simulation of mutual wheel/rail wear

Wear 253 (2002) 100–106

Simulation of mutual wheel/rail wear

Sergey Zakharov∗, Ilya ZharovAll-Russian Railway Research Institute, 10 Third Mytishchinskaya,129851 Moscow, Russia

Abstract

Wheel/rail wear simulation based on mathematical models is one of the instruments of study and prognosis of wheel/rail life. Theoreticalstudy of two mutually wearing bodies has shown that their steady-state worn profile depend on their initial profiles. This conclusion enablesto suggest the possibility of control over wheel/rail stable-state worn profile, which should be optimal on the selected criteria. Equationsderived in this study make it possible to find the optimal wheel flange/rail head profile which provides for minimal total wear rate of wheeltraveling along the selected track section, once the lateral forces, and the angle of attack and the wear model are known. To obtain the wearmodel, i.e. the dependence of the wear rate from the contact parameters, laboratory simulation of wear between wheel flange and side faceof the rail head is used. Laboratory experiments were carried out on the rolling-lateral sliding wear testing machine. Analysis of modeledworn wheel flange/rail head optimal profile is made. Practical application of the suggested approach is discussed.© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Wheel; Rail; Wear; Simulation

1. Introduction

Wheel/rail wear is continuing to be a very important fac-tor limiting asset life. Wheel/rail wear simulation based onmathematical models is one of the instruments of study andprognosis of their life. When simulating wheel/rail wear, itis necessary to consider inter-relation of the wear processwith dynamics of the vehicle/track interaction, contact me-chanics parameters and tribological properties of interactingmaterials[1].

Many authors have performed wheel/rail wear simula-tions on the assumption that only a wheel profile is subjectto wear[2,3]. Dynamic forces acting between wheel and railwere calculated using different codes (Medina, NEWCARS,etc.). Normal and tangential contact stresses were calcu-lated using Hertzian or non-Hertzian theory and Kalker’scodes. The geometric irregularities of the track were ob-tained from experiments and were expressed in excitationsof the wheel set motion[3]. Calculations were performedbased on the linear relation between wear and frictionalwork. Another approach[4] was based on the stochas-tic vehicle dynamics[5], developed contact mechanicpackage.

In work [6], a combined simulation of rail and wheelprofile due to wear was described. The spring structural

∗ Corresponding author. Tel.:+7-95-287-7336; fax:+7-95-287-7236.E-mail address: [email protected] (S. Zakharov).

dynamic model of vehicle track system consisted of thevehicle body, the bogie with wheel sets and the discretemasses representing the track. The stochastic lateral rail ir-regularities are also taken into consideration. Discretizationtechnique was used in the simulation. The stabilized wheeland rail profile distribution were obtained on the basis ofsufficiently long distance covered or wheel sets passed byon straight or curved track sections. A spline method ofwear smoothing over contact patch was applied. The sim-ulation model was tested by using data on the particularline.

When studying wheel/rail profiles evolution due to wear,several important aspects should be taken into consider-ation. If evolution of only one profile (wheel or rail) issimulated, then another asset’s profile is considered as thesteady-state profile. If evolution of both profiles is sim-ulated, then it is implicitly assumed that the steady-stateworn wheel/rail profiles exists independent of the ini-tial profiles. The last assumption has not been theoreti-cally proved. At the same time, the results of combinedwheel/rail wear simulation considerably depend on thefollowing.

• Numerical methods of wear smoothing along the contactpatch. Any method has certain inexactitude and after num-ber of time steps of calculations result in accumulatederror.

• Method of distribution of wear between wheel and rail.This depends on the selection of coefficients in wear

0043-1648/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0043-1648(02)00088-1

S. Zakharov, I. Zharov / Wear 253 (2002) 100–106 101

formulae for wheel and rail and their ratio. The ratio ofwear rate depends not only on wheel and rail material tri-bological properties, but also on the number of wheel setspassed by and some other factors.

As a result, due to accumulated error and uneven distribu-tion of wear, the simulated profile may not be the stable-stateone.

Theoretical study of two mutually wearing bodies[7]has shown that their steady-state worn profile do dependon their initial profiles. This conclusion enables to suggesttwo schemes of control over wheel/rail stable-state wornprofile, that is direct and non-direct control. Direct controlincludes selection of the initial wheel/rail profiles and useof rail grinding and wheel turning. A non-direct control is,for example, the developing of surfaces possessing variablewear resistance properties over profile length by meansof surface treatment techniques, such as plasma, weld-on,induction-metallurgical methods, etc. But, before develop-ing any method of control it is necessary to know whatprofiles provides for best results in terms of the minimumwear rate or the minimum pressure distribution.

Another problem is the wear model which should be usedin simulation. Our study has indicated the necessity in ex-perimentally verified wear model that is a best way whichsuits to the suggested approach.

That is why the objectives of the present study were:

• to develop theoretical approach to wheel/rail profiles evo-lution due to wear,

• to find experimentally verified model of wheel flange/railhead side wear,

• to find method of optimizing wheel flange/rail head wornprofiles upon selected criteria of optimization.

2. Suggested approach

Suggested approach is based on the followingassumptions:

• for particular conditions, the steady-state profile exists anddepends on the initial wheel/rail profiles and on tribolog-ical properties of materials;

• there exists an optimal, upon selected criteria, profile pro-viding the minimum wear rate or the minimum pressuredistribution;

• the optimal profiles are selected for a family of wheelflange/rail head conformal profiles;

• to find optimal profiles the condition of the constant ratioof wear rate of contacting bodies at the contact area is set;

• the vertical and the lateral forces, as well as the angleof wheel to rail attack are known from calculation of thequasi-static movement of a bogie in a curve;

• the dependence of the wear rate from the contact param-eters is derived from the laboratory simulation of wheelflange/rail head wear.

3. Wear model

Most frequently, wear model used in simulation models isformulated in terms of proportionality between the specificenergy dissipated over the contact surface and specific massremoval for the unit distance covered. To obtain the wearmodel, that is the dependence of the wear rate from thecontact parameters, simulation of wear between wheel flangeand side face of the rail head when a vehicle moving ina curve was used[8]. Performing such simulation requiressimilarities of wear mechanisms characterized by the wearrate, worn surface features, size, morphology, and color ofwear debris. Scale effects were also considered.

Laboratory experiments were carried out on therolling-lateral sliding wear testing machine in unlubricatedconditions. The lateral slippage was achieved by settingrollers axes under defined angle in relation to each otheron a changeable angle enabling to simulate the relativeslippage from 2.5 to 25%. The load was applied to theupper roller producing the contact pressure from 800 to1590 MPa. All rollers were machined from rail heads andwheels manufactured from the standard rail and wheelsteels having average surface hardness 290 HB. The totalwear rate of both rollers was used for study and analysesof wear. This is because the total wear is more stable tothe variation of external conditions, its change during thewear process is much less than that of a separate roller.Besides, the total wear is a criterion for the optimization ofwheel/rail as a system, especially when the wear resistanceof both materials are close to each other.

The wear rate was studied as a function ofpλ parameter,wherep is the contact pressure (in MPa) andλ is the relativeslippage in parts ofFig. 1. Fig. 1 shows how the wear ratevaries with in wide range of parameterpλ covering severeand catastrophic wear modes[9]. At pλ about 40–60, there isa bifurcation in a wear rate curve, revealing the heavy wearmode[8], characterized by the considerable decrease of thewear rate. All aspects of this phenomena are not clear yet.One of the hypothesis is that the heavy wear mode is con-nected with wheel/rail working hardening due to dynamicbehavior of the system. It is interesting to note that simi-lar results appear from the data of Danks and Clayton[10]on an Amsler test machine under rolling-longitudinal slid-ing test conditions (Fig. 2). For the purpose of the presentwork, the main dependence of the wear rate from the contactparameters was considered.

Basing on the results of the laboratory study shown inFig. 1, it became possible to suggest the formulae for thetotal rollers’ wear rateI (mg/(m mm))

I = kρ

(F

wE

)λ2 = kρ

(P

E

)λ2, (1)

wherek is the wear coefficient,E Young’s modulus,w widthof the narrower roller,F load,P load per unit length, andλis the relative slippage which varies from 0 to 1.

102 S. Zakharov, I. Zharov / Wear 253 (2002) 100–106

Fig. 1. Dependence of the total wear rate frompλ parameter obtained for different lateral slippage and diameter/width of rollers.

Fig. 2. Total wear rate vs.pλ parameter for diverse longitudinal slippage derived from work[10].


Using well-known Hertz formulae for the contact of twocylinders the relationship between loadF and the maximumpressurep is

p =√

FE

(1 − ν2)wπr,

whereν is the Poisson coefficient,r radius of rollers (effec-tive roller radius:r/2).

The specific volume wear rate in dimensionless form canbe expressed as

I ∗ = k∗p∗λ2, (2)

wherek∗ = k/E andp∗ contact pressure in the correspondingpoints of the contact patch.

4. Model of mutual wheel/rail profile wear

The following analytical approach is suggested to find mu-tual wheel/rail profile evolution due to wear. Assume that,as they interact, two rollers rotating with the angular veloc-ities ω1 andω2 are wearing out. The volume wear rates ofeach roller can be found from the formulae:

Ii = kiϕ(P, λ), (3)

whereP is the roller’s load per unit length,λ the roller’srelative slippage,ϕ(P, λ) is a monotonic function ofP andλ, andki is the wear coefficient.

The linear wear rate bound to the volume wear rate asvi = ωiIi /2π . As the relative slippage is small andω1R1 ∼=ω2R2 = U , then

vi = IiU

(2πRi)≈ ϕ(P, λ). (4)

From Eqs. (1) and (2)follows that the ratio of the linearwear rate:v2

v1= k2R1

k1R2= µ = constant. (5)

When contacting surfaces are wearing out, the surfaces areapproaching each other with the velocityV and the followingequation is realized:

V = (v2 + v1)

cosβ, (6)

whereβ is the angle between vectorV and a normal to thecontact line at the given point.

From Eqs. (3) and (4)follows thatvi /cosβ are constantvalues in the contact area and the profiles of running-in sur-faces are not changing in further wear process. In the coor-dinate system bound to the contacting area, the running-inprofiles do not change, but is spreading in the wear process.

To find worn profiles of two contacting bodies the follow-ing algorithm is suggested:

• to place the source profiles in the position of the initialcontact;

• move bodies in the directionVt on the value of ap-proaching each other due to wear;

• where profiles are crossed, draw a line for every point ofwhich the ratio of distances to the first and the secondprofiles equalµ;

• this line is considered as a worn profile.

It is more complicated if the vector of bodies comingtogetherV is not known and it is necessary to find how itchanges with time. As it is pointed out above, the angle ofwheel to rail attackγ , vertical F and lateralQ loads areknown from a quasi-static model, and the dependencyIi =kiϕ(P, λ) is also known from the laboratory simulation. It isnecessary to find the velocityV(t) of bodies’ coming closer,the evolution of the contact areaΩ, and the distribution ofpressure in this area.

In the coordinate system linked to the contact area, bodiescoming closer with the velocitiesV1 andV2, parallel toV andvi = Vicosβ. WhenV is known, thenVi = Vki /(k1˙+ k2).

The vectorV can be found from the following expression:

Q =∫Ω

P(l) sinα dl =∫Ω

P(z)dz, (7)

whereα is an angle of the tangent inclination to the contactarea,l the coordinate along profile line (Fig. 3), andΩ is thecontact area, which for non-conformal profiles, is changingin the process of wear.

An iteration procedure is required to find dependence ofΩ with time. When wheel and rail profiles are conformal,then the contact areaΩ can be considered unchangeable inthe process of wear.

As it was obtained from the laboratory tests, the totalcontacting bodies wear rate depends on the contact pressurep and the relative slippageλ as shown inEq. (1). That means:

ϕ(P, λ) = Pλ2. (8)

In works[11,12], it was shown that the relative slippage canbe calculated from the expression:

λ2 =( z

R

)2 +( γ

cosα

)2, (9)

wherez is the distance from the given point to the instan-taneous axes of wheel set rotation,γ the angle of wheel torail attack,α the angle of tangent to the given point of theprofile, andR the wheel radius.

As the angle between the wheel set axis and the instanta-neous axis of wheel set rotation and is small, it is assumedthat the latter coincides withY (Fig. 3). It is also assumedthat vectorV is parallel to axisY, which is particularly validfor sharp curves. In this case the angleβ between vector ofsurfaces coming togetherV and the normal to the contactline at the given pointβ = π/(2 − α).

If v = v1 + v2, then

V = v

sinα,

v = KPλ2 = KP

[( z

R

)2 +( γ

cosα

)2].


Fig. 3. The contact area of conformal wheel/rail profile—α: profile angle,z(y): profile function,Y and Z axes.

Q =∫Ω

P(z)dz = V

K

∫Ω

sinα

(z/R)2 + (γ /cosα)2dz. (10)

To decrease the number of variables, denotez = z/γR.ThenEq. (10)can be written as:

Kγ

VQ =

∫Ω

sinα

[z2 + (1/cos2 α)]dz = J, (11)

V = Kγ

JQ. (12)

Eqs. (11) and (12)makes it possible to find the conditionsat which the minimal wear rateV occur. Mathematically, anoptimization problem is formulated in the following way:to find a form of the profilez(y) (Fig. 3) under which theintegralJ will have the maximal value and consequentlyVhas the least value.

Fig. 4. Dependence of the integralJ from the angleα of wheel flange profile for different angles of attack: (1) 2 mrad, (2) 5 mrad, and (3) 10 mrad.

If the optimization is carried out among profiles (α =constant) which are line function from 0 tozk, then integralJ can be found as

J = sinα cosα arctan(zk cosα). (13)

Fig. 4 presents the dependence of the integralJ from theangle of a profile inclinationα at the angles of attack 2, 5,and 10 mrad. The maximal values 0.757, 0.716, and 0.650integralJ reaches at 44, 43, and 42 for the correspondingangles of attack. That means that the minimal profile wearrate is realized at the conformal profile inclination 42–44.If to increase the angle of profile inclination from that rangeto 60, this result in decrease of the integralJ on 15–21%,and corresponding deviation of the wear rate from the opti-mal value. Further enlargement of profile inclination to 70,result in considerable reduction of the integralJ on 38, 42,and 48% and corresponding increase of the wear rate.


Fig. 5. Pressure and unit load distribution obtained for the optimal profile: (1) function of profile, (2) function of loadP per unit length, and (3) functionof pressure.

In the case of an arbitrary form of the profile, it is requiredto find the functionα(z) for which

sinα

[z2 + 1/cos2 α]⇒ maximum. (14)

If to substitutet = sin2 α and make necessary transforma-tions, then

[1 + z2(1 − t)]

(1 − t)√t

⇒ minimum. (15)

To find the minimum of relation (15), derivative of (15) withrespect tot equals to zero and after some transformationsthe following equation is obtained

t = 2z2 + 3 −√(8z2 + 9)

2z2. (16)

When z → 0, thent is tending to 0.33 andα approaches35. When z is equal to 25, 10, and 5, angles are equal to77, 69, and 61 correspondingly.

If to denotey = y/γR, then

dy

dz= cotα =

√1

t− 1. (17)

The functiony(z)is given inFig. 5. When calculating thisfunction, it was considered thaty(0) = 0, (dy/dz)(z, z +z) = constant and equal to an arithmetic mean calculatedaccording toEq. (16)for the ends of interval(z, z+z). Atthe angles of attack equal to 2, 5, and 10 mrad, correspond-ing values ofz are equal to 25, 10, and 5 and the values ofintegralJ are equal to 0.811, 0.757, and 0.676 correspond-ingly. These values are only 7, 6, and 4% more than thosecalculated for the optimal linear profile.

In Fig. 5, the function of the unit load (for conveniencemultiplied by 20) is given as

P ∼ 20

[sinα

z2 + 1/cos2 α

], (18)

showing the character of distributionP alongZ axis.To calculate the maximal pressure distributionP along the

contact line as the first approximation, a formulae for twoinfinite cylinders can be used:

p =√

PE∗

πRz

, (19)

where 1/E∗ = ((1 − υ21)/E1) + ((1 − υ2

2)/E2), Rz ≈R/cosα is the radius of curvature of wheel in a plane normalto the rail surface in a given point, andR is the wheel radius.

Thus, the pressure distribution can be found as

p ≈√

sinα cosα

z2 + 1/cos2 α. (20)

Fig. 5 shows the function (20) also multiplied by 20 forconvenience, giving the character of the maximum pressuredistribution alongz axis. It is also shows the rate of pres-sure concentration near the instantaneous axis of wheel setrotation, which can promote the contact fatigue and plasticflow in this area.

All the above calculations have been performed underan assumption that the load and angle of attack is givenand not changing during a bogie’s movement in a curve.However, it is possible to extend this approach to the casewhen wheel/rail interaction parameters are changing duringbogie movement in different radius curves. For that, it maybe assumed that a wheel set is moving inRi radius curve with


the probabilityqi , the angle of wheel/rail attackγ I , and thelateral forceQi . ThenEqs. (7)–(12)should be transformedand an iteration algorithm set accordingly. Note that, throughthis procedure, the minimal total wear rate at all consideredtrack sections could be found. It may be more appropriateto find the optimal conformal profile which provides for theminimal wear rate in sharp curves.

5. Conclusions

Theoretical study of two mutually wearing bodies hasshown that their steady-state worn profile depend on theirinitial profiles. This conclusion enables to suggest the pos-sibility of control over wheel/rail stable-state worn profile.But, before developing any method of control, it is necessaryto know what profiles should be achieved. Equations derivedin this study enable to find the optimal wheel flange/rail headprofile which provides for minimal total wear rate of wheeltraveling along the selected track section, provided that thelateral forces and the angle of attack and the wear model areknown and that an adequate wear model is obtained fromexperiments. Using these equations, it is possible to evaluatethe wear rate for other then optimal profiles.

Analysis of calculated worn wheel/rail conformal profileshows that wear occur in such a way that the pressure distri-bution along the profile is tending to the concentration in thearea of the instantaneous axis of wheel set rotation (gaugearea for the rail head and the flange bottom for the wheel),thus, promoting fatigue and plastic flow failures.

The wear rate could also be controlled by introducingwheels possessing by a variable wear resistance over theentire profile length. Depending on the objectives of controlthe wear resistance may be increased or decreased. Fur-ther directions of the described model development, makepossible to consider, to a certain extent, the variation ofconditions met in practice, material plastic flow, tangential

forces on the contact patch, which result in the additionalcomponent to the lateral load, change in the position ofthe instantaneous axis of wheel set rotation and variabletribological properties of material along the profile.

References

[1] J.J. Kalker, Wheel–rail wear calculation with the program CONTACT,in: Proceedings of the Conference on Contact Mechanics and Wearof Rail/Wheel System, 1987.

[2] A. Chudzikevicz, Evolution of the simulation study of a railwaywheel through wear, in: Proceedings of the Second Mini Conferenceon Contact Mechanics and Wear of Rail/Wheel Systems, Budapest,1996, pp. 207–214.

[3] C. Linder, H. Brauchil, Prediction of wheel wear, in: Proceedingsof the Second Mini Conference on Contact Mechanics and Wear ofRail/Wheel Systems, Budapest, 1996, pp. 215–220.

[4] I.G. Goryacheva, M.N. Dobychin, I.A. Soldatenkov, V.M. Bogdanov,S.M. Zakharov, Simulation of wheel/rail contact and wear in curvedtrack, in: Proceedings of the IHHA STS Conference on Wheel/RailInterface, Vol. 1, Moscow, 1999, pp. 215–220.

[5] A. Kogan, Modelling of rolling stock–track interaction: analyticalapproach, in: Proceedings of the IHHA STS-Conference onWheel/Rail Interface, Vol. 1, Moscow, 1999, pp. 59–66.

[6] A. Szabo, I. Zobory, On combined simulation of rail/wheel profilewear, in: Proceedings of the Second Mini Conference on ContactMechanics and Wear of Rail/Wheel Systems, Budapest, 1996,pp. 196–206.

[7] I. Zharov, A. Konkova, Steady wear of rubbing bodies, J. FrictionWear 18 (5) (1997) 631–635.

[8] S.M. Zakharov, I.A. Zharov, I.A. Komarovsky, Tribology aspects ofwheel/rail interface, in: Proceedings of the IHHA-STS Conferenceon Wheel/Rail Interface, Vol. 1, Moscow, 1999, pp. 221–228.

[9] S.M. Zakharov, I.A. Komarovsky, I.A. Zharov, Wheel flange/rail headwear simulation, Wear 215 (1998) 18–24.

[10] D. Danks, P. Clayton, Comparison of the wear process for Eutectoidrail steels: field and laboratory tests, Wear 120 (1987) 233–250.

[11] S.M. Adreevsky, Side Wear of Rail Heads in Curves, 1964.[12] V.M. Bogdanov, D.P. Markov, I.A. Zharov, S.M. Zakharov, Relative

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simulation of mutual wheel/rail wear

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