Thermal Effect of the Brake Shoes Friction on the Wheel /Rail Contact

A. TUDOR, C. RADULESCU, I. PETRE<strong>Thermal</strong> <strong>Effect</strong> <strong>of</strong> <strong>the</strong> <strong>Brake</strong> <strong>Shoes</strong><strong>Friction</strong> on <strong>the</strong> Wheel /Rail ContactRESEARCHAn analytical solution for <strong>the</strong> temperature distribution in wheel/ brake shoe contact is presented.The wheel <strong>of</strong> locomotive or wagon is simultaneously heated by <strong>the</strong> friction due to wheel/ contactand two brake shoe contacts.The analysis was applied to calculate <strong>the</strong> partition heat coefficientsbetween wheel and rail and between wheel and left or right brake shoe.These coefficients can be used to calculate <strong>the</strong> temperatures for wheel/rail contact wheel/ rightbrake shoe and wheel/left brake shoe. The maximum temperature occurs towards nearly exit <strong>of</strong>heating contact for <strong>the</strong> left or right brake shoe. It is shown that <strong>the</strong> maximum dimensionless contacttemperatures <strong>of</strong> wheel are induced in <strong>the</strong> brake shoe heating contact. Since <strong>the</strong> <strong>the</strong>rmal penetrationdepth is very small, <strong>the</strong>rmally induced plastic deformations for wheel are restricted to a very thinsurface layer. When <strong>the</strong> brake shoes haven’t similarly efficiency (unequal friction), <strong>the</strong> temperaturefield <strong>of</strong> <strong>the</strong> wheel is modified and <strong>the</strong> temperature <strong>of</strong> <strong>the</strong> best efficiently brake shoe increases.Keywords: Wheel rail contact; <strong>Brake</strong> shoe friction; <strong>Friction</strong>al heating; Temperature1. INTRODUCTIONThe frictional heat is generated when two bodiesslide against each o<strong>the</strong>r with a relative speed and acontact pressure. This frictional heat is generated atsliding or rolling interface <strong>of</strong> wheel/brake shoecontact and wheel/rail contact.The <strong>the</strong>rmal aspect around <strong>the</strong> contact region can bestudied by examining <strong>the</strong> heat transfer betweenstationary break shoe and a moving body (relative to<strong>the</strong> heat source). The slip between wheel andbraking system causes friction heating <strong>of</strong> bothbodies. When a brake is working, <strong>the</strong> transformation<strong>of</strong> kinetic energy <strong>of</strong> moving masses into <strong>the</strong>rmalenergy takes place.This kinetic energy is dissipated between two bodiesand appreciably raises <strong>the</strong>ir temperature at <strong>the</strong> area<strong>of</strong> <strong>the</strong> sliding contact. Two aspects are <strong>of</strong> specialimportance in this analysis - <strong>the</strong> nature anddistribution <strong>of</strong> heat partition into each body at <strong>the</strong>interface and <strong>the</strong> resultant temperature fields in <strong>the</strong>two bodies both at <strong>the</strong> interface and with respect todept.While <strong>the</strong> time for reaching <strong>the</strong> quasi-steady-stateA. TUDOR, Pr<strong>of</strong>essor, “Politehnica” University <strong>of</strong>Bucharest, RomaniaC. RADULESCU, Ph.D. Student, Wagon Depot <strong>of</strong>Craiova, RomaniaI. PETRE, Assist. Pr<strong>of</strong>essor, “Valahia” University<strong>of</strong> Targoviste, Romaniaconditions can be very short for a moving body, itcan be relatively long for a stationary body.Thus, it may require a long time to arrive at steadystateconditions for sliding. In this case, <strong>the</strong> heatpartition fractions for <strong>the</strong> two bodies may also varyfor a long time before reaching steady-state value.While railway wheels are heating by friction in <strong>the</strong>contact patch and in <strong>the</strong> contact brakes, <strong>the</strong>re is alsoheat loss due to conduction through <strong>the</strong> contactpatch into <strong>the</strong> rail and into brake shoe. A literaturesurvey revealed that considerable attention has beendevoted to <strong>the</strong> cooling and heating <strong>of</strong> rollingelements.Using a quasi-steady-state approach, Ling (1970)expressed <strong>the</strong> temperature solution for a cylindersubject to cooling and heating. Patula (1981)provided <strong>the</strong> solution <strong>of</strong> <strong>the</strong> temperature for cylindersubject to cooled and heated on parts <strong>of</strong> its surfacebut insulated on <strong>the</strong> rest Ulyse and Khonsari (1993)used an analytical solution for <strong>the</strong> temperaturedistribution in a cylinder subject to surface heatingand no uniform cooling Ertz and Kno<strong>the</strong> (2002)analyzed a comparison <strong>of</strong> analytical and numericalmethods for <strong>the</strong> calculation <strong>of</strong> temperatures inwheel/rail contact. The friction heat distributionbetween a stationary pin and a rotating disc wasstudied by Yevtushenko et. al. (1996) and Kar andBahadur (1981). Tudor and Radulescu (2003)analyzed <strong>the</strong> heat friction partition coefficients <strong>of</strong>wheel contact with rail and two brake shoes.Tribology in industry, Volume 25, No. 1&2, 2003. 27

α+α oC 5RCαωα’q 3γ 3γ q 2φC 6 γC 2α’Cα+α 1 oαOq 12δFigure 1. Geometry and boundary conditionsIn this paper, an analytical solution for <strong>the</strong> frictionheat temperature distribution in a wheel subject tosurface heating by rolling in rail contact and subjectto two surfaces heating by sliding in brake shoes ispresented. This also nonuniforming cooling isshown. The friction temperature distribution isobtained by a Fourier transform technique.2. THE FRICTION HEAT TRANSFERMODEL OF WHEELThe wheel is considered that a short rotatingcylinder. This wheel, subjected to heating by rollingand sliding friction and no uniform convectivecooling at arbitrary angles (β c ) along <strong>the</strong>circumference, is shown in Fig. 1. In this paper, weused <strong>the</strong> model presented in [9].The relative positions <strong>of</strong> <strong>the</strong> rail and <strong>the</strong> bra-ke shoeswill be analyzed by <strong>the</strong> angular location. Heating isassumed be provided by means <strong>of</strong> rol-ling friction in<strong>the</strong> angular (γ a ) and means <strong>of</strong> sli-ding friction in <strong>the</strong>two intervals γ 2 and γ 3 (Fig. 1).External cooling is assumed to be provided bymeans <strong>of</strong> airflow convective heat transfer(convection coefficient α) in <strong>the</strong> interval (C 1 , C 2 )and (C 3 , C 4 ) and by forced convection heat transfer(convection coefficient α+α o ) in <strong>the</strong> interval (C 4 , C 5 )and (C 6 , 0) (fig. 1).It is used <strong>the</strong> following dimensionless variablesλTr R1 αRθ = ; ρ = ; ρ1= ;Bi= ;q R R R1λ2αoRα'R q2q3Bio= ; Bis= ; qa2= ; q3=λ λ ⋅ 2 ⋅ δ q1q1The heat conduction equation in dimensionlessvariables is2∂ θ 1+2∂ρ ρ∂θ∂ρ1+2ρ2∂ θ= B2∂φisθ(1)The solution <strong>of</strong> differential equation to accepct <strong>the</strong>boundary conditions will be writing as [9]:θ ( ρ,ψ)= θ ε + θ ε + θ ε(2)o11o2The dimensionless parameter θ o1 , θ o2 and θ o3 aredefined in [9]. In order to calculate <strong>the</strong> temperaturerise <strong>of</strong> <strong>the</strong> disc with <strong>the</strong> dimensionless radius ρ = ρ 1and ρ = 1 using equation (1), <strong>the</strong> heat distributioncoefficients (ε 1 , ε 2 , ε 3 ) are evaluated by Tudor andRadulescu [9].3. HEAT ROLLING FRICTIONTEMPERATUREThe heat rolling friction distribution coefficient (ε1)is <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> amount <strong>of</strong> heat energy flowing into<strong>the</strong> wheel, as a disc to <strong>the</strong> total heat generated at <strong>the</strong>rolling interface wheel/rail contact.When wheel and rail are brought into contact under<strong>the</strong> action <strong>of</strong> <strong>the</strong> static wheel load, <strong>the</strong> area <strong>of</strong>contact and <strong>the</strong> pressure distribution are usuallycalculated with <strong>the</strong> Hertz’s <strong>the</strong>ory. If a tangentialforce T is transmitted between wheel and rail, <strong>the</strong>reis always a mean relative velocity in <strong>the</strong> contactpoint. Sliding occurs within <strong>the</strong> whole contact areaand <strong>the</strong> tangential force is T = μNw (Nw – wheelload).The contact patch moves with respect to <strong>the</strong> wheelsurface and <strong>the</strong> friction heating within <strong>the</strong> contactpatch is a time dependent heat source (Fig. 2).With <strong>the</strong> typical values for wheel/rail contact, aH ≅5 mm, ar = 14,2⋅10-6 m2/s and vo=26 m/s, one getsL = 4584. In this case, <strong>the</strong> longitudinal and lateral2o3328Tribology in industry, Volume 25, No. 1&2, 2003.

heat conduction (x – and y – direction), can,<strong>the</strong>refore, be neglected and heat conduction equationis [8]ar2∂ T2∂zr∂T=∂tr(3)Thus, T r represents <strong>the</strong> temperature rail rise due to<strong>the</strong> heat supply within <strong>the</strong> contact patch.Since <strong>the</strong> heat flew is one-dimensional, this problemis similar to a semi-infinite solid with an arbitrarilydistributed heat source q 1 (t) applied to <strong>the</strong> surface z= 0 at t ≥ 0. The solution T r (z,t) has to fulfil <strong>the</strong>differential equation (3), <strong>the</strong> initial conditionT r(z, t = 0) = 0and <strong>the</strong> boundary condition− λr∂Tr(z =∂z0, t)= q1r(t) = (1 − ε1)q1(t)(4)(5)The solution <strong>of</strong> this problem can be found in <strong>the</strong>book <strong>of</strong> Carslaw and Jaeger,a trTr(z, t) = ∫ qλ πr⎛ z(t − t')exp⎜ −⎝ 4a1r0 r2⎟ ⎞t' ⎠dt't'(6)The points on <strong>the</strong> surface <strong>of</strong> wheel and rail passthrough <strong>the</strong> contact patch at different speeds due to<strong>the</strong> sliding velocity.The largest heat flux and <strong>the</strong> highest temperaturesrecur <strong>the</strong> major hertzian axis which is parallel to <strong>the</strong>rolling direction at y=0. It is meaningful to substitute<strong>the</strong> time t elapsed since entering <strong>the</strong> contact patchwith <strong>the</strong> current position x in a coordinate systemfixed to <strong>the</strong> contact patch (Fig. 2)x = vt −a HWith <strong>the</strong> dimensionless coordinatesx =axaHzz a=and zδ ,<strong>the</strong> temperature <strong>of</strong> rail (T r ) can be calculated asT ( zx a∫r−1q1ra,x( x'aa) =λarraHπv'⎛ dxa)exp⎜−'⎝ 2( x − xaoa⎞⎟⎠a(7)'dxa'x − xa(8)The analytical solution <strong>of</strong> <strong>the</strong> integral in equation (8)is quite simple if we assume a constant heat flowrate q 1r at <strong>the</strong> rail surface within <strong>the</strong> contact patch.The frictional power dissipation rate in rollingcontact patch is proportional to <strong>the</strong> pressures, <strong>the</strong>coefficient <strong>of</strong> friction μ and <strong>the</strong> sliding velocity ascan be considered constant values:q21(xa) = μvspz(xa) = μvspo1−xaThe average heat flow at <strong>the</strong> surface11πq1 = ∫ q1(xa)dxa= μvsp2 −14o(9)(10)It is generally assumed that <strong>the</strong> frictional powerdissipation is transformed in heat. With <strong>the</strong> heatpartition factor ε 1 , this can be written asq (x ) qand1wa= ε1q1 r(xa) = (1 − ε1(xa)1)q1(xa)for railfor wheel (11)The maximum temperature occurs at <strong>the</strong> trailingedge <strong>of</strong> <strong>the</strong> contact patch.For this case, dimensionless temperature <strong>of</strong> discsurface iswithCrTr(0,1) λdθ r =qR=8aHar2πRvsλλdr= C.r− ε C1r(12)The part <strong>of</strong> <strong>the</strong> rolling frictional heating (ε 1 ) thatflows into wheel can be defined with equations (2)and (12):θ( 1, β ε1)= Cr−1CrThe angle β 1 = γ 1 = 2 asin (a H /R).(13)For example, <strong>the</strong> normal conditions <strong>of</strong> wheel/railcontact are wheel load Fn= 100 kN, wheel radius R=0,5 m, vehicle speed vo= 26 m/s, sliding velocity(longitudinal), vs= 1 m/s, semi-axis <strong>of</strong> <strong>the</strong> contactellipse in rolling direction a H = 5,88 mm. In this caseβ 1 = γ 1 = 0,024 rad.4. HEAT SLIDING FRICTIONTEMPERATURE DISTRIBUTIONConsidering <strong>the</strong> heat transfer by condition along <strong>the</strong>length <strong>of</strong> <strong>the</strong> brake shoe and by convection from <strong>the</strong>periphery (Fig. 3), <strong>the</strong> differential equation for <strong>the</strong>temperature distribu-tion at any axis distance z isgivenTribology in industry, Volume 25, No. 1&2, 2003. 29

2∂ T αbp−2∂zλ Abbb(T − T )o(14)where λ b is <strong>the</strong>rmal conductivity <strong>of</strong> <strong>the</strong> brake shoematerial, T <strong>the</strong> temperature in <strong>the</strong> brake shoe at anyaxial distance z, T o <strong>the</strong> ambient temperature and, α b<strong>the</strong> heat transfer coefficient for <strong>the</strong> brake shoe, p b ,A b <strong>the</strong> perimeter and cross-sectional area <strong>of</strong> <strong>the</strong>brake shoe.Substituting T b =T-T o and m b =(α b p b /λ b A b ) 1/2 , <strong>the</strong>general solution <strong>of</strong> equation (14) is given bymbz−mzTb= Ae + Be b, A and B are <strong>the</strong> constantswhich are defined by boundary conditions.With <strong>the</strong> boundary conditionsTb= 0, z = g b(15)λb∂T∂z= −q= −(1− ε) qb2b2 2z= 0(16)andTb = Ts2, z = 0(17)<strong>the</strong> solution <strong>of</strong> equation (14) becomes:Tssinh{ mb(gb− z) }Tb(z) =sinh(mbgb)(18)The sliding frictional power dissipation rate in <strong>the</strong>contact patch <strong>of</strong> wheel and brake shoe isq v p (x)proportional to <strong>the</strong> pressure2= μ b o b.We assume that friction coefficient (μ b ) and pressurep b (x) are constants and that all <strong>the</strong> frictional powerdissipation is transformed in heat.Using equations (18) and (16) can be calculated <strong>the</strong>surface brake shoe temperatureTs2(1 − ε=λ2b)q21mtanh(mDimensionless temperature <strong>of</strong> brake shoe surfacewithqθCs2b2a 2Ts2λd=q Rq=q1q=λ12a 2a 2,= Cba2b2−ε C2bdb21tanh(mm Rλλ=λbgb)bgb)and(41).(19)α bγ 2t'R dxFigure 3. Co-ordinate system for temperaturecalculation in wheel/brake shoe contactThe partition coefficient ε 2 will be calculated wi<strong>the</strong>quations (19) and (2)⎛ γ ⎞θ⎜1,βε⎝ 2 ⎠22+ ⎟ = Cb2−2Cb2It is generally assumed that every wheel has twobrake shoes. The relative position is defined by <strong>the</strong>angles β 2 , β 3 , γ 2 and γ 3 for <strong>the</strong> left (b 2 ) and rightbrake shoe (b 3 ) (Fig. 4).The partition heat coefficient for every brake shoe(ε 2 and ε 3 ) are calculated by conditions that <strong>the</strong>temperature for <strong>the</strong> wheel and rail, left brake shoeand right brake shoe are respectively equal.5. STEADY-STATE WHEELTEMPERATUREThe bulk temperature <strong>of</strong> <strong>the</strong> wheel increases withtime due to continuous frictional heating on itsrolling surface.Therefore, <strong>the</strong> temperatures <strong>of</strong> wheel and rail aredifferent when a point on <strong>the</strong> surface <strong>of</strong> <strong>the</strong> wheelcomes into in <strong>the</strong> area <strong>of</strong> contact again. This givesrise to a considerable heat flow from <strong>the</strong> hot wheelinto <strong>the</strong> cold rail due to conduction through <strong>the</strong>contact patch.Outside <strong>the</strong> area <strong>of</strong> contract, frictional heat flowsfrom <strong>the</strong> wheel into ambient air by convection at <strong>the</strong>free surfaces.During <strong>the</strong> very short time period that every point on<strong>the</strong> surface is in rolling contact, he <strong>the</strong>rmalpenetration depth is very small compared to <strong>the</strong> sizezα b30Tribology in industry, Volume 25, No. 1&2, 2003.

<strong>of</strong> <strong>the</strong> contact patch. With <strong>the</strong> boundary conditions,<strong>the</strong> wheel it is considered as a central disc and a thinannulus. This thin annulus <strong>of</strong> <strong>the</strong> disc is affected bytemperature oscillation.bCCCOFigure 4. Relative position <strong>of</strong> heat sourcesThe dimensionless temperature <strong>of</strong> <strong>the</strong> central discTcλdθc= =q R1ε γ +ε γ q +ε γ q λ1 1 2 2 a2 3 3 a3dR R( β − γ − γ ) α+ (2π−β− γ ) α + π α'c 1 2c 3 oδ(21)6. RESULTS AND DISCUSSIONThe dimensionless wheel surface tempe-ratureθ(1,ψ) as <strong>the</strong> angular location is shown in <strong>the</strong> fig. 5for different values <strong>of</strong> Biot’s number (0.2,1,5).The local temperature rise around <strong>the</strong> heat sourcesdecreases with increase <strong>of</strong> <strong>the</strong> parameter B io .On notices that <strong>the</strong> maximum temperature in Fig. 5,when <strong>the</strong> two brake shoes have equals frictionefficiently, is between wheel and <strong>the</strong> left brakeshoes. This can be explained by examining <strong>the</strong>friction <strong>the</strong>rmal history <strong>of</strong> a material element. Apoint uniformly heated by <strong>the</strong> heat sources increasesits temperature significantly to reach a maximum at<strong>the</strong> end <strong>of</strong> <strong>the</strong> heating wits.The cooling and heating less affect points inside <strong>the</strong>wheel. For this cyclically steady-state problem, dueto <strong>the</strong> length <strong>of</strong> time needed for <strong>the</strong> heat to beconducted into <strong>the</strong> inner layers, <strong>the</strong> maximumtemperatures at interior locations are always shiftedin <strong>the</strong> direction <strong>of</strong> rotation.ωCrCCbDimensionless surface wheel temperature0.0020.0010.0010.0020Bi = 0.210 2 4 6 8Angular location (radian)5q a2 = 0.010q a3 = 0.024Figure 5. Surface wheel temperatureThis point is <strong>the</strong>n cooled as <strong>the</strong> results <strong>of</strong> convectivecooling as well as circumferential and radial heatconduction.Figure 6 presents <strong>the</strong> temperature pr<strong>of</strong>ile for <strong>the</strong> casewhere <strong>the</strong> brake shoes haven’t similarly efficiency.The negative values <strong>of</strong> <strong>the</strong> dimensi-onlesstemperatures show that cooling by convec-tion ishigher than <strong>the</strong> rolling friction heating.Dimensionless surface wheel temperatureθ 1 ( ψ)θ 2 ( ψ)θ 3 ( ψ)0.0020.00100.001Bi = 0.210.0020 2 4 6 85q a2 = q a3 = 0.01ψAngular location (radian)Figure 6. Surface wheel temperatureThe data in Table 1 are usual operating conditions <strong>of</strong>European locomotives and wagons at <strong>the</strong> low andhigh speed or carriages at <strong>the</strong> low speed [8]. Themaximum temperatures are evaluated for <strong>the</strong>seconditions.7. CONCLUSIONSAn analytical solution is presented for <strong>the</strong>temperature distribution <strong>of</strong> rotating wheel (cylinder)subjected to no uniform cooling and three uniformfriction heating...Tribology in industry, Volume 25, No. 1&2, 2003. 31

Table 1.Units Low speed High speedNormal load [8] F n (kN) 100 100Vehicle speed [8] v o (m/s) 30 90Sliding velocity (longitudinal) [8] v s (m/s) 1 3Coeficient <strong>of</strong> friction-in wheel / rail contact-in brake shoe0.30.150.10.1Normal load on brake shoe F bs (N) 2230 2230<strong>Friction</strong>al rolling power dissipation [8] P friction (kW) 30 30<strong>Friction</strong>al sliding power in brake shoe P fb (kW) 10 20Geometry <strong>of</strong> brake shoeMaximum temperature for:-left brake shoe-right brake shoe-rolling rail contactμ- transversal section area(m 2 )- perimeter (m)- wheel contact area (m 2 )0.00480.6560.02o C 4453831580.00480.6560.02703656271The analysis was applied to calculate <strong>the</strong> partitionheat coefficients between wheel and rail andbetween wheel and left or right brake shoe.These coefficients can be used to calculate <strong>the</strong>temperatures for wheel/rail contact wheel/ rightbrake shoe and wheel/left brake shoe.The maximum temperature occurs towards nearlyexit <strong>of</strong> heating contact for <strong>the</strong> left or right brakeshoe.It is shown that <strong>the</strong> maximum dimensionless contacttemperatures <strong>of</strong> wheel are induced in <strong>the</strong> brake shoeheating contact. Since <strong>the</strong> <strong>the</strong>rmal penetration depthis very small, <strong>the</strong>rmally induced plastic deformationsfor wheel are restricted to a very thin surface layer.When <strong>the</strong> brake shoes haven’t similary efficiency(inequal friction), <strong>the</strong> temperature field <strong>of</strong> <strong>the</strong> wheelis modified and <strong>the</strong> temperature <strong>of</strong> <strong>the</strong> bestefficiently brake shoe increases.REFERENCES[1] M.K. Kar and S. Bahadur, Heat TransferAnalysis for a Pin-and-Disc Sliding System,WEAR, Vol. 67(1), 1981, pp. 71.[2] M. El-Sherbing and T.P. Newcomb, TheTemperature Distribution due to <strong>Friction</strong>al HeatGenerated between a Stationary Cylinder and aRotating Cylinder, WEAR, Vol. 42(1), 1977,pp. 23.[3] F.F. Ling, Surface Mechanics, WileyInterscience, New York, 1970.[4] E.J. Patula, Steady-State TemperatureDistribution in a Rotating Roll Subject toSurface Heat Fluxes and Convective Cooling,J. Heat Transfer, Vol. 103, 1981, pp. 36-41.[5] P. Ulyse and M.M. Khonsari, <strong>Thermal</strong>Response <strong>of</strong> Rolling Components under MixedBoundary Conditions: An Analytical Approach,J. Heat Transfer, Vol. 115, 1993, pp. 857-865.[6] Yeotushenko, O. Ukhanska, R. Chapovska,<strong>Friction</strong> Heat Distribution between a StationaryPin and a Rotating Disc, WEAR, Vol. 196,1996, pp. 219.[7] K. Kno<strong>the</strong> and S. Liebelt, Determination <strong>of</strong>Temperatures for Sliding Contact withApplications for Wheel-Rail Systems, WEAR,Vol. 189, 1995, pp. 91-99.[8] M. Ertz and K. Kno<strong>the</strong>, A Comparasion <strong>of</strong>Analytical and Numerical Methods for <strong>the</strong>Calculation <strong>of</strong> Temperatures in Wheel / RailContact, WEAR, Vol. 253, 2002, pp. 498-508.[9] Tudor and C. Radulescu, Analysis <strong>of</strong> heatfriction partition in wheel/rail contact andwhell/brake shoe contact. Univ. „Politehnica”Bucharest, Sc. Bull. Series D. Mech. Eng- inpress.32Tribology in industry, Volume 25, No. 1&2, 2003.