Application of the semi-Hertzian method to the prediction of wheelwear in heavy haul freight car

Junjun Ding n, Fu Li, Yunhua Huang, Shulei Sun, Lixia ZhangSouthwest Jiaotong University, School of Mechanics and Engineering, Chengdu 610031, Sichuan, China

a r t i c l e i n f o

Article history:Received 23 November 2013Accepted 25 November 2013Available online 7 December 2013

Keywords:WearSemi-HertzianWheel profileSimulation

a b s t r a c t

In order to simulate the wheel wear behaviour of freight cars, the rail vehicle's multi-body dynamicalmodel and wheel wear simulation programme were combined as a wheel wear simulation tool in thispaper. Multi-body dynamical models of China's heavy haul freight cars such as the C80, C80H, C70 andC70H, were built in SIMPACK software and the track system was built based on the China's Ring-line. Forthe wheel/rail rolling contact, the semi-Hertzian method and FASTSIM algorithm were applied to solvethe normal and tangential problem respectively. The shapes of worn wheel profiles in the simulationagree well with the field measurements, but wear rates from the simulation are larger than those foundin field measurements. This discrepancy is contributed to two factors. The first factor is the wheelmaterial. The CL60 wheel steel used in China's freight car, is harder than the BS11 wheel steel used inZobory's wheel/rail wear experiments. The second factor is the impact of a material's elastic sheardeformation on the slip velocity in contact patch. This impact will increase the wear rate and wasconsidered in the present wear simulation.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

The wear phenomenon which occurs through the wheel/railcontact is currently receiving more and more attention frommanufacturers and operators. The rail vehicle's dynamic behavioursuch as stability, safety and ride comfort are all strongly related towheel profiles, which need replacement once wear reaches itslimitation. Turning repair, however, reduces the service life ofrailway wheels, so manufacturers and operators hope to lengthenthe total running distance of wheels before such replacements areneeded. Due to the intense competition in the railway market, thedevelopment of a tool able to simulate wheel wear would provehighly profitable.

1.1. Wheel wear simulation

Pearce and Sherratt built a simple model connecting wear tothe dissipated energy in the contact area and attempted tosimulate the wheel wear of a railway vehicle [1]. Zobory devel-oped three wear models to formulate the proportionality existingbetween the debris mass flow density and the flow density ofenergy dissipated in the contact patch [2]. Due to the differentregimes on wheel tread and wheel flange, the mild regime on the

tread and severe regime on the flange were introduced. In order tosimulate the wheel wear, Zobory built a multi-body vehicle modelin ELDACW, and used the Hertzian theory and the FASTSIMalgorithm to solve the normal and tangential contact problem. Inthe simulation, wheel profiles are updated every 1000 km, and thesimulation results are compared with various on-line measure-ments related to a vehicle running on the Gotthard line.

In order to simulate the wear evaluation of wheel profiles,Jendel built a wear model with wear coefficients derived from theexperiments on pin-on-disc and disc-on-disc machines [3]. Jendelbuilt the vehicle dynamical model of the X10B in Gensys code andused Kalker's simplified theory to analyse the adhesion/slip regionin the contact patch. The simulation updated wheel profiles wheneither a maximum wear depth of 0.1 mm or a maximum distanceof 1500 km was reached. The simulation results were comparedwith measurements of serviced wheels on the commuter railwaynetwork in Stockholm. Enblom continued Jendel's work, takinginto account the impact of materials' elastic shear deformation onslip velocityand the influence of disk braking and lubricationconditions on wheel wear [4].

Braghin developed a fast and reliable mathematical model topredict wheel profile evolution caused by wear [5]. Instead offriction power, this model utilised wear index, the product of creepforce and creepage. The simulation results were compared withexperimental measurements from a full-scale wear test, and werein good agreement on the wheel tread and yielded an over-estimation on the wheel flange.

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0043-1648/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.wear.2013.11.052

n Corresponding author. Tel.: þ86 2887601321; fax: þ86 2887634632.E-mail address: dingjunjun@swjtu.edu.cn (J. Ding).

Wear 314 (2014) 104–110

1.2. Wheel/rail contact modelling

Wheel wear not only depends on the contact pressure andcreep, but also on the boundary between adhesion and slip regions,which is determined by the wheel/rail rolling contact theory.

In the dynamic simulation and wheel wear prediction ofrailway vehicles, the wheel/rail normal contact is usually modelledusing Hertz's theory. Hertz's theory is based on the assumptionthat the curvatures in the contact zone are constant. When inflange contact cases, even in some tread contact cases, theviolation of Hertz's assumption may occur where the real contactarea is not an ellipse anymore.

In order to solve the problem of inconsistent curvatures in thecontact patch, a semi-Hertzian method was proposed by Ayasse andChollet [6]. In the semi-Hertzian method, the lateral curvatures arenot constant in the contact area, however the longitudinal curva-tures remain constant like in the Hertzian hypothesis.

The semi-Hertzian method, while useful for correcting theaforementioned curvature inconsistencies is still limited due to itsinfinite elastic half space assumption. Therefore, the finite elementmethod (FEM), which is not subject to this limitation, was used forthe wheel/rail normal problem [7]. However, since, the FEMrequires a relatively long computing time, it is not currently suitablefor rail vehicle dynamic simulation or wheel wear prediction.

Since it requires far less calculation time, Kalker's simplifiedtheory (FASTSIM) is usually applied to calculate the creep forcesand boundary of adhesion/slip regions in a wheel/rail tangentialcontact patch [8]. Kalker's exact theory (CONTACT) is more precisethan FASTSIM [9], but is seldom used in rail vehicle dynamicsimulation and wheel wear prediction due to the slow speeds atwhich its calculations can be made.

2. Wheel wear simulation

A wheel profile wear simulation tool consists of four main tasks:multi-body dynamic simulation of vehicle/track system, wheel/railrolling contact analysis, calculation of material removed by wear,and smoothing of wear depth and updating of wheel profile.A schematic representation of a wheel profile wear simulation toolis shown in Fig. 1.

At each integration step of the multi-body dynamic simulation,the outputs wheel/rail contact parameters such as contact posi-tions, creepages and normal forces, are the inputs of the wheel/railrolling contact analysis. From the wheel/rail rolling contact analy-sis, the contact pressures, slip velocities, and the boundary of

adhesion/slip regions in the contact patch can all be obtained.With this data, the removed material in each contact patch canthen be calculated using the wear model, and the total wear depthdistribution in wheel profiles can be obtained by accumulating thewear volume in each contact patch. The information on the wheelprofile will be updated, and then fed back into the multi-bodydynamic model of the rail vehicle. The procedure should becontinuously repeated until the parameters of a wheel profile ortotal mileage reach their respective designated limits.

2.1. Multi-body dynamic model of rail freight car

In this paper, the multi-body dynamical models of differentkinds of rail freight cars were built in the SIMPACK software. Anexample of one such model is shown in Fig. 2. In the multi-bodymodels, car-bodies, bolsters, frames and wheelsets are all consid-ered rigid bodies, with primary and secondary suspensions repre-sented through linear and non-linear viscoelastic elements. Inorder to obtain more accurate results of dynamic simulation, thetrack irregularity is considered in multi-body models. The China'sLM wheel profile and CHN60 rail profile, shown in Fig. 3, are usedin present multi-body models.

2.2. Semi-Hertzian method

In the semi-Hertzian method, the contact patch is divided intoindependent longitudinal strips with width Δy, shown in Fig. 4.A strip is characterised by its position along the y axis (yi) and itsindentation hi. The part of a strip between [�ai, ai] that is incontact with ai is called the participation (interaction) length.

In Hertzian's contact theory, the vertical separation of wheeland rail profiles is approximated by

zðx; yÞ ¼ Ax2þBy2 ð1Þ

where A and B are called relative curvatures.Since the curvatures of wheel and rail profiles in contact

patches are not constant, the vertical separation of wheel and railprofiles can not be defined by Eq. (1). The vertical separation ofwheel and rail profiles at point yi must then be estimated by

zðx; yiÞ ¼ zp;1ðyiÞ�zp;2ðyiÞþAðyiÞx2 ð2Þ

where zp,1(yi) is the wheel profile and zp,2(yi) is the rail profile.The first strip in contact (minima of z(x¼0, yi)) is called the

mother strip, with position yo¼0 and vertical separation zo.

Fig. 1. Scheme of wheel wear simulation.

J. Ding et al. / Wear 314 (2014) 104–110 105

The condition for a strip to be in contact is

yi such that zðx¼ 0; yiÞ�ho�zor0 ð3ÞThe half length of the strip ai is

ai ¼ffiffiffiffiffiffiffiffiffiffiffiffiffihi=Aci

qð4Þ

where hi¼zoþho�z(x¼0, yi), ho is the indentation of the motherstrip, mi and ni are the Hertz contact parameters of the strip, Aci.The relative curvature corrected from A, can be estimated by

Aci ¼Bin2

i

m2i

ð5Þ

The normal force of the ith strip is

Ni ¼12

1niri

E1�v2

1εihiΔy ð6Þ

where Ni is the normal force, E is the Young's modulus, v isPoisson's ratio, ri is the Hertz parameter of the ith strip, and εi can

be expressed as

εi ¼n2i Bi

riðBiþAiÞð7Þ

To estimate the tangential stresses in wheel/rail contact patch,Kalker proposed a simplified theory (FASTSIM). As FASTSIM algo-rithm works along the x-direction with no influence of the ydirection (as the semi-Hertzian method does), it is possible toadopt this method to estimate tangential stresses when normalstresses are calculated with the semi-Hertzian method.

By applying the FASTSIM algorithm, it is possible to obtain anexpression of stresses valid on a single strip. Stresses formulationfor the semi-Hertzian method are [6]

sz;iðxÞ ¼43π

1niri

E1�v2

1� xai

� �2 !

hiai

aia1εi

ð8Þ

sx;i ¼ �38GC11;iξx;i 1� x

ai

� �aia

ð9Þ

sy;i ¼ � 38GC22;iξy;iþ

2π

ffiffiffiffiffiffini

mi

rGC23;iξz;iðaiþxÞ

� �1� x

ai

� �aia

ð10Þ

where sz,j is the traction bound, sx,i and sy,i are tangential contactstresses, a is the half length of the mother strip, G is elastic shearmodulus, ξx,i, ξy,i and ξz,i are the longitudinal, lateral and spincreepages respectively, C11,i, C22,i and C23,i are Kalker's creepcoefficients of the ith strip.

2.3. Wear model

In this paper, the wheel wear model proposed by Zobory isused to calculate the material removed by wear. The contact patchis divided into an adhesion region and a slip region, denoted as Aa

and As respectively. This is shown in Fig. 5. Due to the dependenceof wear on the slip velocity, Zobory considered wear to only occurin the slip region.

The contact patch is divided into nx�ny elements. With thedissipated energy flow density, Ed defined as the specific energyloss due to sliding friction., The Ed of the element [i, j] can then bewritten as [2]

Ed½i; j� ¼sx½i; j�vx½i; j�þsy½i; j�vy½i; j� ½i; j�AAsðtÞ

0 ½i; j�=2AsðtÞ

( )ð11Þ

where vx and vy are longitudinal and transversal slip velocitiesrespectively, i¼1, 2…nx, j¼1, 2…ny.

Fig. 2. Multi-body dynamical model in SIMPACK.

Fig. 3. Wheel and rail profiles in present model.

Fig. 4. Parameters definition of the semi-Hertzian method.

Fig. 5. Adhesion and slip region in contact patch.

J. Ding et al. / Wear 314 (2014) 104–110106

The proportionality between the stochastic debris mass flowdensity md and the dissipated energy flow density at element [i, j]is

md½i; j� ¼ k½i; j�Ed½i; j� ð12Þwhere k is the wear coefficient.

The relation between the wear coefficient k and dissipatedenergy for classical wheel material is shown in Fig. 6. Due to thedifferent regimes on wheel tread and wheel flange, Zoboryintroduced two wear zones in the wear model, zone I and zoneII, representing the mild regime and severe regime respectively.

2.4. Smoothing method and wheel profile updating

The accumulated wear depth, obtained from the multi-bodydynamical model and wear model, is usually distributed in a sawtooth pattern. Wear depth with no smoothing leads to the shortwavelength concavities along the wheel profiles, and these shortwavelength concavities have no physical meaning.

Barbarino [10] compared several different smoothing methodswith the criteria of computational cost and stability and accuracy.It was found the best smoothing strategy was the moving average.The cubic spline algorithm was also used to smooth wear dis-tribution [3].

In this paper, a wavelet filtering method was used to smooththe cumulative wear depth. With the wavelet filtering method,cumulative wear depth is considered a signal with noise, and thede-noising of the signal can smoothen the wear depth. Thewavelet filtering method was compared with the moving averagemethod and cubic spline smoothing method, and the resultsindicated that the wavelet filtering method has a better smoothingeffect on wear depth.

The updating strategy is a key point of the profile wearprediction model. Its purpose is to determine the mileage withupdated wheel profiles and new calculation results of contactforces, tractions and slips. An overly conservative strategy, requir-ing too frequent profile updates, could result in unnecessarycomputational effort. On the other hand, increasing the mileagebetween two calculations too much may lead to inaccuracies inthe final worn wheel profile (or even divergence in the numericalprocedure) due to the non-updated wheel profiles in the multi-body code.

Different wheel profile updating strategies were compared andit was found that the most efficient one is based on the maximumwear depth, i.e. the profile is updated when a given threshold ofthe maximum value of cumulative wear depth is reached [10].A sensitivity analysis showed that a threshold of 0.1 mm is lowenough to guarantee a good accuracy and at the same time doesnot lead to excessive computational effort.

3. Reliability tests of freight car in China's Ring-line

From December of 2003 to August of 2007, the reliability testsof the speed increased freight car carried out in the China's Ring-line, is shown in Fig. 7.

The Ring-line track is 8.5 km long and is composed of a big ringand a small ring. The big ring has a curve radius of 1432 m and acant is 105 mm. The small ring has a curve radius is 1000 m and acant is 150 mm. The planar graph of the Ring-line is shown inFig. 8.

In the reliability tests on the Ring-line, the maximum runningspeed is 120 km/h and the total mileage of each freight car is1,80,000 km. The tested freight cars include the C70, C80, C70Hand C80H, where the C70 and C80 are equipped with cross bracedbogies and the C70H and C80H are equipped with swing motionbogies. Each type of freight car has two load states, emptyand heavy.

In order to research the influence of increasing speed on wheelwear, the parameters of the wheel profiles were tested throughoutthe entirety of the reliability tests. The parameters of the wheelprofiles include tread wear depth (wear depth in rolling circle),flange thickness, and the shape of worn wheel profiles.

When the C80 fright car equipped with the ZK6 bogie, ran onthe Ring-line in the heavy state, it produced wheel profile evolu-tion due to wear data that is shown in Fig. 9. The tread weardepths and flange thicknesses of all wheels are shown in Figs. 10and 11 respectively.

Due to the large curve radius of the Ring-line, we can find thatmost of the wear occurs in the tread while nearly none occurs inthe flange. Most of the wear is distributed in �35 mm to 60 mmon the wheel profile. All of the wheels' tread wear depths arenearly increasing linearly with the mileage, and the largest weardepth occurs in the 4th wheel. Because most of the wear occurs inthe wheel tread, the flange thicknesses of all wheels becamegreater with the increase of mileage. The thickest flange was foundin the 4th wheel.

The wear rates and mean tread wear depths of different typeand different load states of freight cars are shown in Table 1, andthe cumulative tread wear depths are the mean value of all eight ofthe freight car's wheels once the total mileage had reached1,80,000 km. The wear rates of empty cars are larger than thoseof heavy cars and the mean tread wear depths of heavy cars areabout two times those of empty cars. For the heavy cars, the meantread wear depths the C80/C80H are larger than those of the C70/C70H because the C80/C80H weighs about 10 t more than the C70/C70H.

Fig. 6. Wear coefficient of classical wheel material. Fig. 7. Reliability tests of freight car on China's Ring-line.

J. Ding et al. / Wear 314 (2014) 104–110 107

4. Simulation results and comparisons with fieldmeasurements

In order to simulate the wheel profile evolution of rail vehicle, awheel wear simulation programme was implemented in MATLABsoftware according to the scheme in Fig. 1.

The multi-body dynamical models of the C70, C80, C70H, andC80H freight cars were built in SIMPACK software and the tracksystem was built according to the Ring-line. The multi-bodydynamical models of the vehicle/track system and wheel wearsimulation programme can be combined to simulate wheel wearbehaviour.

For the heavy states of the C80 freight car, the measured andsimulation worn wheel profiles are shown in Fig. 12. The wear,obtained by simulation, is mainly distributed in�35 mm to 40 mmon the wheel profile, but for the measured profiles, the wear isdistributed in �35 mm to 60 mm. Within the �35 mm to 30 mmof the wheel profile, the shapes of simulation profile are close tothe measured profile. Since the track's switch and wheel material'splastic deformation were not considered in the wheel wearsimulation, no wear occurs in the 40–60 mm of the wheel profiles.

For the heavy states of the C70, C80, C70H and C80H freightcars, the wheel profile evolution obtained from the simulation areshown in Fig. 13. The wear of all fright cars, obtained by simula-tion, is distributed in �35 mm to 40 mm on the wheel profile,nearly none occurs in the flange.

For the heavy states of freight cars, the tread wear depths andthe flange thicknesses obtained by simulation are shown in Figs. 14and 15 respectively. The tread wear depths of all freight carsincrease with the mileage linearly. The cumulative tread weardepth of the C80H is the largest while that of the C70 is thesmallest. The flange thicknesses of all freight cars became thicker

Fig. 8. Planar graph of the Ring-line.

Fig. 9. Measured wheel profiles of C80 freight car.

Fig. 10. Tread wear depths of C80 freight car.

Fig. 11. Flange thicknesses of C80 freight car.

Table 1Wheel wear rates and tread wear depths of freight car on the Ring-line.

Vehicles Load state Wear rates(mm/104 t km)

Mean tread wear(depths/mm)

C80/C80H Empty 0.0439 2.0Heavy 0.0200 4.6

C70/C70H Empty 0.0394 2.1Heavy 0.0207 4.3

Fig. 12. Comparison of measured and simulation worn profiles of C80 freight car.

J. Ding et al. / Wear 314 (2014) 104–110108

with the increase of mileage with the thickest flange occuring inthe C80 fright car. The trend of change on tread wear depths andthe flange thicknesses with the mileage agree well with the fieldmeasurements.

Although both the wheel profiles' shape and the wheel para-meters' change trend from the simulation agree well with themeasured results, the wear rates from the simulation are largerthan measured results. For the C80 heavy car, when the totalmileage running on the Ring-line reached 1,80,000 km, the cumu-lative tread wear depth was 4.6 mm, but in the wheel wearsimulation, the same tread wear depth was reached after only1,00,000 km. For the C80 empty car, the tread wear depth reached2.0 mm after a total mileage of 1,80,000 km on the Ring-line, butthe same tread wear depth was reached in the simulation afteronly 1,15,000 km.

The wear rates from the simulation and the ratios of the fieldmeasurement to simulation data are shown in Table 2.

The wear rates from the simulation are about 1.394–1.842times the magnitude of the wear rates from field measurement.This result is attributed to two factors.

The first one is a difference in wheel materials. Zobory's wearmodel, used in present simulations, was built according to thewear data from the wheel/rail wear experiments. The wheelmaterial used in these experiments is BS11 steel, however thewheel material of the freight cars running on the Ring-line is CL60steel. The chemical composition and hardness of BS11 and CL60materials are shown in Table 3. We can find that the hardnessof the CL60 material is 2.99 GPa, which is higher than the BS11material with a hardness of 2.4 GPa. The wear rate of CL60material should be less than the BS11 material, and this is reflectedin the simulation data. This accounts for one reason why the wearrate in the simulation is larger than the field measurement rate.

The second factor is the difference of slip velocities in a contactpatch. The impact of material's elastic shear deformation on slipvelocity in a contact patch was considered in the present wearsimulation, but not by Zobory when the wear model was built.Enblom's research indicated that the wear volume calculated takingthe impact of a material's elastic shear deformation on slip velocity

Fig. 13. Simulation worn profiles of different freight cars.

Fig. 14. Tread wear depths of different freight cars.

Fig. 15. Flange thicknesses of different freight cars.

Table 2Wear rate ratios of the measured and simulation results.

Vehicles Wear rates/mm � (104 km)�1 Wear rate ratios

Simulation Field measurement

C70 (empty) 0.1698 0.1167 1.455C70H (empty) 0.1870 1.602

C80 (empty) 0.1549 0.1111 1.394C80H (empty) 0.1713 1.542

C70 (heavy) 0.4012 0.2389 1.679C70H (heavy) 0.4397 1.841

C80 (heavy) 0.4601 0.2556 1.800C80H (heavy) 0.4709 1.842

Table 3Chemistry composition and mean hardness of wheel materials.

Materials Composition (%) Hardness (GPa)

C Si Mn S P

BS11 0.52 0.2 1.07 0.018 0.013 2.40CL60 0.6 0.25 0.62 0.040 0.036 2.99

J. Ding et al. / Wear 314 (2014) 104–110 109

into account is larger than it would be when that influence is notconsidered. [4] This contributes to the reason why wear rates foundin the simulation are larger than the field measurement rates.

5. Conclusions

In order to simulate the evolution of wheel profile due to wear,a wheel wear simulation programme was implemented inMATLAB software according to the semi-Hertzian method. Wornwheel profiles can be obtained by combining the wheel wearsimulation programme and multi-body dynamical model.

The wheel wear behaviours of China's heavy haul freight carssuch as the C80, C80H, C70 and C70H running on China's Ring-linewere simulated in this paper, and the simulation results werecompared with filed measurements. We find that the shape ofworn wheel profiles from the simulation agree well with the fieldmeasurements except for the 40–60 mm of the wheel profilewhere wear occurs in field measurements and not in simulations.The reason is that the track's switch and wheel material's plasticdeformation were not considered in the present wheel wearsimulation.

The wear rates from the simulation are larger than fieldmeasurements for two reasons. Firstly, the CL60 wheel materialused in China's freight car is harder than the BS11 wheel material,used in Zobory's wheel/rail wear experiments. Secondly, theimpact of material's elastic shear deformation on slip velocity ina contact patch was considered in the present wear simulation.

In present simulations, we did not consider the plastic deformationof wheel material. However with the increasing axle load of heavyhaul freight cars, the wheel/rail contact pressure will exceed the yieldstress of wheel materials, resulting in the plastic deformation, which

will change the shapes of wheel profile and affect the precision ofwheel wear simulation. So the plastic deformation of wheel materialshould be considered in future work.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (No. 51305359), the China Postdoctoral ScienceFoundation.

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