slightlysmaller than that with the nonlinear contact model

8 Conclusions

A model for dynamic analysis of the vehicle-track nonlinearcoupling system is established by finite element method Anda cross iteration algorithm is presented to solve the nonlineardynamic equation of the vehicle-track coupling system Insimulation analysis the whole system is divided into two

subsystems that is the vehicle subsystem considered as arolling stock unit with a primary and secondary suspensionsystem and the track subsystem regarded as a three-elastic-beam model Coupling of the two systems is achievedby equilibrium conditions for wheel-rail nonlinear contactforces and geometrical compatibility conditions The testcomputation examples show that a relatively small time stepmust be selected to ensure convergence which will greatlyincrease the computation time In order to solve this problemand accelerate the iterative convergence rate a relaxationtechnique is introduced to modify the wheel-rail contactforce Based on the dynamic analysis of the high speedtrain and slab track nonlinear coupling system the followingconclusions are obtained

(1) The train speed has a significant influence on thedynamic responses of the train and the track structuresRegardless of the wheel-to-rail linear contact model or non-linear contact model the accelerations and the displacementsof the rail the concrete slab and the hydraulically bondedlayer (and the accelerations of the wheel the bogie and thecar body) as well as the wheel-rail contact forces increasewith the increase of the train speeds while the displacementsof the wheel the bogie and the car body vary little with thetrain speed

(2) Results of the dynamic responses for the train and thetrack structure with the wheel-to-rail linear contact model


Without irregularity excitationWith irregularity excitation

0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105

(a) At the speed of 350 kmh


0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105

(b) At the speed of 300 kmh


0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)

(c) At the speed of 250 kmh


0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105

(d) At the speed of 200 kmh

Figure 10 Time history of thewheel-rail contact forcewith the linear contactmodel (a) at the speed of 350 kmh (b) at the speed of 300 kmh(c) at the speed of 250 kmh and (d) at the speed of 200 kmh

are larger than those with the wheel-to-rail nonlinear contactmodel under the same train speed The differences for themaximum and the total amplitudes of the displacements andthe accelerations are in the range of 10 and the differencesfor the wheel-rail contact forces are within 5 Meanwhilethe train speed has a great influence on the difference forrail acceleration The higher the train speed is the larger thedifferences of the rail acceleration will beThe total amplitudeof the rail acceleration with the wheel-to-rail linear contact

model is twice as large as that with thewheel-to-rail nonlinearcontact model at higher train speed whereas the differenceof the rail acceleration is within 5 at lower train speedSummarily it is conservative and safer to make the design ofthe train and the track structure based on the computationalresults with the wheel-to-rail linear contact model

(3) The introduction of the relaxation coefficient tomodify the wheel-rail contact force can evidently acceler-ate the convergence speed of the cross iterative algorithm



0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)


Figure 11 Time history of the wheel-rail contact force with the nonlinear contact model (a) at the speed of 350 kmh (b) at the speed of300 kmh (c) at the speed of 250 kmh and (d) at the speed of 200 kmh

The computation tests show that only 3 to 6 iteration stepsare required to approach the convergence in each iteration

(4) As the vehicle and slab track coupling nonlinearsystem is divided into the vehicle subsystem and the tracksubsystem the dynamic equations of the two subsystemscan be solved independently This method can not onlyreduce the analysis scale of the problem but also reduce thedifficulty of the programming design At the same time thecoefficient matrices of the finite element equation for the

two subsystems are constant and symmetric and invertingis only required once In the computation procedure thesame inverted matrices can be substituted in each iterationand each time step As a result the computation speed isgreatly enhanced In contrast with the existing method ofldquoset-in-right-positionrdquo to dynamic analysis of the vehicle-track coupling system the coefficient matrices of the finiteelement equations change with the change of the positionof the train on the track Therefore inversion calculation is


necessary in every time step which greatly reduces the com-putational efficiency A conclusion by simulating analysisof the dynamic responses for the vehicle and the slab trackstructure (which are induced by the trainmoving on the trackwith the length of 370m) is obtained that the computingtime in the ordinary computer workstation is of only 40-minute durationwith the cross iteration algorithm comparedwith the 150 minutes required by the method of ldquoset-in-right-positionrdquo (details of the computers and programminglanguages and systems employed are as follows (1) CPUIntel(R) core(TM) i5-2400CPU31 GHz 31 GHz (2) RAM400GB (3) programming languages MATLAB R2010b)The computation efficiency of the former method is 3 to5 times higher than the latter under normal conditions Ifthe scale of the analyzed problem is larger the computationefficiency will be higher

It should be pointed out that the proposed algorithm hasa general applicability which can be used to analyze all kindsof the linear and the nonlinear problems undermoving loads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work reported herein was supported by the EngineeringResearch Center of Railway Environment Vibration andNoise the Ministry of Education and the Natural ScienceFoundation of China (51478184)

References

[1] H Huawu ldquoBallastless track shall be developed in great effortson Chinese passenger dedicated linesrdquo Chinese Railways no 1pp 11ndash15 2005

[2] W H Zhang ldquoThe development of Chinarsquos high-speed railwaysystems and a study of the dynamics of coupled systems inhigh-speed trainsrdquo Proceedings of the Institution of MechanicalEngineers Part F Journal of Rail and Rapid Transit vol 228 no4 pp 367ndash377 2014

[3] C Esveld ldquoRecent developments in slab trackrdquo EuropeanRailway Review vol 9 pp 81ndash86 2003

[4] J M Zwarthoed V L Markine and C Esveld ldquoSlab trackdesign flexural stiffness versus soil improvementrdquo in Proceed-ings of the Rail-Tech Europe Conference CDR Utrecht TheNetherlands April 2001

[5] W Zhai W Han C Cai and Q Wang ldquoDynamic propertiesof high-speed railway slab tracksrdquo Journal of the China RailwaySociety vol 21 no 6 pp 65ndash69 1999 (Chinese)

[6] J Xiang D He and Q-Y Zeng ldquoAnalysis method of verticalvibration of train and ballastless track system with the lateralfinite strip and slab segment elementrdquo Journal of the ChinaRailway Society vol 29 no 4 pp 64ndash69 2007 (Chinese)

[7] J Xiang D He and Q-Y Zeng ldquoAnalysis theory of spatialvibration of high-speed train and slab track systemrdquo Journal ofCentral South University of Technology vol 15 no 1 pp 121ndash1262008

[8] L Dong C Zhao D Cai Q Zhang and Y Ye ldquoExperimentalvalidation of a numerical model for prediction of the dynamicresponse of ballastless subgrade of high-speed railwaysrdquo ChinaCivil Engineering Journal vol 41 no 10 pp 81ndash86 2008(Chinese)

[9] X Lei and B Zhang ldquoAnalysis of dynamic behavior forslab track of high-speed railway based on vehicle and trackelementsrdquo Journal of Transportation Engineering vol 137 no 4pp 227ndash240 2011

[10] X Lei and J Wang ldquoDynamic analysis of the train and slabtrack coupling system with finite elements in a moving frameof referencerdquo Journal of Vibration and Control vol 20 no 9 pp1301ndash1317 2014

[11] X Kang X Liu H Li F Yang J Gao and W Zhai ldquoPSD ofballastless track irregularities of high-speed railwayrdquo ScientiaSinica Technologica vol 44 no 7 pp 687ndash696 2014

[12] J Zhang Y Zhao Y-H Zhang et al ldquoNon-stationary randomvibration of a coupled vehiclendashslab track system using a parallelalgorithm based on the pseudo excitation methodrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 227 no 3 pp 203ndash216 2013

[13] J N Varandas P Holscher and M A G Silva ldquoSettlement ofballasted track under traffic loading application to transitionzonesrdquo Proceedings of the Institution of Mechanical EngineersPart F Journal of Rail and Rapid Transit vol 228 no 3 pp 242ndash259 2014

[14] K Nguyen J M Goicolea and F Galbadon ldquoComparison ofdynamic effects of high-speed traffic load on ballasted trackusing a simplified two-dimensional and full three-dimensionalmodelrdquo Proceedings of the Institution of Mechanical EngineersPart F Journal of Rail and Rapid Transit vol 228 no 2 pp 128ndash142 2014

[15] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 2 pp 195ndash2151996

[16] W Zhai H Xia C Cai et al ldquoHigh-speed trainndashtrackndashbridgedynamic interactionsmdashpart I theoretical model and numericalsimulationrdquo International Journal of Rail Transportation vol 1no 1-2 pp 3ndash24 2013

[17] W Zhai S Wang N Zhang et al ldquoHigh-speed trainndashtrackndashbridge dynamic interactionsmdashpart II experimental validationand engineering applicationrdquo International Journal of RailTransportation vol 1 no 1-2 pp 25ndash41 2013

[18] S G M Neves A F M Azevedo and R Calcada ldquoA directmethod for analyzing the vertical vehicle-structure interactionrdquoEngineering Structures vol 34 pp 414ndash420 2012

[19] S G M Neves P A Montenegro A F M Azevedo and RCalcada ldquoA direct method for analyzing the non-linear vehicle-structure interactionrdquo Engineering Structures vol 69 pp 83ndash892014

[20] L Xiaoyan A Khomenko andA Elena ldquoInnovationamp sustain-ability ofmodern railwayrdquo inProceedings of the 3rd InternationalSymposium on Innovation amp Sustainability of Modern RailwayL Xiaoyan Ed Nanchang China September 2012

[21] O C Zienkiewicz The Finite Element Method McGraw-HillNew York NY USA 1977

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developed a vehicleslab track interactionmodel based on thetheory of vehicle-track coupling dynamics and analyzed thedynamic properties of slab tracks used in high speed railwaysThe effects of the elasticity and damping of the CA-layerunder the slab on system dynamics were also investigated[5] Xiang et al established a dynamic analysis model ofthe lateral finite strip and slab segment element based onthe structural characteristics of the ballastless track such asthe Bogl slab track [6] Both the motor car and trailers ofthe high speed train were modeled as a multibody systemwith two suspensions The vertical displacements of the railand the slab were obtained by the traditional static modeland by the dynamic analysis model of the lateral finite stripand slab segment element respectivelyThe calculated resultsdemonstrate that the vertical static and dynamic maximumdisplacements of the rail and the slab are close and the valuesare in general range Next Xiang et al put forward a newspatial vibration model of track segment element of the slabtrack according to structural characteristic of a slab track[7] The spatial vibration equation set of the high speedtrain and slab track system was then established on thebasis of the rule of ldquoset-in-right-positionrdquo for formulatingsystem matrices The equation set was solved by the Wilson-120579 direct integration method The theory was verified bythe high speed running experiment carried out on the slabtrack of the Qinghuang dao-Shenyang passenger transportline Dong et al carried out experimental validation of anumerical model for prediction of the dynamic response ofballastless subgrade of high speed railways [8] 3D consistentviscous-spring artificial boundary elements are introduced inthe ABAQUS software the effects of the distance betweenmultisource loading points and the viscous-spring artificialboundary on the simulation results under train loadingconditions are discussed and 3D finite element ballastlesstracksubgrade models have been developed Comparisonbetween the field measured data of CRH2 at the NationalRailway Test Site and the ABAQUS simulation results showsgood agreement The results indicate that the dynamic stressof the subgrade increases with the speed Lei and Zhangpresented a model for dynamic analysis of vehicle-track-subgrade coupling system to the CRTS II (China RailwayTrack System) ballastless slab track system [9] Based onthe model a new type of ballastless slab track element isdeveloped and the associated stiffness matrix mass matrixand dampingmatrix for the element is deducedThis elementincludes rail rail pad and fastener prefab slab cement-asphalt mortar hydraulically bonded layer and subgrade Bymeans of the Lagrange equation the finite element equationfor analysis of dynamic behavior of the ballastless slab trackis formulated As application examples parameter studieson the track vibration of the ballastless slab track structuresuch as stiffness and damping resulting from the rail padand fastener CA mortar and subgrade are investigated Inorder to investigate the dynamic behavior of the train and slabtrack coupling system Lei andWang studied a new approachwith finite elements in a moving frame of reference basedon conceptions of vehicle element and track element [10]By discretizing the slab track subsystem into track elementsthat flow with the moving vehicle the proposed method

eliminates the need for keeping track of the vehicle positionwith respect to the track model The governing equationsare formulated in a coordinate system moving at a constantvelocity and the associated stiffness matrix mass matrix anddamping matrix for the track element in a moving frameof reference are derived The vehicle element is introducedto model a car with primary and secondary suspensionsystems which has 26 degrees of freedom where 10 degreesof freedom are used to describe the vertical movement ofthe car and 16 degrees of freedom are associated with raildisplacementsThemethod is shown towork for varying trainspeed and track parameters and has several advantages overthe conventional finite element method in a fixed system ofreference (It is known that track irregularity is the principalexciting source inducing vibrations of the train and the track)Kang et al proposed a calculation method fitting functionand fitting spectrum of power spectral density (PSD) oftrack irregularity for high speed railways [11] To improvethe calculation accuracy of PSD of track irregularity a linearinterpolation and a wavelet analysis are given and used toeliminate the outliers and trends in the track irregularitiesBased on the method and the data from the high speedcomprehensive inspection train the track irregularity PSDfitting equation was obtained as well as the frequencymultiplication energy table to reflect the impact of trackperiodical structure of high speed railways PSD of theballastless track irregularities of high speed railway includesfitting spectrum and frequency multiplication energy tablewhich provides the basis for maintenance and optimizationdesign for high speed railways By means of a parallelalgorithm based on the pseudo excitation method (PEM)Zhang et al investigated the nonstationary random responseof a vertically coupled vehicle-slab track system subjectedto random excitation induced by track irregularity [12] Thevehicle is simplified as a multibody system with 10 degreesof freedom and the slab track is represented by a three-layerBernoulli-Euler beam model which includes the rail slaband roadbed Linear wheel-rail contact provides interactionbetween the vehicle and slab track models In the abovecontributions studies are mainly focused on the vehicle-track linear coupling dynamic problems and have their owncharacteristics whereas the analysis of dynamic responsefor the vehicle-track nonlinear coupling system is rarelygiven attention as is the corresponding algorithm Varandaset al presented a methodology to predict the settlementof railway track in transition zones due to train loading[13] The methodology is based on dynamic calculationsusing a (non-linear) train-track interaction model and anincremental settlement model Nguyen et al carried outcomparison of dynamic effects of high speed traffic loadon ballasted track using a simplified two-dimensional andfull three-dimensional model [14] In these models thevehicle and track are coupled via a nonlinear Hertz contactmechanism The method of Lagrange multipliers is usedfor the contact constraint enforcement between the wheeland rail Due to the contact nonlinearities the numericalsimulations are performed in the time domain using a directintegration method for the transient problem Yang andFonder presented an iterative scheme to analyze the dynamic



Mc Jc120593c

c

Bogie 2 Bogie 1Mt

Mw4 Mw3

Ful4 Ful3

Mw2 Mw1

Ful2 Ful1

kw

Ks1 Cs1

Jt

Ks2 Cs2t1

w1

120593t1











3 Vehicle Subsystem





119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (1)


119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906 (2)

where 119872119906 119862119906 and 119870



119872119906

= diag (119872119888 119869119888 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084)

(3)

119862119906and119870



119876119906


(4)


119865119906119897119894

=

1

11986632


0 V119908119894 minus (V119897119888119894 + 120578119894) ge 0

(5)




1

2

3

4

5

6

1

2

3

4

5

6

1205791

1205792

1205793

1205794

1205795

1205796

cy1ky1

cy2ky2

cy3ky3


cy1ky1

cy2ky2

cy3ky3

Ful4 Ful3 Ful2 Ful1




119886119890

119897= V1 1205791 V2 1205792 V3 1205793 V4 1205794 V5 1205795 V6 1205796

119879 (6)


119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876119897 (7)

where 119872119897 119862119897 and 119870



119872119897= sum

119890

119898119890

119897

119862119897= sum

119890

119888119890

119897

119870119897= sum

119890

119896119890

119897

(8)

where 119898119890

119897 119888119890119897 and 119896

119890



119898119890

119897= 119898119890

119903+ 119898119890

119904+ 119898119890

ℎ

119896119890

119897= 119896119890

119903+ 119896119890

119904+ 119896119890

ℎ+ 119896119890

1119888+ 119896119890

2119888+ 119896119890

3119888

119888119890

119897= 119888119890

119903+ 119888119890

1119888+ 119888119890

2119888+ 119888119890

3119888

(9)


where 119898119890119903 119898119890119904 and 119898

119890



119896119890

119904 and 119896

119890



1119888 1198961198902119888 and

119896119890





119890

3119888are damp-




119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906119892

+ 119865119906119897 (10)

where 119876119906119892




119865119906119897

=


119906minus 119886119897119888minus 120578)119894lt 0

0 (119886119906minus 119886119897119888minus 120578)119894ge 0

(11)



119876119906119892

= minus119892 119872119888 0 119872119905 119872119905 0 0 1198721199081 1198721199082 1198721199083 1198721199084119879

(12)

119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (13)

119886119897119888= 0 0 0 0 0 0 V1198971198881 V1198971198882 V1198971198883 V1198971198884

119879 (14)

119870119908= diag 0 0 0 0 0 0 119896119908 119896119908 119896119908 119896119908 (15)

119896119908 =3

2119866119901013

(Ncm) (16)



119872119906119906+ 119862119906119906+ (119870119906+ 119870119908) 119886119906= 119876119906119892

+ 119870119908(119886119897119888+ 120578) (17)


119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876lg minus 119865

119906119897 (18)






0

119897of the



value of 1198860





119906 velocity

0


0





and 119905119886119896minus1119897119888


119896


119905119865119896

119906119897119894=

1

11986632

100381610038161003816100381610038161003816

119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905120578119896minus1

119894)100381610038161003816100381610038161003816

32119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) lt 0

0119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) ge 0

(19)


119896

119906119897119894(119894 = 1 2 3 4) that is

119905119865119896

119906119897119894=119905119865119896minus1

119906119897119894+ 120583 (119905119865119896

119906119897119894minus119905119865119896minus1

119906119897119894) (20)






21198881199041(KNsdotsm) 100


21198881199042(kNsdotsm) 120


1(m) 250




119888(MNm) 1325 times 103






(3) Assembling 119905119865119896


119905119865119896



119897 and


solving (18)(4) Based on 119905119886119896


119897119888119894(119894 = 1 2 3 4) of


119896



119896




119906 and



Δ119905119886119896

119897=119905119886119896

119897minus119905119886119896minus1

119897 (21)

where 119905119886119896119897and 119905119886119896minus1




Norm Δ119905119886119896

119897

Norm (119905119886119896

119897)

le 120576 (22)

where

Norm Δ119905119886119896

119897=

119899

sum

119894=1

Δ1199051198862

(119894)119896

119897

Norm (119905119886119896

119897) =

119899

sum

119894=1

119905119886119896

119897(119894)2

(23)



reasonable accuracy



119905+Δ1199051198860

119906=119905119886119896

119906

119905+Δ1199050

119906=119905

119896

119906

119905+Δ1199050

119906=119905119896

119906

119905+Δ1199051198860

119897=119905119886119896

119897

119905+Δ1199050

119897=119905119896

119897

119905+Δ1199050

119897=119905119896

119897

(24)


6 Verification





Parameters Value




Concrete slab













08

06

04

02

0

Rail

disp

lace

men

t (m

m)

1 2 3 40



Rail

disp

lace

men

t (m

m)

09

06

03

00

1 2 3 40













21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)



minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number




minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number











































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Mc Jc120593c

c

Bogie 2 Bogie 1Mt

Mw4 Mw3

Ful4 Ful3

Mw2 Mw1

Ful2 Ful1

kw

Ks1 Cs1

Jt

Ks2 Cs2t1

w1

120593t1











3 Vehicle Subsystem





119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (1)


119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906 (2)

where 119872119906 119862119906 and 119870



119872119906

= diag (119872119888 119869119888 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084)

(3)

119862119906and119870



119876119906


(4)


119865119906119897119894

=

1

11986632


0 V119908119894 minus (V119897119888119894 + 120578119894) ge 0

(5)




1

2

3

4

5

6

1

2

3

4

5

6

1205791

1205792

1205793

1205794

1205795

1205796

cy1ky1

cy2ky2

cy3ky3


cy1ky1

cy2ky2

cy3ky3

Ful4 Ful3 Ful2 Ful1




119886119890

119897= V1 1205791 V2 1205792 V3 1205793 V4 1205794 V5 1205795 V6 1205796

119879 (6)


119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876119897 (7)

where 119872119897 119862119897 and 119870



119872119897= sum

119890

119898119890

119897

119862119897= sum

119890

119888119890

119897

119870119897= sum

119890

119896119890

119897

(8)

where 119898119890

119897 119888119890119897 and 119896

119890



119898119890

119897= 119898119890

119903+ 119898119890

119904+ 119898119890

ℎ

119896119890

119897= 119896119890

119903+ 119896119890

119904+ 119896119890

ℎ+ 119896119890

1119888+ 119896119890

2119888+ 119896119890

3119888

119888119890

119897= 119888119890

119903+ 119888119890

1119888+ 119888119890

2119888+ 119888119890

3119888

(9)


where 119898119890119903 119898119890119904 and 119898

119890



119896119890

119904 and 119896

119890



1119888 1198961198902119888 and

119896119890





119890

3119888are damp-




119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906119892

+ 119865119906119897 (10)

where 119876119906119892




119865119906119897

=


119906minus 119886119897119888minus 120578)119894lt 0

0 (119886119906minus 119886119897119888minus 120578)119894ge 0

(11)



119876119906119892

= minus119892 119872119888 0 119872119905 119872119905 0 0 1198721199081 1198721199082 1198721199083 1198721199084119879

(12)

119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (13)

119886119897119888= 0 0 0 0 0 0 V1198971198881 V1198971198882 V1198971198883 V1198971198884

119879 (14)

119870119908= diag 0 0 0 0 0 0 119896119908 119896119908 119896119908 119896119908 (15)

119896119908 =3

2119866119901013

(Ncm) (16)



119872119906119906+ 119862119906119906+ (119870119906+ 119870119908) 119886119906= 119876119906119892

+ 119870119908(119886119897119888+ 120578) (17)


119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876lg minus 119865

119906119897 (18)






0

119897of the



value of 1198860





119906 velocity

0


0





and 119905119886119896minus1119897119888


119896


119905119865119896

119906119897119894=

1

11986632

100381610038161003816100381610038161003816

119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905120578119896minus1

119894)100381610038161003816100381610038161003816

32119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) lt 0

0119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) ge 0

(19)


119896

119906119897119894(119894 = 1 2 3 4) that is

119905119865119896

119906119897119894=119905119865119896minus1

119906119897119894+ 120583 (119905119865119896

119906119897119894minus119905119865119896minus1

119906119897119894) (20)






21198881199041(KNsdotsm) 100


21198881199042(kNsdotsm) 120


1(m) 250




119888(MNm) 1325 times 103






(3) Assembling 119905119865119896


119905119865119896



119897 and


solving (18)(4) Based on 119905119886119896


119897119888119894(119894 = 1 2 3 4) of


119896



119896




119906 and



Δ119905119886119896

119897=119905119886119896

119897minus119905119886119896minus1

119897 (21)

where 119905119886119896119897and 119905119886119896minus1




Norm Δ119905119886119896

119897

Norm (119905119886119896

119897)

le 120576 (22)

where

Norm Δ119905119886119896

119897=

119899

sum

119894=1

Δ1199051198862

(119894)119896

119897

Norm (119905119886119896

119897) =

119899

sum

119894=1

119905119886119896

119897(119894)2

(23)



reasonable accuracy



119905+Δ1199051198860

119906=119905119886119896

119906

119905+Δ1199050

119906=119905

119896

119906

119905+Δ1199050

119906=119905119896

119906

119905+Δ1199051198860

119897=119905119886119896

119897

119905+Δ1199050

119897=119905119896

119897

119905+Δ1199050

119897=119905119896

119897

(24)


6 Verification





Parameters Value




Concrete slab













08

06

04

02

0

Rail

disp

lace

men

t (m

m)

1 2 3 40



Rail

disp

lace

men

t (m

m)

09

06

03

00

1 2 3 40













21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)



minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number




minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number











































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (1)


119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906 (2)

where 119872119906 119862119906 and 119870



119872119906

= diag (119872119888 119869119888 119872119905 119872119905 119869119905 119869119905 1198721199081 1198721199082 1198721199083 1198721199084)

(3)

119862119906and119870



119876119906


(4)


119865119906119897119894

=

1

11986632


0 V119908119894 minus (V119897119888119894 + 120578119894) ge 0

(5)




1

2

3

4

5

6

1

2

3

4

5

6

1205791

1205792

1205793

1205794

1205795

1205796

cy1ky1

cy2ky2

cy3ky3


cy1ky1

cy2ky2

cy3ky3

Ful4 Ful3 Ful2 Ful1




119886119890

119897= V1 1205791 V2 1205792 V3 1205793 V4 1205794 V5 1205795 V6 1205796

119879 (6)


119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876119897 (7)

where 119872119897 119862119897 and 119870



119872119897= sum

119890

119898119890

119897

119862119897= sum

119890

119888119890

119897

119870119897= sum

119890

119896119890

119897

(8)

where 119898119890

119897 119888119890119897 and 119896

119890



119898119890

119897= 119898119890

119903+ 119898119890

119904+ 119898119890

ℎ

119896119890

119897= 119896119890

119903+ 119896119890

119904+ 119896119890

ℎ+ 119896119890

1119888+ 119896119890

2119888+ 119896119890

3119888

119888119890

119897= 119888119890

119903+ 119888119890

1119888+ 119888119890

2119888+ 119888119890

3119888

(9)


where 119898119890119903 119898119890119904 and 119898

119890



119896119890

119904 and 119896

119890



1119888 1198961198902119888 and

119896119890





119890

3119888are damp-




119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906119892

+ 119865119906119897 (10)

where 119876119906119892




119865119906119897

=


119906minus 119886119897119888minus 120578)119894lt 0

0 (119886119906minus 119886119897119888minus 120578)119894ge 0

(11)



119876119906119892

= minus119892 119872119888 0 119872119905 119872119905 0 0 1198721199081 1198721199082 1198721199083 1198721199084119879

(12)

119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (13)

119886119897119888= 0 0 0 0 0 0 V1198971198881 V1198971198882 V1198971198883 V1198971198884

119879 (14)

119870119908= diag 0 0 0 0 0 0 119896119908 119896119908 119896119908 119896119908 (15)

119896119908 =3

2119866119901013

(Ncm) (16)



119872119906119906+ 119862119906119906+ (119870119906+ 119870119908) 119886119906= 119876119906119892

+ 119870119908(119886119897119888+ 120578) (17)


119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876lg minus 119865

119906119897 (18)






0

119897of the



value of 1198860





119906 velocity

0


0





and 119905119886119896minus1119897119888


119896


119905119865119896

119906119897119894=

1

11986632

100381610038161003816100381610038161003816

119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905120578119896minus1

119894)100381610038161003816100381610038161003816

32119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) lt 0

0119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) ge 0

(19)


119896

119906119897119894(119894 = 1 2 3 4) that is

119905119865119896

119906119897119894=119905119865119896minus1

119906119897119894+ 120583 (119905119865119896

119906119897119894minus119905119865119896minus1

119906119897119894) (20)






21198881199041(KNsdotsm) 100


21198881199042(kNsdotsm) 120


1(m) 250




119888(MNm) 1325 times 103






(3) Assembling 119905119865119896


119905119865119896



119897 and


solving (18)(4) Based on 119905119886119896


119897119888119894(119894 = 1 2 3 4) of


119896



119896




119906 and



Δ119905119886119896

119897=119905119886119896

119897minus119905119886119896minus1

119897 (21)

where 119905119886119896119897and 119905119886119896minus1




Norm Δ119905119886119896

119897

Norm (119905119886119896

119897)

le 120576 (22)

where

Norm Δ119905119886119896

119897=

119899

sum

119894=1

Δ1199051198862

(119894)119896

119897

Norm (119905119886119896

119897) =

119899

sum

119894=1

119905119886119896

119897(119894)2

(23)



reasonable accuracy



119905+Δ1199051198860

119906=119905119886119896

119906

119905+Δ1199050

119906=119905

119896

119906

119905+Δ1199050

119906=119905119896

119906

119905+Δ1199051198860

119897=119905119886119896

119897

119905+Δ1199050

119897=119905119896

119897

119905+Δ1199050

119897=119905119896

119897

(24)


6 Verification





Parameters Value




Concrete slab













08

06

04

02

0

Rail

disp

lace

men

t (m

m)

1 2 3 40



Rail

disp

lace

men

t (m

m)

09

06

03

00

1 2 3 40













21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)



minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number




minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number











































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where 119898119890119903 119898119890119904 and 119898

119890



119896119890

119904 and 119896

119890



1119888 1198961198902119888 and

119896119890





119890

3119888are damp-




119872119906119906+ 119862119906119906+ 119870119906119886119906= 119876119906119892

+ 119865119906119897 (10)

where 119876119906119892




119865119906119897

=


119906minus 119886119897119888minus 120578)119894lt 0

0 (119886119906minus 119886119897119888minus 120578)119894ge 0

(11)



119876119906119892

= minus119892 119872119888 0 119872119905 119872119905 0 0 1198721199081 1198721199082 1198721199083 1198721199084119879

(12)

119886119906= V119888 120593119888 V1199051 V1199052 1205931199051 1205931199052 V1199081 V1199082 V1199083 V1199084

119879 (13)

119886119897119888= 0 0 0 0 0 0 V1198971198881 V1198971198882 V1198971198883 V1198971198884

119879 (14)

119870119908= diag 0 0 0 0 0 0 119896119908 119896119908 119896119908 119896119908 (15)

119896119908 =3

2119866119901013

(Ncm) (16)



119872119906119906+ 119862119906119906+ (119870119906+ 119870119908) 119886119906= 119876119906119892

+ 119870119908(119886119897119888+ 120578) (17)


119872119897119897+ 119862119897119897+ 119870119897119886119897= 119876lg minus 119865

119906119897 (18)






0

119897of the



value of 1198860





119906 velocity

0


0





and 119905119886119896minus1119897119888


119896


119905119865119896

119906119897119894=

1

11986632

100381610038161003816100381610038161003816

119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905120578119896minus1

119894)100381610038161003816100381610038161003816

32119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) lt 0

0119905V119896minus1119908119894

minus (119905V119896minus1119897119888119894

+119905

120578119896minus1

119894) ge 0

(19)


119896

119906119897119894(119894 = 1 2 3 4) that is

119905119865119896

119906119897119894=119905119865119896minus1

119906119897119894+ 120583 (119905119865119896

119906119897119894minus119905119865119896minus1

119906119897119894) (20)






21198881199041(KNsdotsm) 100


21198881199042(kNsdotsm) 120


1(m) 250




119888(MNm) 1325 times 103






(3) Assembling 119905119865119896


119905119865119896



119897 and


solving (18)(4) Based on 119905119886119896


119897119888119894(119894 = 1 2 3 4) of


119896



119896




119906 and



Δ119905119886119896

119897=119905119886119896

119897minus119905119886119896minus1

119897 (21)

where 119905119886119896119897and 119905119886119896minus1




Norm Δ119905119886119896

119897

Norm (119905119886119896

119897)

le 120576 (22)

where

Norm Δ119905119886119896

119897=

119899

sum

119894=1

Δ1199051198862

(119894)119896

119897

Norm (119905119886119896

119897) =

119899

sum

119894=1

119905119886119896

119897(119894)2

(23)



reasonable accuracy



119905+Δ1199051198860

119906=119905119886119896

119906

119905+Δ1199050

119906=119905

119896

119906

119905+Δ1199050

119906=119905119896

119906

119905+Δ1199051198860

119897=119905119886119896

119897

119905+Δ1199050

119897=119905119896

119897

119905+Δ1199050

119897=119905119896

119897

(24)


6 Verification





Parameters Value




Concrete slab













08

06

04

02

0

Rail

disp

lace

men

t (m

m)

1 2 3 40



Rail

disp

lace

men

t (m

m)

09

06

03

00

1 2 3 40













21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)



minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number




minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number











































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













21198881199041(KNsdotsm) 100


21198881199042(kNsdotsm) 120


1(m) 250




119888(MNm) 1325 times 103






(3) Assembling 119905119865119896


119905119865119896



119897 and


solving (18)(4) Based on 119905119886119896


119897119888119894(119894 = 1 2 3 4) of


119896



119896




119906 and



Δ119905119886119896

119897=119905119886119896

119897minus119905119886119896minus1

119897 (21)

where 119905119886119896119897and 119905119886119896minus1




Norm Δ119905119886119896

119897

Norm (119905119886119896

119897)

le 120576 (22)

where

Norm Δ119905119886119896

119897=

119899

sum

119894=1

Δ1199051198862

(119894)119896

119897

Norm (119905119886119896

119897) =

119899

sum

119894=1

119905119886119896

119897(119894)2

(23)



reasonable accuracy



119905+Δ1199051198860

119906=119905119886119896

119906

119905+Δ1199050

119906=119905

119896

119906

119905+Δ1199050

119906=119905119896

119906

119905+Δ1199051198860

119897=119905119886119896

119897

119905+Δ1199050

119897=119905119896

119897

119905+Δ1199050

119897=119905119896

119897

(24)


6 Verification





Parameters Value




Concrete slab













08

06

04

02

0

Rail

disp

lace

men

t (m

m)

1 2 3 40



Rail

disp

lace

men

t (m

m)

09

06

03

00

1 2 3 40













21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)



minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number




minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number











































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Parameters Value




Concrete slab













08

06

04

02

0

Rail

disp

lace

men

t (m

m)

1 2 3 40



Rail

disp

lace

men

t (m

m)

09

06

03

00

1 2 3 40













21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)



minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number




minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number











































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













21 30 2505 15


80

90

100

110

120

Whe

el-r

ail f

orce

(kN

)


1 2 30


80

100

120

Whe

el-r

ail f

orce

(kN

)



minus75

minus70

minus65

minus60

minus55

minus50

Whe

el-r

ail f

orce

(kN

)

1 2 3 4 50

Iteration number




minus05

minus049

minus048

minus047

minus046

minus045

Rail

disp

lace

men

t (m

m)

1 2 3 4 50

Iteration number











































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















































The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











The r

ail d

ispla

cem

ent (

mm

)

minus1

minus2

0

2

1

3

250 300 350200Speed (kmh)

(a)

minus1

minus05

0

05

1

15

The c

oncr

ete s

lab

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(b)

minus1

minus05

0

05

1

15

The h

ydra

ulic

ally

bon

ded

layer

disp

lace

men

t (m

m)

250 300 350200Speed (kmh)


(c)

The r

ail a

ccel

erat

ion

(ms

2 )

minus2000

minus1000

0

1000

2000

3000

250 300 350200Speed (kmh)


(d)

Figure 8 Continued


200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










200 250 300 350minus50

0

50

100

Speed (kmh)

The c

oncr

ete s

lab

acce

lera

tion

(ms

2 )


(e)

200 250 300 350minus40

minus20

0

20

40

60

80

Speed (kmh)

The h

ydra

ulic

ally

bon

ded

laye

r acc

eler

atio

n (m


(f)





350 17238 16631300 16005 15216250 14241 14195200 12253 12580




350 6958 6954300 6968 6963250 6982 6979200 7017 7009








0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

250 300 350200Speed (kmh)

minus400

minus300

minus200

minus100

100

The c

ar b

ody

disp

lace

men

ts (120583

m)


(a)

0

250 300 350200Speed (kmh)

minus80

minus60

minus40

minus20

20

The b

ogie

disp

lace

men

ts (120583

m)


(b)

250 300 350200Speed (kmh)

minus10

minus5

0

5

10

15

The w

heel

disp

lace

men

ts (120583

m)


(c)

The c

ar b

ody

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)

minus1

minus05

0

05

1

15


(d)

Figure 9 Continued


minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus15

minus10

minus5

0

5

10

15

20

The b

ogie

acce

lera

tions

(ms

2 )

250 300 350200Speed (kmh)


(e)

0

250 300 350200Speed (kmh)

minus100

minus50

50

100

150

The w

heel

acce

lera

tions

(ms


(f)





8 Conclusions







0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

05 1 15 2 250Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 30Time (s)

times105



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

1 2 3 4 50Time (s)

times105








0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

05

1

15

2

25W

heel

-rai

l con

tact

forc

e (N

)times105

05 1 15 2 250Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 30Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 40Time (s)



0

05

1

15

2

25

Whe

el-r

ail c

onta

ct fo

rce (

N)

times105

1 2 3 4 50Time (s)











Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Dynamic Analysis of the High Speed Train and … · 2019. 7. 30. · Bernoulli-Euler beam model which includes the rail, slab, and roadbed. Linear wheel-rail contact