Track dynamic behavior at rail welds at high speed · Several studies focused on the defects of...

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Acta Mech Sin (2010) 26:449–465 DOI 10.1007/s10409-009-0332-9 RESEARCH PAPER Track dynamic behavior at rail welds at high speed Guangwen Xiao · Xinbiao Xiao · Jun Guo · Zefeng Wen · Xuesong Jin Received: 19 January 2009 / Revised: 13 March 2009 / Accepted: 22 July 2009 / Published online: 13 January 2010 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2010 Abstract As a vehicle passing through a track with different weld irregularities, the dynamic performance of track com- ponents is investigated in detail by using a coupled vehi- cle–track model. In the model, the vehicle is modeled as a multi-body system with 35 degrees of freedom, and a Timoshenko beam is used to model the rails which are dis- cretely supported by sleepers. In the track model, the sleepers are modeled as rigid bodies accounting for their vertical, lat- eral and rolling motions and assumed to move backward at a constant speed to simulate the vehicle running along the track at the same speed. In the study of the coupled vehicle and track dynamics, the Hertizian contact theory and the theory proposed by Shen–Hedrick–Elkins are, respectively, used to calculate normal and creep forces between the wheel and the rails. In the calculation of the normal forces, the coefficient of the normal contact stiffness is determined by transient contact condition of the wheel and rail surface. In the calcu- lation of the creepages, the lateral, roll-over motions of the rail and the fact that the relative velocity between the wheel and rail in their common normal direction is equal to zero are simultaneously taken into account. The motion equations of the vehicle and track are solved by means of an explicit integration method, in which the rail weld irregularities are modeled as local track vertical deviations described by some ideal cosine functions. The effects of the train speed, the axle load, the wavelength and depth of the irregularities, and the weld center position in a sleeper span on the wheel–rail This project was supported by the National Basic Research Program of China (2007CB714702) and the National Natural Science Foundation of China (50821063, 50675183 and 50875221). G. Xiao · X. Xiao · J. Guo · Z. Wen · X. Jin (B ) State Key Laboratory of Traction Power, Southwest Jiaotong University, 610031 Chengdu, China e-mail: [email protected] impact loading are analyzed. The numerical results obtained are greatly useful in the tolerance design of welded rail pro- file irregularity caused by hand-grinding after rail welding and track maintenances. Keywords Rail weld · Irregularity · Vehicle–track coupling dynamics 1 Introduction Most damages to the vehicle and track components are caused by high impact forces. Wheel and rail surface irreg- ularities such as wheel flats, wheel shells, rail joints and rail corrugation are major sources that cause severe impact loading. In particular, owing to the existence of rail gap, height difference and dip angle in the traditional track [1], the dynamic interaction between wheel and rail joints gives rise to high, localized stresses and failure of track compo- nents and increases the cost of track maintenance. Many track maintenance problem can be traced back to the rough-ride conditions caused by rail joints. Fortunately, the continu- ously-welded rail (CWR) has increasingly replaced tradi- tional fish-plated joints as a preferred method of joining rails in track all over the world since 1930s [2]. The CWR efficiently reduces the level of wheel/rail impact loading at rail joints and extends the service life of wheels and rails. However, rail welds still represent a discontinuity on the rail running surface, because of variation in material characteris- tics (strength, hardness and microstructure), misalignment at the rail ends prior to welding and subsequent weld batter. In addition, welded rail may contain weld defects and high level residual stresses. All of these factors are combined to produce higher, localised wheel/rail impact loading. The increases of impact loading are prone to generate fatigue failure of rail 123

Transcript of Track dynamic behavior at rail welds at high speed · Several studies focused on the defects of...

Page 1: Track dynamic behavior at rail welds at high speed · Several studies focused on the defects of rail welds on rail-way site, ... Track dynamic behavior at rail welds at high ... rail

Acta Mech Sin (2010) 26:449–465DOI 10.1007/s10409-009-0332-9

RESEARCH PAPER

Track dynamic behavior at rail welds at high speed

Guangwen Xiao · Xinbiao Xiao · Jun Guo ·Zefeng Wen · Xuesong Jin

Received: 19 January 2009 / Revised: 13 March 2009 / Accepted: 22 July 2009 / Published online: 13 January 2010© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2010

Abstract As a vehicle passing through a track with differentweld irregularities, the dynamic performance of track com-ponents is investigated in detail by using a coupled vehi-cle–track model. In the model, the vehicle is modeled asa multi-body system with 35 degrees of freedom, and aTimoshenko beam is used to model the rails which are dis-cretely supported by sleepers. In the track model, the sleepersare modeled as rigid bodies accounting for their vertical, lat-eral and rolling motions and assumed to move backward at aconstant speed to simulate the vehicle running along the trackat the same speed. In the study of the coupled vehicle andtrack dynamics, the Hertizian contact theory and the theoryproposed by Shen–Hedrick–Elkins are, respectively, used tocalculate normal and creep forces between the wheel and therails. In the calculation of the normal forces, the coefficientof the normal contact stiffness is determined by transientcontact condition of the wheel and rail surface. In the calcu-lation of the creepages, the lateral, roll-over motions of therail and the fact that the relative velocity between the wheeland rail in their common normal direction is equal to zeroare simultaneously taken into account. The motion equationsof the vehicle and track are solved by means of an explicitintegration method, in which the rail weld irregularities aremodeled as local track vertical deviations described by someideal cosine functions. The effects of the train speed, theaxle load, the wavelength and depth of the irregularities, andthe weld center position in a sleeper span on the wheel–rail

This project was supported by the National Basic Research Programof China (2007CB714702) and the National Natural ScienceFoundation of China (50821063, 50675183 and 50875221).

G. Xiao · X. Xiao · J. Guo · Z. Wen · X. Jin (B)State Key Laboratory of Traction Power,Southwest Jiaotong University, 610031 Chengdu, Chinae-mail: [email protected]

impact loading are analyzed. The numerical results obtainedare greatly useful in the tolerance design of welded rail pro-file irregularity caused by hand-grinding after rail weldingand track maintenances.

Keywords Rail weld · Irregularity · Vehicle–track couplingdynamics

1 Introduction

Most damages to the vehicle and track components arecaused by high impact forces. Wheel and rail surface irreg-ularities such as wheel flats, wheel shells, rail joints andrail corrugation are major sources that cause severe impactloading. In particular, owing to the existence of rail gap,height difference and dip angle in the traditional track [1],the dynamic interaction between wheel and rail joints givesrise to high, localized stresses and failure of track compo-nents and increases the cost of track maintenance. Many trackmaintenance problem can be traced back to the rough-rideconditions caused by rail joints. Fortunately, the continu-ously-welded rail (CWR) has increasingly replaced tradi-tional fish-plated joints as a preferred method of joiningrails in track all over the world since 1930s [2]. The CWRefficiently reduces the level of wheel/rail impact loading atrail joints and extends the service life of wheels and rails.However, rail welds still represent a discontinuity on the railrunning surface, because of variation in material characteris-tics (strength, hardness and microstructure), misalignment atthe rail ends prior to welding and subsequent weld batter. Inaddition, welded rail may contain weld defects and high levelresidual stresses. All of these factors are combined to producehigher, localised wheel/rail impact loading. The increases ofimpact loading are prone to generate fatigue failure of rail

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450 G. Xiao et al.

Fig. 1 Surface deterioration at a: a Flash-butt welds; b Squats at a thermite weld [3]; c Squats at a thermite weld [4]; d Flash-butt weld [4]

welds, and to accelerate the settlement and deterioration ofthe track components. This is a feedback process betweenthe increasing track deformation and the increasing impactloading. Since 1970s the CWR had been widely used in mainrailway lines in China, and it was recently found that veryserious cracks, squats, spalling and fractures occurred at railwelds of main railway lines in China after the increase oftrain speed. Surface damages at welds were also found at therailway sites of other countries. Figure 1a shows the weld sur-face deterioration at China railway site and Fig. 1b–d showsquats at thermite weld and flash butt-weld in Netherlands[3,4]. Furthermore, the welds imply a high maintenance costand represent one of the main risk for catastrophic derailment.

Today, the most common rail welding processes are flash-butt welding, thermite welding, gas pressure welding andenclosed arc welding [5], and many papers have been pub-lished on the issues of rail welds. According to a reviewof recently published papers on the effect of out-of-roundwheel on track and vehicle components [6], the welds arethe weakest points in the continuous rail [7]. Aluminother-mic welds are of poorer quality than flash butt and gas pres-sure welds [8]. Service performance of an aluminothermicrail weld depends on the weld’s structural integrity and itsresistance to “batter” [9]. Batter is the process of deteriora-tion of the longitudinal profile arising from differential wearand plastic deformation of the weld and parent materials [10].Several studies focused on the defects of rail welds on rail-way site, the fatigue crack nucleation and the propagation atflash-butt welding and thermite welding as well as mechan-ical properties.

Different damage types at welds were investigated all overthe world. Abe et al. [11] investigated the rail surface irreg-ularities of welded part, in which the growth of rail surfaceirregularities of welded part was represented as a functionof passing tonnage. The forms and causes of damaged railwelds from 1985 to 2001 in Japan were investigated inRef. [12]. Shitara et al. [13] introduced nondestructivetesting and evaluation methods for rail welds in Japan.Mutton and Alvarez [14] presented the failure modes inthermite rail welds under high axle load and experimentallystudied the influence of the longitudinal weld profile andtrain speed on the dynamic impact load and the deteriora-tion occurring in the underlying track structure. From mea-surements an approximately linear relationship was foundbetween the train speed and the dynamic impact factor occur-ring at welds. A high percentage of squats at welds werefound in Netherlands [4].

In the field of vehicle–track coupling dynamics, it is nec-essary to understand the root causes of the surface dam-ages of rail welds. However, so far few studies were focusedon the vehicle–track coupling dynamics at rail welds. Zhaiet al. [15] used a vehicle–track coupling dynamic model tostudy the dynamical effect of welds irregularities on trainand track interaction caused by raising train speed. In themodel, the rail was modeled as an Euler beam, which is notsuitable for simulating high frequency impact between thewheel and rail weld irregularities. The relationship betweenthe geometry of rail welds and the dynamic wheel–rail inter-action has been dealt with extensively by Steenbergen andEsveld on the basis of finite element simulations [16,17].

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Track dynamic behavior at rail welds at high speed 451

According to the simulation results, they elaborated a newassessment method for the rail weld geometry based onthe gradient, which is more adequate than the traditionalmethod which is based on the principle of vertical tol-erance. Steenbergen [3] investigated dynamic wheel–railinteraction at rail welds using an analytical model, whichwas formulated in the frequency domain and not suitable foraccurate, quantitative predictions due to its simplicity andlinearity.

In the present study, a coupled vehicle–track model isdeveloped to analyze the effect of rail welds with differentwave-depths, wavelengths and positions in a sleeper spanon the track’s dynamical behaviour. Also investigated is theeffect of vehicle speed, axle load and different type weldrail irregularities on the wheel/rail impact loading. In themodel, the vehicle is treated as a multi-body system with35 degrees of freedom, and the structural elastic deforma-tion of the vehicle components is neglected. The rail is mod-eled by a Timoshenko beam resting on discrete sleepers, inwhich the rail’s lateral, vertical and torsional deformationsare simultaneously taken into account. The sleepers are mod-eled as rigid bodies to account for their vertical, lateral androlling motions. A moving sleeper support model is intro-duced to simulate the excitation from discrete sleeper sup-porters, in which the sleepers are assumed to move backwardat a constant speed to simulate the vehicle running alongthe track at the assumed speed. The ballast bed is replacedwith equivalent masses where only their vertical motion isaccounted for. The motion of the subgrade is ignored, andthe equivalent dampers and springs are used to replace theconnections between different parts of the system. The Her-tizian contact theory and the theory proposed by Shen–Hed-rick–Elkins are respectively used to calculate normal andcreep forces between the wheel and the rails. In the calcu-lation of normal forces, the coefficient of the normal con-tact stiffness is calculated by using the transient contactgeometry condition of wheel/rail surfaces. In the calcula-tion of creepages, the lateral, roll-over motions of the railand the fact that the relative velocity between the wheel andrail in their common normal direction is equal to zero areconsidered simultaneously. A local track vertical deviationdescribed by some ideal cosine functions are used to modelthe welded rail irregularities. The numerical results obtainedare greatly useful in the tolerance design of welded rail pro-file irregularity caused by damage and hand-grinding afterrail welding.

2 Modeling of coupled vehicle–track

Since 1970s, several studies have been published on the vehi-cle–track interaction model to investigate the behaviour ofrailway vehicle and track system [18,19]. The vehicle–track

system is usually divided into three subsystems: the vehi-cle, the track and the wheel/rail contact, as described inSects. 2.1–2.3, respectively. The interaction between thevehicle subsystem and the track subsystem takes placethrough the wheel/rail contact.

2.1 Vehicle model

The calculation model of the coupled vehicle–track is illus-trated in Fig. 2, in which the passenger car considered istreated as a full rigid multi-body dynamic model equippedwith a pair of two-axle bogies with double suspension sys-tems. The wheelset and the bogie are connected by the pri-mary suspension, while the car body is supported on thebogie through the secondary suspension. The structural elas-tic deformation of the vehicle components is neglected. Itshould be noted that the longitudinal motion variation of thevehicle is ignored in the simulation, namely, it is assumed thatthe acceleration of the vehicle in the forward direction alwaysequals zero. For convenience, the front bogie and rear bogieare numbered by 1 and 2, respectively, the leading wheelsetand the trailing wheelset of the front bogie by 1 and 2, respec-tively, and the corresponding wheelsets of the rear bogie by3 and 4. In Fig. 2, the variables Z , Y ,� and β with subscriptsindicate the vertical displacement, the lateral displacement,the rolling angle and the pitching angle of concerned compo-nents of the vehicle. The subscripts L and R denote the leftand right sides, respectively, and C and K with subscriptsstand for the coefficients of the equivalent dampers and thestiffness coefficients of the equivalent springs, respectively.The equivalent dampers and springs are used to replace theconnections between components of the passenger car andthe track. Therefore the total degree of freedom of the vehicleis 35, as listed in Table 1.

The motion equations of the vehicle subsystem are sec-ond order ordinary differential equations. The lateral, verti-cal, roll, pitch, and yaw differential equations of the carriageare

McYc = FybL1 + FybL2 + FybR1 + FybR2, (1)

Mc Zc = −FzbL1 − FzbL2 − FzbR1 − FzbR2 + Mcg, (2)

Icx φc = −(FybL1 + FybL2 + FybR1 + FybR2)HcB

+(FzbL1 + FzbL2 − FzbR1 − FzbR2)ds, (3)

Icy βc = [FzbL1 − FzbL2 + FzbR1 − FzbR2]lc−[FxbL1 + FxbL2 + FxbR1 + FxbR2]HcB, (4)

Iczψc = (FyfL1 − FyfL2 + FyfR1 − FyfR2)lc

+(−FxbL1 − FxbL2 + FxbR1 + FxbR2)ds, (5)

The equations of motion of the bogie i (i = 1, 2) in thelateral, vertical, roll, pitch and yaw directions are

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452 G. Xiao et al.

Fig. 2 Coupling vehicle/track model: a Elevation; b Side elevation

MbYbi = FyfL(2i−1) + FyfL(2i) − FytLi

+FyfR(2i−1) + FyfR(2i) − FybRi , (6)

Mb Zbi = FzbLi − FzfL(2i−1) − FzfL(2i)

+FzbRi − FzfR(2i−1) − FzfR(2i), (7)

Ibx φbi = −[FyfL(2i−1) + FyfL(2i) + FyfR(2i−1)

+FyfR(2i)]Hbw + [FzfL(2i−1) + FzfL(2i)

−FzfR(2i−1) − FzfR(2i)]dw + (FzbRi − FzbLi )ds

−(FybLi + FybRi )HBb, (8)

Iby βbi = [FzfL(2i−1) − FzfL(2i) + FzfR(2i−1) − FzfR(2i)]lb−[FxfL(2i−1) + FxfL(2i) + FxfR(2i−1)

+FxfR(2i)]Hbw − (FxbLi + FxbRi )HBt , (9)

Ibzψbi = [FyfL(2i−1) + FyfR(2i−1) − FyfL(2i)

−FyfR(2i)]lb + [FxfR(2i−1) + FxfR(2i)

−FxfL(2i−1) − FxfL(2i)]dw + (FxbLi − FxbRi )ds,

(10)

The different equations of the wheelset i (i = 1, 4) in thelateral, vertical, roll, pitch and yaw directions are

MwYwi = −FyfLi − FyfRi + FwryLi + FwryRi , (11)

Mw Zwi = −FwrzLi − FwrzRi + FzfLi + FzfRi + Mwg,(12)

Iwx φwi = a0(FLzi + NLzi − FwrzRi )− rLi FwryLi

−rRi FwryRi + dw(FzfRi − FzfLi ), (13)

Iwy βwi = rLi FLxi + rRi FRxi + rRiψwi FwryRi

+rLiψwi FwryLi + MLyi

Table 1 Variables of the vehicle model

Vehicle components Variables

Lateral Vertical Roll Pitch Yaw

Car body Yc Zc φc βc ψc

Bogie frame (i = 1, 2) Ybi Zbi φbi βbi ψbi

Wheelsets (i = 1, 4) Ywi Zwi φwi βwi ψwi

+MRyi + rLi FwryLi + rRi FwryRi , (14)

Iwzψwi = a0(FwrxLi − FwrxRi )+ a0ψwi (FwryLi − FwryRi )

+MLzi + MRzi + dw(FxfLi − FxfRi ), (15)

The explanation of some symbols in Eqs. (1)–(15) aregiven in Table 2. The detailed derivation of these motionequations of the vehicle system is very tedious, and is there-fore neglected for simplicity. For detailed expressions ofthe forces between the vehicle components, please refer toRef. [20].

2.2 Track model

The calculation model of tracks is shown in Fig. 2, andtangent tracks are considered. The gauge of the track is1,435 mm, the rail cant is 1/40 and the sleeper pitch is600 mm. The type of the rails laid has mass of 60 kg m−1

(CN60), which is widely used in the high speed line ofChina. The track, except for the rails, is also treated as a

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Table 2 Notations for vehicleequation parameters

Symbols Properties

Mc Car body mass

Mw Mass of wheelset

Ibx Roll inertia moment of bogie

Iby Pitch inertia moment of bogie

Ibz Yaw inertia moment of bogie

Iwx Roll inertia moment of wheelset

Iwy Pitch inertia moment of wheelset

Iwz Yaw inertia moment of wheelset

Mb Bogie mass

r0 Nominal rolling radius of wheelset

Icx Roll inertia moment of car body

Icy Pitch inertia moment of car body

Icz Yaw inertia moment of car body

rL, rR Left and right instant rolling radius of wheelset

g Gravity acceleration

Hbw Height from bogie centre to wheelset centre

lb Half distance between the two axles of the bogie

HcB Height from the carriage centre to the secondary suspension

ds Lateral half distance between the secondary suspension systems ofthe bogie

lc Lateral half distance between the second suspensions of the bogie

HBb Height from the secondary suspension to the bogie centre

dw Lateral half distance between the two primary suspensions of thebogie

a0 Lateral half distance between wheel/rail contact points

Fxb j i ,Fyb j i , Fzb j i Left and right forces between the car body and the i th bogie in x ,(i = 1, 2; j = L or R) y, and z directions

Fxf j i , Fyf j i , Fzf j i Left and right forces between the bogie and the i th wheelset in x , y(i = 1, . . . , 4; j = L or R) and z directions

Fwrx ji ,Fwry ji , Fwrz ji Left and right forces between the i th wheels and rails in x , y, and z(i = 1, . . . , 4; j = L or R) directions

Mwry ji ,Mwrz ji Left and right spin moment components between the i th wheel and(i = 1, . . . , 4; j = L or R) rail in y and z directions

rigid multi-body dynamic model. The calculation model ofthe track used in Ref. [21] is adopted in the present study.Each rail in the track shown in Fig. 3 is modeled as a Tim-oshenko beam resting on discrete sleepers. The Timoshenkobeam theory considers both shear effect and rotational iner-tial effect of the beam, which can characterize the dynamicbehavior of the rail more clearly and correctly in the highfrequency range, than the Euler beam model [21,22]. Thevertical and lateral bending deformations and rotation of therail are simultaneously taken into consideration, in whichboth ends of the calculated rail are hinged as shown in Fig. 3.

In order to simulate the effect of discrete sleeper supportson the coupling dynamics of the vehicle and track, the sleeperunder the beam is assumed to move backward at a constant Fig. 3 Calculation model of wheels rolling over a Timoshenko beam

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454 G. Xiao et al.

speed v and the vehicle is static with respect to the rails. Whenthe sleepers (1 to i+1), as shown in Fig. 3, start to move back-ward, the sleeper i + 2 appears from end B. Before sleeper 1reaches end A the beam is supported by i + 2 sleepers. Aftersleeper 1 disappears at end A, i+1 sleepers support the beam,the same as in the initial state. This process is repeated, andthus the behavior of discrete supports presents a periodicalaction on the beam. This model can be used to efficientlystudy the effect of different positions of the weld joint ina sleeper span. In order to eliminate the effect of hangingend of the beam, ltim = 52.8 m is selected, which covers 88sleeper bays and is long enough to eliminate the effect of thehanging end. xwk (k = 1, 2) is the distance from end A ofthe beam to wheelset k. The bogie is located at the centre ofthe calculated rail (the beam). xsi is the distance from end Ato sleeper i , and can be written as

xsi = ils−vt, (0 ≤ t ≤ iTs = ils/v, i =1, 2, . . . , Ns).

(16)

Based on the Timoshenko beam theory [22], the beammotion equations are obtained in different directions as fol-lows: for the lateral direction

m∂2 y(x, t)

∂t2 + κyG A

[∂ψy(x, t)

∂x− ∂2 y(x, t)

∂x2

]

= −Ns∑

i=1

Ryi (t)δ(x − xsi )+Nw∑j=1

Fwry j (t)δ(x − xw j ),

(17)

ρ Iz∂2ψy(x, t)

∂t2 + κyG A

[ψy(x, t)− ∂y(x, t)

∂x

]

−E Iz∂2ψy(x, t)

∂x2 = 0,

for the vertical direction

m∂2z(x, t)

∂t2 + κzG A

[∂ψz(x, t)

∂x− ∂2z(x, t)

∂x2

]

= −Ns∑

i=1

Rzi (t)δ(x − xsi )+Nw∑j=1

Fwrz j (t)δ(x − xw j ),

(18)

ρ Iy∂2ψz(x, t)

∂t2 + κzG A

[ψz(x, t)− ∂z(x, t)

∂x

]

−E Iy∂2ψz(x, t)

∂x2 = 0,

and for the rotation

ρ I0∂2φ(x, t)

∂t2 − G K∂2φ(x, t)

∂x2 = −Ns∑

i=1

Msi (t)δ(x − xsi )

+Nw∑j=1

MG j (t)δ(x − xw j ). (19)

The notations of y, z, and φ in Eqs. (17)–(19) are, respec-tively, the lateral, vertical and roll displacements of the rail,ψy and ψz are the slope of the deflection curve of rail withrespect to z and y axis.

The deflection and rotation of the rail are obtained usingmodal superposition principle, and written as

y(x, t) =NMY∑k=1

Yk(x)qyk(t), (20)

ψy(x, t) =NMY∑k=1

yk(x)wyk(t), (21)

z(x, t) =NMZ∑k=1

Zk(x)qzk(t), (22)

ψz(x, t) =NMZ∑k=1

zk(x)wzk(t), (23)

φ(x, t) =NMT∑k=1

�k(x)qT k(t), (24)

where qyk(t), qzk(t), and qT k(t) are the generalized coordi-nates in the lateral, vertical and roll directions, respectively.wyk(t) and wzk(t) are the generalized coordinates in the railcross-section rotation about y and z axis, respectively. Yk(x),Zk(x), and�k(x) are the kth mode shape functions of lateralbending, vertical bending and torsion of the rail, respectively,and yk(x) and zk(x) are the kth mode shape functions ofthe rail cross-section rotation about y and z axis. NMY =NMZ = NMT = 120, the total number of the rail mode func-tions selected in the calculation. The selection of NMY, NMZand NMT was discussed in detail in Ref. [20]. By substitut-ing Eqs. (20)–(24) into Eqs. (17)–(19), the partial differentialequations (17)–(19) are converted into second-order ordinaryequations as follows:for the lateral direction

qyk(t)+ κyG A

m

(iπ

ltim

)2

qyk(t)+κyG Aiπ

ltim

√1

mρ Izwyk(t)

= −Ns∑

i=1

Ryi (t)Yk(xsi )+Nw∑j=1

Fwry j (t)Yk(xw j ), (25)

wyk(t)+[κyG A

ρ Iz+ E Iz

ρ Iz

(iπ

ltim

)2]wyk(t)

−κyG Aiπ

ltim

√1

mρ Izqyk(t) = 0, (k = 1, . . . ,NMY),

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for the vertical direction

qzk(t)+ κzG A

m

(iπ

ltim

)2

qzk(t)+ κzG Aiπ

ltim

√1

mρ Iywzk(t)

= −Ns∑

i=1

Rzi (t)Zk(xsi )+Nw∑j=1

Fwrz j (t)Zk(xw j ), (26)

wzk(t)+[κzG A

ρ Iy+ E Iy

ρ Iy

(iπ

ltim

)2]wzk(t)

−κzG Aiπ

ltim

√1

mρ Iyqzk(t) = 0, (k = 1, . . . ,NMZ),

and for the rotation

qT k(t)+ G K

ρ I0

(iπ

ltim

)2

qT k(t) = −Ns∑

i=1

Msi (t)�k(xsi )

+Nw∑j=1

MG j (t)�k(xw j ), (k = 1, . . . ,NMT). (27)

The denotations of some symbols in Eqs. (17)–(27) can befound in Ref. [21]. qyk(t), qzk(t), and qT k(t) were obtainedfrom dynamic calculation, and thus the vertical, lateral andtorsional displacements of the rail, based on the superposi-tion principle of Timoshenko theory, were easily determined.However, their coupling interaction is ignored.

Each sleeper is represented by a rigid rectangular beam.Its calculation model and motion equations were presentedin Ref. [23], in which the yaws, longitudinal and pitchingmotions of the sleeper were neglected.

For the ballast bed, it is very difficult to satisfactorilycharacterize it with the help of any existent complicatedmathematical model. The calculation model of ballast bedin Ref. [20] is adopted in the present study, the feasibilityof which was discussed in Ref. [23]. Here only the verticalmotion is taken into account, and the motion of the subgradeis ignored.

2.3 Model of wheel/rail interaction

In the modeling of vehicle and track, the vehicle is coupledwith the track via a wheel/rail contact model. The numericalmethod for calculating the contact geometry was discussedin detail in Ref. [24]. The wheel/rail contact model includesa normal model and a tangential one. For considering theinfluence of the weld irregularities of the rail surface on thenormal load, the usual model of the wheel–rail normal loadreads

Fwrnk(t)=

⎧⎪⎪⎨⎪⎪⎩

{1

G[Zwk(t)−Zr(xw j , t)+δ0 − Z0(t)]

}3/2

,

when Zwk(t)−Zr(xw j , t)− Z0(t) > 0,0, when Zwk(t)−Zr(xw j , t)−Z0(t)≤0,

(28)

where k = 1, . . . , 8 indicate wheel 1, . . ., and 8, respectively;Zwk(t) and Zr(xw j , t) are the normal displacements of wheelk and the rail, respectively, at their contact point, which aredetermined by solving Eqs. (1)–(28) and calculating the con-tact geometry of the wheelset and the rail; δ0 is the approachbetween the wheel and rail caused by the static normal load;Z0(t) is the time function of the weld irregularity and G isthe coefficient of the normal contact stiffness concerning theHertizian contact condition of the wheel and rail, which is aconstant related to the wheel and rail contact surface curva-ture. Published studies usually adapted the value of G fromRef. [25], as shown in Eq. (29), which is based on a worn railhead profile (Rr = 230 mm).

G = 3.86R−0.115 × 10−8. (29)

Equation (29) is suitable for the worn rail profile, where R isthe wheel radius at the contact point. However this equationis not available for the new rail of CN60, because the newhead profile radii consist of three circular arcs with radii of300, 80 and 13 mm. In this study the transient G is calculatedusing the wheel and rail surface curvatures, and is determinedby calculating the contact geometry of the wheel/rail.

Shen–Hedrick–Elkins’ model [26] is used as the tangentialmodel to determine the relationship between the creepagesand the total creep forces of the wheel/rail. In the calculationof tangential creep forces, the creepages of wheel and railshould be calculated through transient analysis of the vehi-cle–track coupling dynamics. In the creepages calculation,the lateral and roll-over motions of the rail are taken intoaccount simultaneously, while it is assumed that the relativevelocity between the wheel and rail in their common normaldirection is equal to zero.

For the left wheel/rail, in the local coordinate system ofthe contact patch, the projection components of the relativevelocity read [20]

�V1L = �VxL cosψ +�VyL sinψ,

�V2L = −�VxL cos(δL + φ) sinψ +�VyL

× cos(δL + φ) cosψ +�VzL sin(δL + φ), (30)

�V3L = �VxL sin(δL + φ) sinψ

−�VyL sin(δL + φ) cosψ +�VzL cos(δL + φ),

where �VxL, �VyL and �VzL are components of the rela-tive velocity difference of the left wheel/rail in x , y and zdirections, respectively, also in the global coordinate system.For detail expressions, refer to Ref. [20]. ψ is the yaw angleof the wheelset, δL is the contact angle of the left wheel and

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456 G. Xiao et al.

rail; and φ is the roll angle of the wheelset. We use the factthat the relative velocity between the wheel and rail in theircommon normal direction is equal to zero [27], namely,

�V3L =�VxL sin(δL+φ) sinψ−�VyL sin(δL+φ) cosψ

+�VzL cos(δL + φ) = 0. (31)

Therefore, the components of Eq. (30) can be written as

�V1L = �VxL cosψ +�VyL sinψ,

�V2L = −�VxL sinψ/ cos(δL + ϕ) (32)

+�VyL cosψ/ cos(δL + ϕ),

�V3L = 0.

According to the definition of the creepages, the lateral creep-age component is expressed by

ξyL = �V2L

V, (33)

where V is the nominal forward speed of the wheelset. FromEq. (32) we can find that only the lateral creepage is changedand different from the lateral creepage formulae used in thepast. Therefore, the lateral creepage calculation is improvedin the present curving simulation.

It is noted that the mathematical expressions for the con-cerned forces in Eqs. (1)–(27) include the coefficients of thecorresponding equivalent springs and dampers, as well asthe displacement and velocity differences of the correspond-ing neighboring components in the system, furthermore, thewheel/rail interaction introduces some non-linear factors.Zhai [28] developed a numerical method especially to solvethe coupling dynamics equations of the railway vehicle andtrack. The stability, calculation efficiency and accuracy of thenumerical method were also discussed in detail.

2.4 Initial and boundary conditions of the coupledvehicle–track system

Before solving the motion equations of the dynamic system,the initial and boundary conditions should be given. Bothends of the Timoshenko beam modeling the rails are hinged,and the deflections and the bending moments at the beamhinged ends are assumed to be zero. The deflection of theroad bed is neglected, and only the vertical deflections of theequivalent ballast bodies are taken into account. The verticalmotion of the ballast bodies at both ends of the calculationlength is assumed to be always zero, and the static state ofthe systems is regarded as the original point of reference. Theinitial displacements and velocities of all components of thetrack are set to zero. The initial displacements and the initialvertical and lateral velocities of all components of the vehicleare also set to zero, and the initial longitudinal velocity is therunning speed of the vehicle, which is a constant.

2.5 Comparison between the results by SIMPACKand VTM

In order to verify the accuracy and reliability of the pres-ent model that is briefly called as VTM, the commercialcode SIMPACK for railway vehicle dynamics simulation andVTM are, respectively, used to calculate the dynamic behav-ior of a vehicle passing over a curved track at the speed of120 km h−1. The gauge of the track is 1,435 mm, its superel-evation is 120 mm, the rail cant is 1/40 and the radius of thecurved track is 800 m.

Figures 4, 5 and 6 are, respectively, the lateral displace-ments, the normal loads and the lateral forces of wheelset1, which are calculated by SIMPACK and VTM, respec-tively. Figure 4 shows good agreement between the lateraldisplacements calculated with the two models. When thewheelset passes over a circular curve, however, the lateraldisplacement obtained by SIMPACK is a little larger than thatobtained by VTM because the track flexibility considered inVTM is larger and also more reasonable than that in SIM-PACK. When the wheelset passes through the circular curvethe structure deformation of the track in the lateral directioncalculated with VTM is larger than that calculated with SIM-PACK. The lateral displacement of the wheelset obtained byusing VTM is larger than, of course, that obtained by usingSIMPACK. The lateral displacement of wheelset indicatesthat the lateral displacement is measured from the wheelsetcenter to the track center line.

Figure 5 illustrates the normal loads of wheelset 1, inwhich the solid lines and the broken lines indicate respec-tively the results given by VTM and SIMPACK. The normalload of the wheel on the high rail calculated by VTM issmaller than that by SIMPACK, and the situation is just the

Fig. 4 Lateral displacements of wheelset 1 versus traveling distance,calculated by SIMPACK and VTM

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Track dynamic behavior at rail welds at high speed 457

Fig. 5 Normal loads of wheelset 1 versus traveling distance, calculatedby SIMPACK and VTM

Fig. 6 Lateral forces of wheelset 1 versus traveling distance, calcu-lated by SIMPACK and VTM

opposite for wheels on the low rail. VTM is more reasonablethan SIMPACK because VTM considers the flexibility of thetrack. Furthermore, VTM can reveal the fact that the flexibletrack efficiently reduces the dynamic loading and the load ofthe wheelset is reduced when the wheelset is curving.

Figure 6 illustrates the lateral forces of the wheels. Thereis a good agreement between the lateral forces calculated byusing the two models for wheels on the low rail. The reason isthat the lateral forces between the wheel and the low rail arevery small, leading to small deformation of the curved trackin the lateral direction, and thus there is no much difference inusing rigid track models (SIMPACK) or using flexible trackmodels (VTM). However, for large lateral forces, the flexi-ble track models (VTM) have superiority over the rigid trackmodels (SIMPACK). Figure 6 shows that when the wheelset

Fig. 7 Wheel flange contact and contact forces

is passing through a circular curve, the lateral force betweenthe wheel and the high rail given by VTM is larger than thatby SIMPACK. The reason can be explained as follows. Whenthe wheelset passes over a circular curve the flange root ofthe outside wheel of the wheelset contacts the gauge cornerof the high rail of the curved track, as shown in Fig. 7.

The dashed arrow indicates the direction of the lateralcreepage ξτ of the wheel, which denotes the wheel contactingsurface slips with respect to the rail gauge corner. ξτ equalsthe rigid slip minus the slip caused by elastic deformationbetween the wheel and the high rail. In the case indicatedin Fig. 7, VTM calculates the slip arising from the elasticdeformation, which is larger than that given by SIMPACK.Therefore the total slip ξτ given by VTM is smaller thanthat by SIMPACK. According to the creep force theory ofwheel/rail system, the larger total slip can cause larger totalcreep force between the wheel and the rail. The creep forceFτ calculated by VTM is smaller than that by SIMPACKbecause the slip due to elastic deformation given by VTM islarger than that by SIMPACK. Using Fig. 7 the total lateralforce between the wheel and the high rail is written as

Fy = Fn sin δ − Fτ cos δ. (34)

Fn is the normal load, δ is the contact angle of the wheel andthe high rail. We should notice that, of course, the value oflateral force Fy depends not only on Fτ , but also on Fn andδ, as shown in Eq. (34).

2.6 Model of weld rail joint irregularity

According to the statistics of Japanese Railways (JR) [11],an irregularity, as shown in Fig. 8a, exists widely on the sur-faces of rail welds on Shinkansen railway lines. Figure 8billustrates a type of step irregularity (TSI), which is causedby misalignment at the rail ends prior to welding or after

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458 G. Xiao et al.

Fig. 8 Irregularities at welded rail joint. a Type of convexity irregu-larity (TCI) [24]. b Type of step irregularity (TSI)

welding. This type of convexity irregularity (TCI), as shownin Fig. 8a, consists of three curved segments, and can bedescribed as

Z0(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1

2δ1(1 − cos 2πvt), when 0 ≤ t ≤ 1 − λ

2v,

1

2δ2

[1 − cos

2πv

λ

(t − 1 − λ

2v

)]

+1

2δ1[1 − cosπ(1 − λ)],

when1 − λ

2v≤ t ≤ 1 + λ

2v,

1

2δ1(1 − cos 2πvt), when

1 + λ

2v≤ t ≤ 1

v,

(35)

where δ1 and δ2 are the depths of long and short wavelengthirregularity, respectively. The long and the short wavelengthsare, respectively, 1 m and λ, and t is the time.

Similarly, TSI can be determined by

Z0(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, when 0 ≤ t ≤ 1 − l

2v,

h

[1 − cos

πv

l

(t − 1 − l

2v

)],

when1 − l

2v< t <

1

2v,

h, when1

2v≤ t ≤ 1

v

(36)

where h and l are the height and length of the step irregularity,respectively.

3 Numerical results and discussions

In the present study, a tangent track with different types ofrail welds is considered, as described in Sect. 2.2. The trackgauge is 1,435 mm, the rail cant is 1/40, the rail is CN60and the sleeper pitch is 600 mm. The vehicle speed is rang-ing from 200 to 300 km h−1, and the assumed position of theweld varies from 0.3 m before and after a sleeper. The weldjoints on both side rails are assumed to be symmetrical aboutthe central line of the track. The vehicle and track parametersare listed in Table 3.

Table 3 Structural parameters of passenger car and track

Parameters Values Parameters Values

Mc/kg 4.820×104 Mt /kg 3.086×103

Mw/kg 1.675×103 Iwx /(kg m2) 0.900×103

Icx /(kg m2) 8.167×105 Itx /(kg m2) 2.312×103

Iwy /(kg m2) 0.108×103 Iwz /(kg m2) 0.900×103

Icy /(kg m2) 2.999×106 Ity /(kg m2) 4.730×103

Ktz /(N m−1) 0.178×106 Ctz /(N s m−1) 8.000×104

Icz /(kg m2) 2.999×106 Itz /(kg m2) 4.730×103

Kpz /(N m−1) 1.067×106 Cpz /(N s m−1) 6.000×103

Ktx /(N m−1) 4.500×106 Kty /(N m−1) 8.465×105

r0/m 457.5×10−3 Hbw/m 0.140

G/(N m−2) 7.919×1010 A/m2 7.753×10−3

mb/kg 4.660×102 hR/m 9.453×10−2

Ctx /(N s m−1) 0.000 Cty /(N s m−1) 3.000×104

Kpx /(N m−1) 8.788×106 Kpy /(N m−1) 2.984×106

Cpx /(N s m−1) 0.000 Cpy /(N s m−1) 0.000

g/(m s−2) 9.810 a0/m 746.5×10−3

HcB/m 1.792 HBb/m 0.078

I0/(m4) 3.740×10−5 KbhL/(N m−1) 3.000×107

κy 0.4507 κz 0.5329

CbhL/(N s m−1) 6.000×104 KbhR/(N m−1) 3.000×107

ds/m 1.120 dw/m 0.978

CbhR/(N s m−1) 6.000×104 Kw/(N m−1) 7.800×107

Cw/(N s m−1) 8.000×104 ls/ mm 600.0

Nw 4 lc/m 8.500

mr /(kg m−1) 60.64 ms/kg 3.490×102

E /(N m−2) 2.059×1011 ρ/(kg m−3) 7.800×103

Iy /m4 3.217×10−5 Iz /m4 5.240×10−6

KphL/(N m−1) 2.947×107 KpvL/(N m−1) 7.800×107

CphL/(N s m−1) 5.000×104 CpvL/(N s m−1) 5.000×104

KphR/(N m−1) 2.947×107 KpvR/(N m−1) 7.800×107

CphR/(N s m−1) 5.000×104 CpvR/(N s m−1) 5.000×104

KbvL/(N m−1) 7.000×107 KbvR/(N m−1) 7.000×107

CbvL/(N s m−1) 5.000×104 CbvR/(N s m−1) 5.000×104

KfL/(N m−1) 6.500×107 CfL/(N s m−1) 3.100×104

KfR/(N m−1) 6.500×107 CfR/(N s m−1) 3.100×104

Nt 2 ν 0.3

lt /m 1.200

3.1 Track dynamic behavior for vehicle passing on weldirregularities

3.1.1 Effect of the rail welds with different typesof irregularities

In order to investigate the effect of different types of weldirregularities on the wheel/rail interaction, three types ofthe weld irregularities are, respectively, considered in this

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Track dynamic behavior at rail welds at high speed 459

Fig. 9 Histories of wheel/rail normal force. a SWI versus CWI. b SWI versus TSI

section. The first type is the case of λ = 0.1 m, δ1 = 0.1 mmand δ2 = 0.3 mm that is the compound weld irregular-ity (CWI), as shown in Fig. 8a. The second is the case ofλ1 = 0.1 m, δ1 = 0 and δ2 = 0.3 mm, which is the singleweld irregularity (SWI). The third is TSI with h = 0.3 mmand l = 0.05 m, as shown in Fig. 8b.

Figure 9 shows time histories of the wheel/rail normalforces when the vehicle passes through three types of railweld irregularities at the speed of 200 km h−1. In Fig. 9, theblack boxes denote the sleepers, the center of the rail weldis above the sleeper at a distance of 3.0 m from the origin.It is observed from Fig. 9a that SWI and CWI give rise toapproximately the same normal load peaks, and the peaksof wheel/rail normal forces occur when the wheel passesthrough the short-wave weld irregularity. However, there issome difference between the time histories of the normalforces caused by CWI and SWI. Figure 9b illustrates thetime histories of the normal wheel/rail forces when the vehi-cle passes through SWI and TSI. It is worthy noticing that thenormal loads excited by SWI and TSI are in close agreement,and Fig. 9b is similar to Fig. 9a. The loss of contact betweenthe wheel/rail occurs in the three cases. The maximum ofthe dynamical normal loads is about three times of the staticnormal load. From Fig. 9 we can see that the short wave-length irregularity plays a key role in the wheel/rail impact,as described in Refs. [3,16,17,29].

The short-wave contribution was almost entirely deter-mined by the rail welded geometry, especially in the caseof new tracks. It is directly correlated with the rate of trackcomponents deterioration [3]. Hence, in the following study,only the type of single short-wave weld irregularities (SWI)should be taken into account.

3.1.2 Comparison of two types of SWI

The rail weld standards of world railways consider the ver-tical tolerances. The current Chinese standard states that

the vertical deviation of convexity irregularities should beless than 0.3 mm in the 1 m range for the railway lines of200 km h−1 and 0.2 mm for the lines of more than 200 km h−1.The weld concavity of rail surface should be avoided.Steenbergen and Esveld [16,17] stated that the current weldstandards were not adequate because they did not take intoaccount the geometrical shapes of the welds which weredirectly correlated with dynamic wheel/rail contact forces.In order to testify whether the current standards are reason-able or not, the effect of two types of SWI on wheel/rail nor-mal forces is investigated when the vehicle passes over suchrail weld irregularities at 200 km h−1. Figure 10 illustratestwo types of SWI with the same depth of 0.3 mm and twodifferent wavelengths, i.e. 1 m and 0.1 m. It is obvious thattheir depths do not exceed the maximum vertical toleranceof the weld standard of China Railway. The time historiesof wheel/rail normal forces when the vehicle passes throughthese two types of SWI and their corresponding linear spec-trums are given in Fig. 11a and b. From Fig. 11a we can seethat there is a big difference between the dynamical normalloads of the two cases, and the irregularity with λ = 0.1 m(dotted line) has greater dynamic effect than the irregularitywith λ = 1 m (solid line). The maximum impact force causedby the weld irregularity with 0.1 m wavelength is 205.9 kN,which is approximately 2.8 times the static wheel load and2.5 times the maximum load caused by the irregularity withλ = 1.0 m (dotted line). The loss of contact occurs in thecase of λ = 0.1 m, while in the case of λ = 1.0 m the wheeland rail keep in contact all the way.

The frequencies of the fluctuation of wheel/rail normalforces are illustrated in Fig. 11b, from which it is found thatthe two types of the irregularities excite the same two reso-nant frequencies which are, respectively, fA = 33.9 Hz andfB = 84.7 Hz. The peak for the case of λ = 0.1 m is muchhigher than that for the case of λ = 1 m at fA = 33.9 Hz. Theirregularity with λ = 0.1 m excites higher resonant frequen-cies, fC = 406.9 Hz, fD = 1203.7 Hz and fE = 1729.3 Hz.

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460 G. Xiao et al.

Fig. 10 Assumed irregularities at welds

However, the long wavelength irregularity does not exciteresonant frequencies larger than fB. The resonant frequencyfA is probably the resonant frequency of the system ofwheel/rail in contact which is related to the wheel mass andthe contact stiffness of the wheel/rail system. But this facthas not been verified through tests so far. fB approaches tothe passing frequency of the sleepers when the vehicle passesthe track at 200 km h−1 [23], corresponding to a passing fre-quency of 92.5 Hz. The higher frequencies fC , fD and fE

are certainly the resonant frequencies of the track which maybe easily excited by the short wavelength irregularity. fC isusually the resonant frequency for the rails to move relativeto the sleepers in vertical direction [18]. Maybe fD and fE

are the first and the second pinged-pinned resonant frequen-cies of the track [18]. In order to fully understand fC, fD

and fE it is necessary to make further investigation into thetrack structure through tests.

Hence, according to the above results, it can be predictedthat the service life of the welded rail would be much short-ened due to the high frequencies and larger impact forcescaused by the short-wavelength weld irregularities.

3.1.3 Impact forces between the rail and the sleeper

When the vehicle passes through the rail weld irregularitieswith three different wavelengths, i.e. λ = 0.1, 0.5 and 1.0 mat 200 km h−1, power spectral densities (PSD) of the verti-cal accelerations of the rail and the sleeper are illustrated inFig. 12a, b, respectively. From Fig. 12 we can see that thePSD of the vertical acceleration of the rail is larger than that ofthe sleeper at high frequencies. The PSD of the vertical accel-eration of the rail and the sleeper increases with decreasingwavelength for frequencies lager than 100 Hz. When the fre-quency is less than 100 Hz, the situation is just the opposite.Figure 12 indicates that shorter wavelength weld irregulari-ties easily excite higher frequencies dynamical behavior.

The impact force of the rail/sleeper (FRS) is an importantfactor to be considered in the analysis of dynamic behaviourof the track components. Negative FRS indicates compres-sive force between the rail and the sleeper, and positive onerepresents tensile force. When the vehicle passes through atrack without any irregularity at 200 km h−1, the time his-tory of the FRS is given in Fig. 13, in which the verticaldotted line or the gray box indicates the position of sleeperS48, the coordinate of which is 28.4 m along the track. Itis obvious that when the vehicle passes through the sleeperFRS has four peaks with negative values, which are excitedby the four wheelsets passing over the sleeper S48, respec-tively. The distance between the first and the second peaks is2.4 m that is just the wheelbase of the bogie. The maximumsof the positive and the negative FRS are 0.85 and −26.1 kN,respectively.

Figure 14 shows time histories of FRS when the vehiclepasses through the rail weld irregularity with wavelengthλ =0.1 m and depth δ = 0.3 mm. The FRS excited by the frontbogie passing over the weld irregularity is similar to thatby the rear bogie passing over the same irregularity. Hence,the FRS excited by the rear bogie was not shown here for

Fig. 11 a Histories of wheel/rail normal forces. b Linear spectrums of the corresponding wheel/rail normal forces

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Track dynamic behavior at rail welds at high speed 461

Fig. 12 Power spectral densities of the vertical acceleration of rail and sleeper. a Rail. b Sleeper

Fig. 13 Time history of FRS without any irregularity

simplicity. Figure 14a shows vertical forces between therail and sleepers S10, S11, S12 and S13, caused by the frontbogie. Sleeper S10 is at the coordinate of 6 m and the rail weldjoint is over sleeper S10. The results in the dashed frame inFig. 14a indicate the FRS generated from wheelset 1 pass-ing through the dashed frame, part of which is amplified inFig. 14b. The weld center coordinate is 6.0 m. It is obviousthat the maximim negative FRS occurs when the weld irregu-larity is above the sleeper, the maximum positive and negativeFRS are 6.0 and −87.2 kN, which are approximately 7.1 and3.3 times the FRS under the normal condition. These largeFRS may accelerate fatigue failure of the rail pad, the fastensystem and the sleeper.

Figure 15 illustrates FRS for vehicle passing through thewelded rail with three different wavelengths (λ = 0.1, 0.5and 1.0 m) and depth δ = 0.3 mm. The weld center locationhas the coordinate of 3.0 m along the track, and is locatedover sleeper S6. It is noted that the FRS variation tendency,as shown in Fig. 15, is similar to the wheel/rail normal forcesshown in Fig. 9a. FRS increases rapidly, in particular in the

case of short wavelength irregularity. Figure 15 indicates thatthe maximum negative values of FRS are −87.2, −34.3 and−27.8 kN for the three cases of λ = 0.1, 0.5 and 1.0 m,respectively. They are approximately 3.3, 1.7 and 1.07 timesof the static FRS. For the case of λ = 0.1 m, large transienttensile force is generated between the rail and sleeper S6.

It can be concluded from the above analysis that short-wavelength irregularity easily excites higher resonant fre-quencies of the track, which could lead to severe damageand noise of the wheel and track components.

3.2 Effect of the vehicle speed

Figure 16 shows the influence of the train speed on thepeak of wheel/rail normal forces for the four irregulari-ties with the same wavelength λ = 0.1 m and two depthsδ = ±0.3 and ±0.2 mm, where notations “+” and “−” rep-resent the convexity and concavity of the weld irregularity,respectively. It can be found from Fig. 16 that the peak ofwheel/rail normal forces increases linearly with increasingspeed. When the weld irregularities are over the sleeper,only the normal forces of the first wheelset is present. Forthe cases of δ= ±0.3 mm, the impact force peak caused bythe concave rail weld irregularity (δ= − 0.3 mm) is alwayslarger than that caused by the convex one (δ= 0.3 mm)as the speed increases. For the cases of δ= ± 0.2 mm,when the speed is less than 230 km h−1, the impact forcepeak caused by the convex rail weld is larger than thatcaused by the concave one. When the speed is larger than230 km h−1, the situation is just the opposite. The increasingrates of the impact force peaks for the concave rail weldsare higher than those for the convex ones when the speedincreases.

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462 G. Xiao et al.

Fig. 14 Time histories of FRS of S10 to S13

Fig. 15 Time histories of FRS of S6

Fig. 16 Peak of wheel/rail normal forces versus train speed

3.3 Effect of the weld irregularity position

Actually, different positions of the weld joints in a sleeperspan were found in railway sites, as shown in Fig. 17.Figure 17a shows that it is about 0.2 m from the weld joint tothe center of the nearest sleeper. Figure 17b indicates 0.3 m(at about the mid span); and Fig. 17c illustrates that the weldjoint is just above the sleeper. To investigate the effect of theweld joint position on the impact force, the vehicle speed of200 km h−1 and the weld joint irregularities used in Sect. 3.2are considered. Figure 18 shows variation curves of the peakvalues of the wheel/rail normal forces versus the weld jointposition. The grey box in the figure denotes the center ofa sleeper. For the excitation of the convex or concave weldjoints the peak normal force reaches its maximum when theweld rail joint is set to be over the sleeper. The peak val-ues of the normal load over the sleeper for δ = ±0.3 and±0.2 mm increase by about 16.0–18.1% and 12.6–26.3% ,respectively, with respect to those at the mid span.

3.4 Effect of the irregularity wavelength

The results in Fig. 19 exhibit the influence of the irregu-larity wavelength at the rail weld on the impact force. Thetrain speed of 200 km h−1 and the depths of δ = ±0.3 and±0.2 mm are considered. Figure 19 indicates that the peakof wheel/rail normal forces decrease quickly when the wave-length increases from 0.1 to 0.3 m. However, when the wave-length exceeds 0.3 m, the peak values of the normal contactforces decrease slowly. It is suggested that the irregularityat rail welds with its wavelength less than 0.3 m and depthlarger than 0.2 mm should be avoided on the high speed lines.There is no significant difference between the results for thetwo cases of concave and convex weld joint, as plotted inFig. 19.

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Fig. 17 Different positions of welded joint in a sleeper span

Fig. 18 Peak wheel/rail normal forces versus weld joint position

Fig. 19 Peak wheel/rail normal forces versus wavelength of the irreg-ularity

3.5 Effect of the irregularity depth

Figure 20 illustrates that the influence of the irregularity wavedepth at rail welds on the impact force. The train speed of200 km h−1 and the three wavelengths of λ = 0.1, 0.2, 0.3 m

Fig. 20 Peak wheel/rail normal forces versus wave depth of the irreg-ularity

are considered. It can be found from Fig. 20 that the peakimpact forces increase linearly as the wave depth increases.The increasing rate of the normal force for the concave railwelds is slightly larger than that for the convex welds, espe-cially for irregularity welds with their wavelength λ less than0.2 m.

3.6 Effect of the axle load

The vehicle axle load has a great influence on the normalforce of the wheel/rail. Figure 21 shows the influence of theaxle load on the peak of wheel/rail normal force fluctuation.The train speed of 200 km h−1 and the depths of δ = ±0.3and ±0.2 mm are considered. From Fig. 21 we can see thatthe wheel/rail normal force fluctuation decrease with increas-ing axle load for the case of concave welded irregularities.However, there is no significant difference for the cases ofconvex welded irregularities. It is noted that when the axleload is 15 ton, the peak of wheel/rail normal force fluctuationreaches its maximum.

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464 G. Xiao et al.

Fig. 21 Peak of fluctuate wheel/rail normal forces versus axle load

3.7 Combined effects of the wavelength and the depth

In the results described above, the effect of one single fac-tor, i.e. wavelength or wave depth of the irregularity, onthe wheel/rail normal forces is discussed. Figure 22 sum-marizes the combined effects of both wavelength and wavedepth of convex rail welds on the dynamic impact factorat 200 km h−1. The dynamic impact factor η is defined asthe ratio of the impact force peak to the static wheel load.Figure 22a uses the three-dimensional surface to describe theinfluence and Fig. 22b indicates the corresponding contourof the surface. It can be found that the dynamic impact fac-tor changes drastically with increasing wave depth when thewavelength is short. When the wavelength is large, the wavedepth does not significantly affect the dynamic impact factor.The dark area in the Fig. 22a denotes that the ratio η exceeds2, and the grey area indicates that η varies between 1.4 and 2.

From Fig. 22b it is known that for a given dynamic impactfactor the relationship between the wave depth and wave-length is approximately linear. This further indicates that theratio of the wave depth to wavelength (δ/λ) is more ade-quate to assess the rail weld geometry than the wave depththat is adopted in the current standard. For example, if weguaranty that the dynamic impact factor is less than 1.4 or 2,the ratio δ/λ should be less than 0.86 × 10−3 or 2.6 × 10−3,respectively. For different train speed and axle load, the aboverelationship needs to be recalculated.

4 Conclusions

A dynamics model for a vehicle coupled with a tangent trackis developed to investigate the effect of a variety of potentialuneven longitudinal profile of tangent rail weld on the trackdynamic behavior. In the investigation, three types of weldirregularities are considered, and the effects of train speed,axle load, wavelength, wave-depth and the position of thewelded joint in a sleeper span are analyzed. From the numer-ical results conclusions are drawn as follows:

(1) CWI, SWI and TSI give rise to similar wheel/railimpact loads. Shorter wavelength weld irregularity eas-ily excites higher resonant frequencies of the vehi-cle/track dynamic system, leading to severe damagesof the wheel and track components.

(2) The wheel/rail impact load caused by the rail weld irreg-ularity with 0.1 m wavelength and 0.3 mm depth is about2.3 times that caused by the irregularity with 1 m wave-length and the same depth at 200 km h−1. Such two typesof irregularities don’t exceed the tolerance of rail weldirregularities in the maintenance standard issued by the

Fig. 22 Influence of wavelength and wave depth of the irregularity on dynamic impact factor. a Surface plot. b Contour plot

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Track dynamic behavior at rail welds at high speed 465

Railway Minister of China. It can be predicted that theservice life of the rail weld joint in the case of the formertype irregularity is certainly much shorter that of thelatter type irregularity. Hence, the present maintenancestandard for weld rail joints is not reasonable and needsto be further revised.

(3) The wheel/rail impact force increases nearly linearlywith increasing depth of the irregularity at a welded railjoint under the condition that the wavelength does notvary. The maximum of the impact normal load causedby the weld joint irregularity over a sleeper is largerthan that caused by the weld joint irregularity in asleeper span. The wheel/rail normal forces fluctuationdecreases with increasing axle load for the cases of con-cave weld irregularities, however there is no significantdifference between the wheel/rail normal forces fluctu-ations in the cases of convex welded irregularities.

(4) The depth to wavelength ratio of the weld rail irregular-ity δ/λ can more reasonably describe the tolerance ofweld rail irregularities. Controlling δ/λ can control theincrease of the wheel/rail normal impact load causedby weld rail irregularities, and thus δ/λ is very useful inthe tolerance design of welded rail profile irregularitycaused by hand-grinding after rail welding.

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