Temperature rise of double-row tapered roller bearings analyzedwith the thermal network method

Siyuan Ai, Wenzhong Wang n, Yunlong Wang, Ziqiang ZhaoSchool of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e i n f o

Article history:Received 26 October 2014Received in revised form10 December 2014Accepted 7 February 2015Available online 17 February 2015

Keywords:Double-row tapered roller bearingThermal network methodTemperaturePower loss

a b s t r a c t

Based on generalized Ohm's law, the thermal network model (TNM) is developed for double-rowtapered roller bearing lubricated with grease, which is commonly used in high-speed railway. The loaddistribution and kinematic parameters in bearing are obtained by developing a quasi-static model. Thetemperature of bearing at different speeds, filling grease ratios and roller large end radius areinvestigated. The results show that large rotating speed and filling grease ratio result in hightemperature rise, especially at roller large end/flange contacts. Besides, an optimal roller large endradius is presented and its mechanism has been explored.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Double-row tapered roller bearing is widely used in high-speedrail for its capacity of carrying very large radial and thrust loads.Owing to different contact angles of roller-inner and -outer ring, acomponent force exists between roller large-end and flange,which can cause a sliding friction. The relative sliding velocitybetween roller large-end and flange becomes large under high-speed conditions, which will result in high sliding friction loss andtemperature rise on contact regions. The thermal network methoddivides bearing systems into isothermal elements connected bythermal resistances; then the temperature can be obtainedthrough generalizing Ohm's law. Thus, the thermal networkmethod can be applied to predict the temperature and to analyzethermal behavior of bearing.

Internal load distribution and kinematics are important to theestablishment of thermal network model of rolling bearing. Harris[1] established the quasi-dynamic model of double-row taperedroller bearing and calculated the kinematic and dynamic para-meters of bearing. Rydell [2] found that the point contact isappropriate for the friction characteristic at the region betweenroller large-end and flange. The load and velocity at differentcontact points have to be determined in advance, which can beobtained through the dynamic model of tapered roller bearing.Kleckner [3] calculated skew, radial and axial displacements, aswell as the position of flange contact of a cylindrical roller bearingby theoretical analysis. Zhang et al. [4] obtained full numerical

solution of pressure and film thickness distribution by forwarditerations for elastohydrodynamic lubrication (EHL) problem ofelliptical contact between rib face and roller end in tapered rollerbearings. In addition, he also investigated the effects of ratios ofcurvature in both principal planes and position of nominal point ofcontact on minimum film thickness and friction.

Cretu et al. [5–8] built the quasi-static and quasi-dynamicmodel of tapered roller bearing through theoretical analysis. Basedon that, an improved quasi-static model was established by Huet al. [9], which can obtain the accurate kinematic and dynamicparameters such as speed and load of bearing's elements. Xia [10]employed the information poor theory to investigate the relationbetween the inner ring rib roughness and the vibration velocity oftapered roller bearing.

The thermal network method is commonly used to investigatethe thermal behavior of bearing. Shaberth initially developed thecomputer program for US Army in 1974 to calculate the thermalperformance of ball bearing and analyze the thermo-mechanicalperformance of load support systems consisting of a shaft sup-ported by up to five rolling-elements; the program [11] wasupgraded in 1981 by adding new capabilities to improve itsexecuting performance. Parker et al. [12,13] initially verified thevalidity of the thermal network method in predicting bearing heatgeneration, inner-race and outer-race temperatures and oil-outtemperatures through computer program, which agreed very wellwith the experimental data obtained from three different sizes ofball bearing. Following that, the lubricant volume fraction (X) inthe drag force expression [14] was then adjusted based on thehypothesis that one single sphere immersed in an infinite fluid, sothat the calculated global loss agrees with the experimental data.For medium rotational speeds, several models aimed at predicting

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Tribology International

http://dx.doi.org/10.1016/j.triboint.2015.02.0110301-679X/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ86 10 68911404.E-mail address: wangwzhong@bit.edu.cn (W. Wang).

Tribology International 87 (2015) 11–22

global losses have been proposed [15], but they gave no informa-tion on the locations of the sources of power loss. Harris [16]detailedly formulated the sliding friction force, viscous drag forceand sliding force between cage and bearing rings to calculate thetemperature of bearing network nodes under the condition of oil-bath lubrication. Pouly et al. [17,18] presented a thermal networkmodel of ball bearing and clarified thermal resistances betweenthermal elements; the results show that bearing temperaturedistribution is very sensitive to the localization of the heat sources.Apart from geometry, the drag coefficient and the fraction of air inthe lubricant are the main parameters to control their intensity;their results show a good agreement with the experimental resultscarried out by NASA [19] on a jet-lubricated high-speed ballbearing subjected to a pure axial load.

As the tapered roller bearing in railway is lubricated with grease,the grease lubrication has to be considered during establishment of

the thermal network model. Kauzlarich et al. [20] published thefirst theoretical analysis of EHL with grease in 1972; they formu-lated the Reynolds equation with the Herschel–Bulkley model andexamined the validity of this model. Cann [21] researched themechanism of grease lubrication and the replenishment problem,and studied the non-Newtonian rheology of grease, surface chem-istry and capillary flow. Jin-Gyoo Yoo et al. [22] conducted ananalysis of grease thermal elastohydrodynamic lubrication pro-blems based on the Herschel-Bulkley model.

It can be found that the studies focused on thermal behaviors ofdouble-row tapered roller bearing are very limited; however,double-row tapered roller bearing is now frequently used inhigh-speed railway and its thermal behaviors may greatly influ-ence the bearing performance. In this paper, using a quasi-staticmodel and the Herschel-Bulkley model of grease, a thermal net-work model of double-row tapered roller bearing is established

Nomenclature

A Areaa Semi-major axis of ellipse in rolling direction at

contactb Semi-minor axis of ellipse in the direction transverse

to the rolling at contactc 0:5dm tan θþ0:5l sin ðαf þαi þαe

2 ÞsecθCm Dimensionless torquedext External diameterdint Internal diameterdm The pitch diameter of bearingdn The distance between two columnsD Shear ratioDh Outside diameter of the tubeDω1 Diameter of roller small-endDω2 Diameter of roller large-endE Young modulusF0 PreloadFa Radial forceFc Centrifugal forceFr Thrust forceh0 Central lubricant thicknessHd Heat flow by heat conductionHg Heat generationHv Heat flow by heat convectioni The sequence number of columnsj The sequence number of rollers at each columnk Thermal conductivityke EllipticityK Load-displacement coefficientl Length of rollerm MassM Tilting momentn Flow indexNu Nusselt numberp PressurePe Peclet number_q Heat flowQ LoadQe Contact force of roller-outer ringQf Contact force of roller large-end-flangeQi Contact force of roller-inner ring_Q Sliding friction loss_Qc Viscous drag lossR Thermal resistance

Re Reynolds numberRm Pitch radius of bearingRs Radius of sphereRx Equivalent radius of contactS Conductance matrixSm Immersed surface of a rollerT TemperatureT0 Atmospheric temperatureT1,T2,T3 Coordinates of the center of the sphere of roller large-

endTa Taylor numberU Relative sliding velocityU1 Sliding velocity of Pi on the roller large-endU2 Sliding velocity of Pi on the flange of inner ringX Volume fraction of greasex,y,z Coordinatesα Pressure-viscosity coefficientαe Contact angle of roller-outer ringαf Contact angle of roller large-end-flangeαi Contact angle of roller-inner ringβ Angleγ Temperature-viscosity coefficientδ0 Axial displacement caused by preloadδr Axial displacementδr Radial displacementδθ Angular displacementε Relative errorεR Radial clearanceΔt Temperature differenceΔv Relative sliding velocityη Viscosity of lubricant at pressure p and temperature Tη0 Viscosity of lubricant at atmospheric pressure and

temperature T0ηair Viscosity of airηeff Viscosity of grease-air mixtureηgrease Viscosity of greaseθ 0:5π�αf

μ Dynamic viscosity of lubricantμs Velocity of forced flow of airν Kinematic viscosity of airρeff Density of grease-air mixtureτ Shear stressτy Yield stressωc Orbital angular velocity of rollerωi Angular velocity of inner ringωR Spinning angular velocity of roller

S. Ai et al. / Tribology International 87 (2015) 11–2212

according to generalized Ohm's law. This model considered thefriction losses between roller and inner ring, roller and outer ring,roller large-end and flange, and also the drag viscous loss; thentemperatures of bearing nodes and thermal behaviors at differentspeeds, filling grease ratios and roller large-end radii are investi-gated. This study provided theoretical support for determining thetype of grease and lubrication properties of bearing.

2. Quasi-static model of double-row tapered roller bearing

It is assumed that the inner ring flange is a cone-shaped partwhile the roller large-end has a spherical shape. As shown in Fig. 1,x-axis is the axis of bearing, Pi is the point of tangency at the rollerlarge-end-flange contacts in the coordinate system oxyz. C is theapex of cone, θ is the angle between line PiC and y-axis and Oi (O1,O2,O3) is the center of the roller spherical large-end.

Because the tangent point Pi lies on cone and its normalsurface, the coordinate of the contact point Pi can be obtained byEq. (1) through geometrical analysis as follows:

x¼O1 sin2θþc cos 2θ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiO22þO2

3

qsin θ cos θ

y¼ O2 cos 2θ� sin θ cos θ O1 � cð ÞffiffiffiffiffiffiffiffiffiffiffiffiO22 þO2

3

p� �

z¼ O3 cos 2θ� sin θ cos θ O1 � cð ÞffiffiffiffiffiffiffiffiffiffiffiffiO22 þO2

3

p� �

8>>>>>>><>>>>>>>:

ð1Þ

where O1, O2 and O3 are coordinate values of the center of rollerspherical large-end.

We assume that the bearing is under zero skew angular, i.e., thecenter of each roller is in the same circle. Thus the coordinate ofthe sphere center of each roller large-end can be obtained by

O1 ¼ � Rs�0:5lð Þ cos αi þαe2

O2 ¼ 0:5dm� Rs�0:5lð Þ sin αi þαe2

� �cos φi

O3 ¼ 0:5dm� Rs�0:5lð Þ sin αi þαe2

� �sin φi

8>><>>: ð2Þ

where Rs is the radius of sphere, l is the length of roller, dm is thepitch diameter of bearing, αi and αe are contact angles of roller-inner ring and roller-outer ring, respectively. θ and c can becalculated according to geometry of bearing structure.

Double-row tapered roller bearing at back-back arrangement isinvestigated in this paper. The bearing is subjected to a combina-tion of radial and thrust loads, tilting moment, and the initial axialpreload. As a real tapered roller bearing system is generallycomplicated, it is assumed that the outer ring of double-rowtapered roller bearing is fixed rigidly; pure rolling occurs betweenthe rollers and outer ring, and thus the friction between them isnegligible in force analysis.

According to kinematic analysis, orbital angular velocity ofroller ωc and its spinning angular velocity ωR can be written as

ωc ¼ ωi

2dmðdm�Dω1þDω2

2cos

αiþαe

2Þ ð3Þ

ωR ¼dm

Dω1þDω2ð1�γ2Þωi ð4Þ

where ωi is the angular speed of inner ring; Dω1 and Dω2 are thesmall- and large-end diameters of tapered roller, respectively; dmis the pitch diameter of bearing; αe, αi, and αf are the contactangles between roller-outer raceway, roller-inner raceway androller large-end-flange, respectively.

γ ¼ Dω1þDω2

2cos

αiþαe

2

� �=dm

For Pi on the roller large-end, the sliding velocity is

U1 ¼ωRR sin 0:5 αiþαeð Þþαf �0:5π þωc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2i þz2i

qð5Þ

The sliding velocity of Pi on the flange of inner ring can becalculated by

U2 ¼ωi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2i þz2i

qð6Þ

As shown in Fig. 2, under the combination of external radialand thrust loads, the conical surface and large-end of taperedroller bearing will support different forces due to the structure ofbearing. Besides, at high-speed condition, the centrifugal forcecannot be neglected, which would change the load distributionsbetween rollers and raceway, as well as between roller large-endand guide flange.

The centrifugal force of roller Fc can be determined by

Fc ¼ 0:5mdmωc2 ð7Þ

The forces acting on a tapered roller should be balanced asfollows:

Q e sin αe�Q i sin αi�Q f sin αf ¼ 0�Qe cos αeþQ i cos αi�Q f cos αf þFc ¼ 0

(ð8Þ

where Qe, Qi, and Qf respectively represent the contact forces ofroller-outer raceway, roller-inner raceway and roller large-end-flange.

The radial, thrust force and tilting moment will cause a smallradial, axial and angle displacement δr, δa, and θ on the inner ringof bearing. When a radial displacement happens in the inner ring,the radial displacement component at position angle φi is

δri ¼ δr cos φi ð9Þ

Fig. 1. Diagram for the calculation of roller end-flange contact location. Fig. 2. Tapered roller under applied loads and inertial forces.

S. Ai et al. / Tribology International 87 (2015) 11–22 13

When an axial displacement happens in the inner ring, theaxial displacement component at position angle φi is

δai ¼ 7δa ð10Þ

where “þ” and “�” represent the first and second column,respectively.

When a deflection angle δθ happens in the inner ring atposition angle φ¼0, the deflection angle of inner ring at positionangle φi relative to outer ring is

θi ¼ 7δθ cos φi ð11Þ

The radial and axial displacement components caused by thedeflection angle of inner ring can be expressed as

δrθi ¼ lDHθi sin βδaθi ¼ lDHθi cos β

(ð12Þ

where β is the angle between the vertical axis and line connectingthe center of bearing and the center of roller, and lDH is thedistance between points D and H as shown in Fig. 1.

Axial displacement δ0 caused by preload F0 is determined by

F0�ZK δ0 sin αe 1:11 ¼ 0 ð13Þ

where Z is the number of rollers and K is the load–displacementcoefficient.

According to geometrical relationship of bearing deformation,the reduced distance between centers of inner and outer racewaygrooves at any roller position φi is given by

δnj ¼ δrjþδrθj

cos αeþðδajþδ0þδaθjÞ sin αe ð14Þ

The relationship of load–displacement is

Qej ¼ Knδ1:11nj ð15Þ

Furthermore, the load Qej can be decomposed into thrust andradial components

Q rj ¼ Qej cos αe cos φj

Qaj ¼Q ej sin αe

(ð16Þ

Tilting moment about the bearing center synthesized by thrustand radial load components is

Mj ¼ 0:5Q ajdm cos φjþ0:5Q rjdnþMdj cos φj ð17Þ

where dn is the distance between two columns of rollers and isequal to dn¼2lDHsinβ.

To balance the external thrust force Fa, radial force Fr andmoment M, the static equilibrium equations of the whole bearingare established as follows:

Fr�X2i ¼ 1

XZj ¼ 1

Q rij ¼ 0

Fa�X2i ¼ 1

XZj ¼ 1

Qaij ¼ 0

M�X2i ¼ 1

XZj ¼ 1

Mij ¼ 0

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð18Þ

where i is the sequence number of columns and j is the sequencenumber of rollers at each column.

The set of Eq. (18) is solved using the Newton–Raphsonmethod, so that the contact forces between rollers and rings canbe obtained.

3. Thermal network method of double-row tapered rollerbearing

Because of the discontinuities of bearing structure that com-prises a rolling bearing assembly, a tapered roller bearing unit with22 isothermal elements (as shown in Fig. 3 and Table 1) connectedby thermal resistance is used to determine the temperature in thegrease-lubricated condition. The thermal resistance is defined bygeneralized Ohm's law as

R¼Δt_Q

ð19Þ

where R is the thermal resistance which depends on the heattransfer mechanism (i.e. conduction, free or forced convection, andradiation), Δt is temperature difference and _Q is the heat flow.

The following sources of dissipation, i.e. power inputs in themodel, are considered as shown in Fig. 3:

(a) Roller/ring friction ( _Q1, _Q3for the outer ring and _Q2, _Q4for theinner ring)

(b) Large-end/flange friction ( _Q6, _Q7)(c) Viscous drag loss ( _Q5)

Because of the symmetry of double-row tapered roller bearingstructure, only one column of thermal relations will be discussednext.

3.1. Power losses

The temperature rise is related to power losses, which consistof relative sliding friction loss between rollers and raceway,relative sliding friction loss between large-end and flange, rolling

Fig. 3. Thermal network of double-row tapered roller bearing.

Table 1Heat transfer diagram.

Node Description Node Description

1 Mass housing 12 Shaft2 Mass housing-cup contacts 13 Ambient air3 Cup(1) 14 Inlet lubricant4 Cup-rollers contacts (1) 15 Lubricant into bearing5 Rollers (1) 16 Large-end-flange contacts (1)6 Rollers-cone contacts (1) 17 Large-end-flange contacts (2)7 Cup-rollers contacts (1) 18 Cone-face (1)8 Rollers (2) 19 Cone-face (2)9 Rollers-cone contacts (2) 20 Cone-shaft contacts (1)

10 Cone(1) 21 Cone-shaft contacts (2)11 Cone(2) 22 Mass housing-cup contacts (2)

1: First column, 2: Second column.

S. Ai et al. / Tribology International 87 (2015) 11–2214

friction loss by elastic hysteresis and viscous drag loss. Because therolling friction is small compared to other losses, in this model it isneglected; thus, the power losses can be classified into two asp-ects: sliding friction loss and viscous drag loss.

3.1.1. Sliding friction lossSliding friction loss between rollers and raceway and between

large-end and flange can be obtained as follows.Contact regions between Lundberg profile tapered rollers and

ring are elliptical; thus the central lubricant thickness can beobtained by the Hamrock–Dowson formula [23]

h0 ¼ 2:69α0:53ðη0uÞ0:67R0:464

x

E0:0730 Q0:067 ð1�0:61e�0:73ke Þ ð20Þ

According to [24], an experiment about the film thickness ofgrease-lubricated contacts against different rolling speeds rangingfrom (0.001 to 10 m/s) has been investigated. It shows that, at ahigher speed (over 0.01 m/s), the grease film thickness approachesthe one calculated based on base oil. Therefore, it is much reas-onable to use the viscosity parameter of base oil to determine thefilm thickness at contacts in this model; thus, the viscosity inEq. (20) is for base oil at the inlet. Then Herschel-Bulkley model isused as the rheological model of greases to determine the shearstress of grease

τ¼ τyþηDn ð21Þ

D¼ ∂v∂z

j ðx;yÞ ¼Δvh0

ð22Þ

where τy is yield stress, η is a viscosity parameter, D is shear ratioand n is flow index, Δv is relative sliding velocity and h0 islubricant film thickness. Owing to high pressure and temperaturevariation in EHL contacts, the change of the viscosity and densityof grease is very large. It is assumed that the viscosity–pressure–temperature relationship is determined by the Roelands formula-tion [25] as follows:

η¼ η0expfðlnη0þ9:67Þ½ð1þ5:1� 10�9pÞz�γ T�T0ð Þ�1�g ð23Þwhere η is the viscosity of lubricant at pressure p and temperatureT, η0 is the viscosity of lubricant at atmospheric pressure andtemperature T0, γ is the temperature-viscosity coefficient of thelubricant, z can be expressed as follows by means of pressure-viscosity coefficient of grease α:

z¼ α5:1� 109½lnðη0Þþ9:67�

ð24Þ

Therefore, sliding friction loss can be obtained by integration onthe entire contact region

_Q1 ¼∬sΔvUτ x; yð Þ dx dy ð25Þwhere Δv is obtained by the above-mentioned quasi-static modelof the double-row tapered roller bearing.

3.1.2. Viscous drag lossThe viscous drag loss generated by the motion of rollers in a

grease-air mixture is taken into account using the results for apinion-gear pair churning in a mixture medium which is in termsof a dimensionless churning torque Cm [26]

_Qc ¼12ρef fωc

3SmR3mCm ð26Þ

where ρeff is the density of grease-air mixture,ωc is orbital angularvelocity of roller, Sm is the immersed surface of a roller and Rm isthe pitch radius of bearing.

The dimensionless churning torque Cm depends on Reynoldsnumber Re and reads as follows according to [26]:

For laminar flows (Reo2000)

Cm ¼ 20Re

For intermediate flow regimes (2000oReo100,000)

Cm ¼ 8:6� 10�4Re1=3

For turbulent flows (100,000oRe)

Cm ¼ 5� 108

Re2

where Reynolds number Re¼ ρef f =ηef f Uωcðdm=2Þ2.Because of the mixture of grease and air, the formulations

proposed by Isbin et al. [27] have been used to estimate thegrease-air mixture physical properties out of contact area as afunction of the lubricant volume fraction X.

ρef f ¼ Xρoilþ 1�Xð Þρair ð27Þ

ηef f ¼ηoil

ηgrease=ηair 1�Xð ÞþXð28Þ

3.2. Thermal resistance

3.2.1. Thermal resistance of conductionThere are many thermal nodes in the model, to express it clearly,

“-” represents thermal relations between two related thermalnodes, for example as shown in Figs. 4, 2-3 represents the thermalrelation between nodes 2 and 3. The number represents the seq-uence number of each thermal node in this model.

(a) The outer and inner rings, housing and shaft are simulatedas cylindrical bodies and the corresponding resistance of conduc-tion is given as in [28]

R¼ lnðdext=dintÞ=2πkL ð29Þwhere dext is the external diameter, dint is the internal diameter, kis thermal conductivity and L is characteristic length. Eq. (29) isused to determine the resistance of conduction between the nodes1–2, 2–3, 10–20, and 12–20.

(b) At roller-ring contacts, heat is generated by a moving sourcelocalized on a very small area which is much smaller than thebearing dimensions. The heat transfer of small moving ellipticalheat source at contacts has been developed by Muzychka et al.[29]. For heat conduction between the node pairs such as 3–4, 4–5,5–6, 6–10, 5–17, and 10–17, one node is associated with the surfaceand another with the bulk, so resistance of constriction (R) can beused as in Ref. [29]

R¼ 1π

ab

� � 1ka

ffiffiffiffiffiffiPe

p ð30Þ

where a is semi-major axis of ellipse in rolling direction, b is semi-minor axis of ellipse in the direction transverse to the rolling dir-ection and Pe is Peclet number.

3.2.2. Thermal resistance of convectionIn most cases, the convection heat transfer coefficient can be

expressed as follows in terms of the dimensionless Nusselt num-ber Nu, the fluid thermal conductivity k and a characteristic length

2Q. .

2T

3Q

3T

2 3R −−

Fig. 4. Heat transfer diagram between nodes 2 and 3.

S. Ai et al. / Tribology International 87 (2015) 11–22 15

L [28] as follows:

Rth ¼1Ahc

¼ 1A

LkNu

� �ð31Þ

where Nusselt number Nu is determined by different convectionconditions as follows.

When rollers are immersed in an infinite medium, its interac-tions with all the surrounding elements (other rollers, rings, cages,etc.) are ignored. So Nusselt number Nu in Eq. (31) between rollerand surrounding grease-air mixture, such as 4–15,6–15,15–17, canbe obtained by [28]:

Nu¼ ð1:2þ0:53Re0:64ÞPr0:3 μj Tbulk

μj Twall

� �ð32Þ

where Re¼ρeff/ηeffVD, V is the orbital linear speed of roller, D is thediameter of roller and Pr is Prandtl number, Pr¼ηeff/(αρeff), and αis thermal diffusivity.

The theory of two rotating concentric cylinders [30] is used inthe convection heat transfer between the mixture of grease-airand rings used as the relative motions between the cage and therings are separated by a fluid. Then the Nusselt number Nu can be

Start

Input the structural parameters of bearing and external loads Fr, Fa, M

Calculate bearing dynamic and kinematic parameters

Input the thermal parameters of grease and airInitialize the temperature of bearing nodes and error ε

<0.001

End

Y

Calculate power losses of bearing,

Calculate associated thermal resistances of bearing nodes, R

i=i+1

N

i=0

Calculate the temperature of bearing nodes, Ti+1

i

ε

<Z

Calculate relative error ε=abs((∑Tnew-∑Told)/∑Told)

Tnew=Told+ω(Tnew-Told)

Q

Fig. 5. Calculation flowchart for thermal network model.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

2000

4000

6000

8000

10000

12000

daoLtcatno

Ceno

C-re lloR

Qi,

N

Orbital Position of Rolling Element

Row 1 My=0 Row 2 My=0 Row 1 My=550Nm Row 2 My=550Nm Row 1 My=1200Nm Row 2 My=1200Nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

100

200

300

400

500

600

700

800

900

1000

daoLtcatno

CegnalF-rello

RQ

f, N

Orbital Position of Rolling Element

Row 1 My=0 Row 2 My=0 Row 1 My=550Nm Row 2 My=550Nm Row 1 My=1200Nm Row 2 My=1200Nm

Fig. 6. Load distribution of inner raceway contact under different moments with thrust load of 2000 N, radial load of 7000 N and speed of 6000 rpm. (a1) (b1) the presentresults; (a2) (b2) results from Ref. [8].

S. Ai et al. / Tribology International 87 (2015) 11–2216

determined according to [31]Nu¼ 2 Tao41Nu¼ 0:167Ta0:69Pr0:4 41oTao100Nu¼ 0:401Ta0:50Pr0:4 100oTa

8><>: ð33Þ

where

Ta¼ Re

ffiffiffiffiffiεRR

rð34Þ

Re¼ ρεRðωRÞμ

ð35Þ

and εR is radial clearance and R is the radius of inner part.The surface of rotating inner ring can be simulated as a rotating

disk. Forced convection between rotating disk and external atmo-sphere is considered due to the high rotating speed of inner ring.Accordingly, rotating disk convection model used on node pair(10–13) can be expressed as [32,33]

Nu¼ 0:4ffiffiffiffiffiffiRe

pPr

13 Reo2:5� 105

Nu¼ 0:0238Re0:8Pr0:6 Reo3:2� 105

(ð36Þ

Because of the rotation of shaft, forced convection relationsbetween the rotating shaft and external atmosphere (node pair12–13) can be simulated as rotating cylinder surface model asderived by [34]

Nu¼ 30:5Re�0:0042 ReZ9600Nu¼ Re0:37 96004ReZ7300Nu¼ 0:00308Reþ4:432 Reo7300

8><>: ð37Þ

where Re¼VD/ν, ν is kinematic viscosity of air, V is the speed ofshaft and D is the diameter of shaft.

According to [16], when forced flow of air with velocity us overthe housing and cup, the outside heat transfer (node pairs 1–13,

3–13) may be approximated by

hc ¼ 0:03kDh

usDh=ν 0:57 ð38Þ

where Dh is the outside diameter of the cup, k is the thermalconductivity of air and ν is kinematic viscosity of air.

3.2.3. Numerical solutionAccording to generalized Ohm's law, thermal nodes are con-

nected by thermal resistances. At each node, heat influx should beequal to heat efflux in a steady-state heat transfer. Therefore, theheat sum flowing towards a temperature node is equal to zero,namely,

HgþX

HdþX

Hv ¼ 0 ð39Þ

where Hg is heat generation, Hd is heat flow by heat conductionand Hv is the heat flow by heat convection.

In order to facilitate solving, it is assumed that the thermalrelation between any two nodes exists but the thermal resistancewill be infinite if they are indeed uncorrelated. The boundarycondition of the thermal network model is that the temperature ofambient air (node 13) and inlet lubricant (node 14) is 40 1C. Heatgeneration [Hg] and conductance matrix terms [S] are determinedat certain conditions. Accordingly, at a certain node, the thermalrelation can be expressed as

XNi ¼ 1

XNj ¼ 1;ja i

ðTi�TjÞSijþHg ¼ 0 ð40Þ

Therefore, the system of linear equations based on thermalrelation between nodes is established and solved by the Gaussianelimination method.

Because the thermal conductance matrix S and power lossesare unknown initially, the model is solved iteratively by the Gauss-Seidel iterative method as shown in Fig. 5. Because the thermalconductance matrix and power losses are related to the tempera-ture of certain nodes, the thermal matrix and power losses areupdated after each iteration. The relaxation iteration factor is setas 0.01. In this model, convergence is reached when the relativeerror is less than 0.1% and the relative error is defined as

ε¼P

Tnew�Told

PTold

ð41Þ

4. Results and discussion

The contact force distributions in different areas of a double-row tapered roller bearing are important to heat calculation indeveloped thermal network model; the comparisons with

Table 2Structure parameters of double-row tapered roller bearing.

Structure parameters of the bearing Value

Young’s modulus of bearing materials (E/N mm�2) 2.06�105

Poisson’s ratio of the bearing materials (ν) 0.3Roller-outer raceway contact angle (αe/deg) 15.11Roller-outer raceway contact angle (αi/deg) 11.11Roller-flange contact angle (αf/deg) 78.35Small-end diameter of taper roller (Dω1/mm) 16.15Large-end diameter of taper roller (Dω2/mm) 17.56Pitch diameter of the bearing (dm/mm) 155.0Center distance of two column rollers (dc/mm) 40.0Center length of taper roller (l/mm) 18.97Number of rollers in each column (Z) 39

0

2000

4000

60000

30

60

90

120

150180

210

240

270

300

330

0

2000

4000

6000

N,ecroftcatnocya

wretuo-relloR

Qe1Qe2

0

2000

4000

60000

30

60

90

120

150180

210

240

270

300

330

0

2000

4000

6000

N,ecroftcatnocya

wrenni-relloR

Qi1Qi2

0

200

400

6000

30

60

90

120

150180

210

240

270

300

330

0

200

400

600

N , ec ro ft cat no ceg nal f- d ne

e gralr ello

R

Qf1Qf2

Fig. 7. Contact forces between roller-outer, roller-inner raceway and flange for a double-row taper roller bearing with a radial load of 75,000 N, thrust load of 25,000 N,torque of 100 N m, preload of 1000 N and inner ring speed of 2160 r/min.

S. Ai et al. / Tribology International 87 (2015) 11–22 17

reference [8] are conducted under the same conditions to verifythe present model for the calculation of contact force distributions.As shown in Fig. 6, a good agreement can be observed between thecurrent results and reference [8], although the model in reference[8] is based on the vector relation. The good agreement verifies thecurrent model for the calculation of force distributions in bearing.

Next, the present model is used to investigate the thermalbehaviors of double-row tapered roller bearing under the follow-ing conditions: the radial load Fr is 75,000 N, the thrust load Fa is25,000 N, torque M is 100 N m and preload F0 is 1000 N. Therelated structural parameters are listed in Table 2 and lubricantproperties used are given in Table 3. Results of roller-outer, roller-inner raceway and roller flange contact forces are shown in Fig. 7.

As shown in Fig. 7, the contact forces at each column aredifferent. For one column, the contact forces are much larger andall the rollers are in the loading region while for the other column,the loading region and the non-loading region both exist. Toinvestigate the largest amount of heat flow and the highest contacttemperature in tapered roller bearing, the roller with the largestcontact forces in the loading region is analyzed by the thermalnetwork method.

4.1. Effects of speed

The speeds of high-speed railway car 250 km/h, 350 km/h and500 km/h correspond to the rotational speeds of double-rowtapered roller bearing 1543 r/min, 2160 r/min and 3086 r/min,respectively; it is assumed that the filling grease ratio of bearingis 0.5 in this section.

Fig. 8 shows the temperature of each node against rotatingspeeds. It can be found that, at the low speed of 100 km/h, thetemperatures at different nodes tend to be uniform; with the

increase of rotating speed, the temperature of each node increasesgradually, which is caused by two sources as shown in Fig. 9. Oneis the increase of relative sliding speed of contacts leading to alarger amount of sliding loss at contacts and the other is a largerviscous drag loss. However, with the increase of speed, tempera-ture rise rate of each node slows down; this can be explained bythe fact that the high temperature rise results in a lower viscosityof lubricant, and thus low frictional loss. This compensation oftemperature affects the bearing and causes a negative effect onitself; thus it is inferred that the increasing temperature of bearingwith speeds have a limiting value. However, the limit cannot bereached because the load-carrying capacity of lubricant decreaseswith the decrease of viscosity and at an excessively high-speedcondition wear tends to occur.

The temperature at roller large-end-flange contacts is the high-est of all nodes due to a very high relative sliding velocity as shownin Fig. 10. In many bearing applications, the performance of grease issubject to operating temperatures. Over an extended operation, thegrease will undergo mechanical and chemical degradation, whichwill finally result in the degradation and failure of the lubricationaction. According to [35], there is a 50% reduction in grease life forevery 10 1C increase, so high operating temperatures are seen to bethe most important factor influencing the grease performance. Thedropping point of calcium-based grease is 70–100 1C and lithium-based grease is 130–160 1C. Normally, the working temperature

Table 3Lubricant properties.

Structure parameters of the bearing Value

Inlet temperature of lubricant (K) 313Inlet density of lubricant (kg/m3) 919Pressure-viscosity coefficient of lubricant (Pa�1) 1.136�10�8

Temperature-viscosity coefficient of lubricant (K�1) 0.04666Coefficient of lubricant thermal expansivity (K�1) 6.5�10�4

Thermal conductivity of lubricant (W/m K) 0.1457Specific heat of lubricant (J/kg K) 2306Specific heat of air, J/kg K 1000Yield stress of grease (Pa) 139.3Viscosity parameter of grease 21.98Flow index 0.63

0 2 4 6 8 10 12 14 16 18 20 22 2410

20

30

40

50

60

70

80

Tem

pera

ture

,°C

The number of thermal nodes

100km/h 250km/h 350km/h 500km/h

Fig. 8. The temperature of bearing nodes against rotating speeds under a filling grease ratio of 0.5.

80 120 160 200 240 280 320 360 400 440 480 5200

10

20

30

40

50

60

70

80

90

cQ

(left column) (right column) (left column)

(right column)fQfQiQiQ.

.

.

.

.

Hea

t,w

Speed,km/h

Fig. 9. Heat generation in bearing against rotating speeds under filling grease ratioof 0.5. ( _Qi-Sliding friction loss between roller and inner raceway; _Qf -Sliding frictionloss between roller large-end and flange; _Qc-viscous drag loss).

S. Ai et al. / Tribology International 87 (2015) 11–2218

should be 20–30 1C lower than the dropping point. When the speedof high-speed rail reaches 500 km/h, the temperature of rollerlarge-end-flange contacts exceeds 70 1C which is higher than thepermitted temperature of calcium-based grease. Accordingly,choosing lithium-based grease is secured.

4.2. Effects of filling grease ratio

In this section, the effect of different filling grease ratios ontemperature rise of bearing is investigated. The speed of high-speed rail is set as 350 km/h which corresponds to the rotatingspeed of double-row tapered roller bearing of 2160 r/min, whilethe filling grease ratios are prescribed as 0.1, 0333, 0.5, 0.667and 0.75.

As shown in Fig. 11, the temperature of each node against thefilling grease ratio is represented. The filling grease ratio greatlyinfluences the temperature rise of bearing. The node temperaturesincrease with the increase of filling grease ratio and this is due tothe significant increase of viscous drag loss. It can be seen fromFig. 12 that at low filling grease ratio, the main frictional loss isfrom the sliding friction at the roller large-end and flange;however, with the increase of filling grease ratio, the viscous dragloss significantly increases in almost a linear fashion and becomesthe main source of temperature rise, while the sliding friction atcontact area rarely changes and even slightly decreases due to thereduction of viscosity at high temperature.

On the other hand, filling grease ratio has an impact on thelubrication of bearing. A larger filling grease ratio leads to a highertemperature of grease, and thus lower viscosity, resulting in decreaseof lubricant thickness. It is known that different precision grade and

tolerance class of bearing determine different surface roughness ofraceway, flange and roller large-end. Here, two situations of manu-facturing grade are exemplified to demonstrate the relation betweensurface roughness of contacts and filling grease ratio.

Usually, the bearing is chosen according to the specific applica-tion; here we consider two situations. Based on JB/T 10235-2001,situation 1 is for precision grade II of rollers and tolerance class Gof flange, the surface roughness of roller large-end and flangeshould not be larger than 0.4 and 0.5 μm, respectively; situation2 is for precision grade I of rollers and the surface roughness ofroller large-end is 0.16 μm. According to the 3σ principle, filmparameter should be larger than 3.0 to ensure full fluid filmlubrication; otherwise, metal-to-metal contact may occur. There-fore, from Fig. 13, it can be concluded that the filling grease ratioshould be less than 0.35 in situation 1, while in situation 2 thesurfaces of bearing elements are finished more smoothly to ensurea lower surface roughness of roller large-end; therefore, a fillinggrease ratio that should be no more than 0.65 can ensure full fluidlubrication. Based on the two examples above, it can be found thatthe influence of filling grease ratio on lubrication should be con-sidered according to machining accuracy of contacts.

4.3. Effects of roller end spherical radius

The speed of high-speed rail is set as 350 km/h, whichcorresponds to 2160 r/min for double-row tapered roller bearing;

80 120 160 200 240 280 320 360 400 440 480 5200

4

8

12

16

20

24

Roll large end- flange contacts Rollers-cone contacts

Speed, km/h

Rel

ativ

e sl

idin

g ve

loci

ty, m

/s

0.00

0.05

0.10

0.15

0.20

0.25

Rel

ativ

e sl

idin

g ve

loci

ty, m

/s

Fig. 10. The relative sliding velocity against rotating speeds.

0 2 4 6 8 10 12 14 16 18 20 22 2435

40

45

50

55

60

65

70

Tem

pera

ture

,°C

The number of thermal nodes

0.1 0.333 0.5 0.667 0.75

Fig. 11. The nodes temperature of bearing against filling grease ratios for 350 km/h.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

40

50

60

70

80

cQ

(left column) (right column) (left column)

(right column)fQfQiQiQ.

.

.

.

.

Hea

t,w

Filling grease ratio

Fig. 12. Heat generation in bearing against filling grease ratios for 350 km/h.( _Qi-Sliding friction loss between roller and inner raceway; _Qf -Sliding friction lossbetween roller large-end and flange; _Qc-viscous drag loss).

S. Ai et al. / Tribology International 87 (2015) 11–22 19

the filling grease ratio is fixed as 0.5. The temperature of large-end-flange contacts at different roller end spherical radii rangingfrom 50 mm to 500 mm is investigated.

Friction loss and high temperature affect the performance ofbearings. To avoid an excess of heat generation, the radius of rollerlarge-end should be optimized. As shown in Fig. 14, the temperaturedecreases with the radius of roller spherical end within the range(50–400 mm), and then increases gradually, leaving a minimumtemperature at the roller end spherical radius of 400 mm. It showsthat an optimal roller end spherical radius exists in this model.

As shown in Fig. 15, the minimum temperature rise is attributed tothe least heat generation at the roller end spherical radius of 400mmcompared with other spherical radii. Temperature at this contact spotis mainly determined by the velocity, film thickness and lubricantviscosity. As shown in Fig. 16, the relative sliding velocity at this contactspot varies greatly with the roller end spherical radius due to thechange of contact point according to Eq. (5) and Eq. (6). It decreasesfirst and then increases quickly while the central film thicknesscontinuously increases with the increase of the roller end sphericalradius. As a consequence, the shear rate decreases greatly first andthen increases with the increase of the roller end spherical radiusaccording to Eq. (22), resulting from the least heat generation at certainradius in light of Eq. (25). Fig. 17 also shows the friction coefficientbetween the roller large-end and flange. The trend indicates that theminimum friction coefficient also exists at the radius of 400mm andshows excellent agreement with temperature rise and heat generation.The result shows that the roller large-end radius should also beoptimized from the viewpoint of temperature as shown in this section.

5. Conclusion

(a) A thermal network model of double-row tapered roller bear-ing based on generalizing Ohm's law is established using the

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8500

1000

1500

2000

2500

3000 Roller large end- flange contacts Rollers- cone raceway contacts lubricant temperature Film parameter in situation 1 Film parameter in situation 2

Filling grease rate

Min

imum

film

thic

knes

s, nm

40

45

50

55

60

65

70

Tem

pera

ture

,°C

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Film

par

amet

er

Fig. 13. Minimum film thickness and lubricant temperature against filling greaseratios for 350 km/h.

0 100 200 300 400 500 60050

52

54

56

58

60

62

64

66

68

Tem

pera

ture

,°C

Roller end spherical radius, mm

Large end-flange contact(left column) Large end-flange contact(right column)

Fig. 14. The temperature of large-end flange contacts against different roller endspherical radiuses with speed of 350 km/h and filling grease ratio of 0.5.

0 100 200 300 400 500 6000

4

8

12

16

20

Hea

t, w

Roller end spherical radius, mm

Heat generation of left column Heat generation of right column

Fig. 15. The heat generation of large-end flange contacts against different rollerend spherical radii with speed of 350 km/h and filling grease ratio of 0.5.

0 100 200 300 400 500 600 7000

2

4

6

8

10

12

Roller end spherical radius, mm

Cen

tral

film

thic

knes

s, μm

Central film thickness (left column) Central film thickness (right column) Relative sliding velocity Shear rate

0

2

4

6

8

10

Rel

ativ

e sl

idin

g ve

loci

ty, m

/s

0

1

2

3

4

5

6

Shea

r ra

te, 1

06 /s

Fig. 16. The central film thicknesses, relative sliding speeds and shear rates atlarge-end flange contacts against different roller end spherical radiuses with speedof 350 km/h and filling grease ratio of 0.5.

0 100 200 300 400 500 6000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Slide-roll ratio Friction coefficient

Roller end spherical radius, mm

Slid

e-ro

ll ra

tio

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

Fric

tion

coef

ficie

nt

Fig. 17. The slide-roll ratio and friction coefficient at large-end flange contacts (leftcolumn) against different roller end spherical radii with speed of 350 km/h andfilling grease ratio of 0.5.

S. Ai et al. / Tribology International 87 (2015) 11–2220

Herschel-Bulkley model as a rheological model of greases,friction power losses between rollers-inner ring, rollers-outerring and roller large-end-flange, and viscous drag loss are allconsidered.

(b) The temperatures and thermal behaviors of double-rowtapered roller bearing against different rotating speeds, fillinggrease ratios and roller large-end radii are investigated. Theresults show that temperature in bearing increases with speedand filling grease ratio. At low filling grease ratio, the powerloss is mainly from the contact; however, at high filling greaseratio, the viscous drag loss becomes the main source of totalheat generation, which implies that filling grease ratio shouldbe considered to avoid high viscous drag loss, and thus hightemperature rise in bearing. Besides, the radius of roller large-end influences the location of contacts, thus affecting thethermal performance of tapered roller bearings, and an opti-mized roller large-end radius can be obtained through thepresent model.

Acknowledgment

The work was supported by the National Key Basic ResearchProgram of China (973) (2011CB706602), the National NaturalScience Foundation of China (Project no. 51275045, 51405017) andProgram for New Century Excellent Talents in University. Theauthors also thank Mr. Nkurunziza Isaac for his edit in English.

Appendix A. Quasi-static model

It is assumed that the inner ring flange is a part of a cone, whilethe roller large-end is spherical. As shown in Fig. 1, Pi(x, y, z) is thetangent point between roller end and flange, the equation forinner ring flange surface can be obtained by the revolvinggeneratrix PiC around the x-axis.

F ¼ xþffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2þz2

qtan θ�c¼ 0 ðA1Þ

where θ is the included angle between PiC and y-axis, θ¼0.5π�αf.The normal vector n! of the flange surface can be expressed as

n!¼ 1;y tan θffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2þz2

p ;z tan θffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2þz2

p !

ðA2Þ

Since the normal of tangent point Pi goes through the center ofroller end spherical surface, the equation of the surface normal atPi can be expressed as

O1�x∂F=∂x

¼O2�y∂F=∂y

¼ O3�z∂F=∂z

ðA3Þ

where O1, O2 and O3 are the coordinates of the center of rollerlarge-end spherical surface.

Solving Eqs. (A1) and (A2) simultaneously, the coordinates oftangent point Pi can be obtained as Eq. [1].

Assume that the sphere end roller contacts the inner ringflange under zero skew angular; therefore, the center D of eachroller is on the pitch circle. Thus, the coordinate of the sphereorigin of each roller can be expressed by the known structuralparameters of the bearing and its azimuth angular, as is shown inFig. 1, the relationship between the lengths can be obtained as thefollowing expressions:

OH,

¼ GOi,

¼ ðRs�0:5lÞ cos αiþαe

2ðA4Þ

OiH,

¼ OD

,

� GD,

¼ 0:5dm�ðRs�0:5lÞ sin αiþαe

2ðA5Þ

where Rs is the radius of the sphere, l is the length of the roller, dmis the pitch diameter of the bearing, and αe and αi are the outerand inner raceway-roller contact angular, respectively.

Therefore, the coordinate of the sphere origin Oi of each rollercan be expressed as

O1 ¼ � OH,

¼ � Rs�0:5lð Þ cos αiþαe

2ðA6Þ

O2 ¼ OiH,

cos φi ¼ 0:5dm� Rs�0:5lð Þ sin αiþαe

2

h icos φi ðA7Þ

O3 ¼ OiH,

sin φi ¼ 0:5dm� Rs�0:5lð Þ sin αiþαe

2

h isin φi ðA7Þ

where φi is the azimuth angular of each roller.Assuming that outer ring is fixed and inner ring rotates at a

speed of ni. As shown in Fig. 1, line segment MN passes throughthe center of roller and is parallel to the small end, the linearvelocity of point N on inner ring can be expressed as

vN ¼ 0:5ωi dm�Dw1þDw2

2cos

αiþαe

2

� �ðA8Þ

where ωi is the angular speed of the inner ring.Assume that there is no gross slip at the roller-raceway contact,

as the linear velocity of the point M on outer ring is zero;therefore, the velocity of the cage or orbital linear speed of theroller is the mean of the inner and outer raceway velocities at Mand N. Hence,

vcage ¼ 0:5 vMþvNð Þ ðA9ÞConsequently, the orbital angular speed of the roller can be

obtained as

ωc ¼vcage0:5dm

ðA10Þ

Assuming the spinning angular speed of the roller is ωR, thelinear velocity vNc of the roller relative to the cage at contact pointN is identical to that of the inner raceway; hence,

vNc ¼ 0:5ωRDw1þDw2

2¼ 0:5 ωi�ωcð Þ dm�Dw1þDw2

2cos

αiþαe

2

� �ðA11Þ

After algebraic reduction, the spinning angular speed ωR of theroller relative to the cage is obtained as

ωR ¼dm

Dw1þDw21�γ2

ωi ðA12Þ

where

γ ¼ ððDw1þDw2Þ=2Þ cos ððαiþαe=2ÞÞdm

ðA13Þ

After the orbital and relative spinning angular speeds of rollerare obtained, one can easily obtain the sliding velocities of thetwo components at the contact point using simple kinematicalrelationships.

For the sphere end roller, the absolute velocity of the contactpoint is the sum of relative velocity and following velocity, whichcan be expressed as Eq. (5), and Eq. (6) for inner ring in Section 1.

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