Tribology International 35 (2002) 137–144www.elsevier.com/locate/triboint

Analysis of the thermal elastohydrodynamic lubrication of a finiteline contact

Xiaoling Liu *, Peiran YangDepartment of Mechanical Engineering, Qingdao Institute of Architecture and Engineering, Qingdao 266033, PR China

Received 2 May 2001; received in revised form 4 October 2001; accepted 5 November 2001

Abstract

A complete numerical solution for the thermal elastohydrodynamic lubrication (EHL) problem of a finite line contact betweenan infinite plane and an axially profiled cylindrical roller is obtained. The pressure profile and the film shape in the middle part ofthe roller are very different from those at the ends. However, the finite line contact results in the middle part of the roller are nearlythe same as those of the corresponding infinite line contact. It is revealed that, in a finite line contact, the maximum pressure, thehighest film temperature and the minimum film thickness all occur at the end regions of the roller. A comparison is made betweenisothermal and thermal solutions of a finite line contact. In addition, variations in the minimum film thickness, the maximumpressure and the highest temperature with the dimensionless velocity parameter, the slide/roll ratio and the radius of the end profilesof the roller are discussed. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Thermal EHL; Finite line contact; Multilevel method

1. Introduction

Finite line contact elastohydrodynamic lubrication(EHL) problems, such as those of a cylindrical rollerbearing, a pair of involute gear teeth, a disk cam and itsroller follower, are of great importance but little workhas been done on them. Thermal EHL (TEHL) solutionsfor such problems are still absent. Wymer and Cameron[1] obtained the first experimental EHL result for a finiteline contact by using optical interferometry. Bahadoranand Gohar [2] offered a solution for an unprofiled cylin-drical roller from a modification of the infinite solution.The first complete numerical analysis for the EHL of anaxially profiled roller was performed by Mostofi andGohar [3]. More recently, Park and Kim [4] obtained anumerical solution of the EHL of a finite line contactusing the finite difference method and the Newton–Raphson method, whose numerical procedure is fullysystematic. All the above theoretical studies dealt withonly isothermal problems.

Although TEHL analysis is now common, it is only

* Corresponding author. Tel.:+86-532-5071283.E-mail address: liuFbetty@263.net (X. Liu).

0301-679X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reservedPII: S0301-679X(01 )00107-4

common for the traditional infinite line contact and pointcontact problems. The complete TEHL solution of thefinite line contact problem has never been achievedbefore. The present study shows that the lubricatingproperties in the end part of the roller are significantlydifferent from those in the middle part of the roller, andthe thermal solution is significantly different from thecorresponding isothermal solution. The new results,therefore, are very important in the design of cylindricalrollers in engineering practices.

The main purpose of the present study is to proposea complete TEHL solution that can be used generally inthe analysis of the TEHL of finite line contacts. As apreliminary study, in this paper we did not consider anynon-Newtonian effects, partly because we hope to showthe significance of the thermal effect separately, and alsobecause the influence of the non-Newtonian behavior ofthe lubricant is not as important as that of the thermalbehavior [5,6].

In this paper we analyze the TEHL problem of a finiteline contact by using a multilevel method. A completenumerical solution is obtained for the axially profiledcylindrical roller. Comparisons are made between thethermal finite solution, the isothermal finite solution, andthe thermal infinite solution.

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Nomenclature

bH Hertzian contact radius, �8wRx / (pLE�), mc, ca, cb specific heat of lubricant and solids, J/kg KE� reduced elastic modulus, PaH dimensionless film thickness for results, 105×h/Rx

h film thickness, mh00 rigid central film thickness, mk, ka, kb thermal conductivities of lubricant and solids, W/m KL length of roller, ml length of the straight part of the roller, mp film pressure, PapH maximum Hertzian contact pressure, 2w/(pbHL)Rx radius of cylindrical roller, mRy radius of the end profile of the roller, mt temperature, Kt0 ambient temperature, Ku, v film velocities, m/sua, ub velocities of surfaces a and b, m/sue entrainment velocity, (ua+ub)/2, m/sw applied load, Nx, y coordinates (see Fig. 1), mxin, xout, yout domain boundaries, mz coordinates across the film, mza, zb coordinates in solids a and b, ma Barus viscosity–pressure coefficient, m2/Nb Reynolds viscosity–temperature coefficient, K�1

x slide/roll ratio, (ua�ub)/ue

h viscosity of lubricant, N s/m2

h0 ambient viscosity, N s/m2

r, ra, rb densities of lubricant and solids, kg/m3

r0 ambient density of lubricant, kg/m3

ma frictional coefficient of surface a

Fig. 1. Geometry of the roller and coordinate system.

2. Governing equations

The geometry of the roller is shown in Fig. 1. A finiteline contact is formed between the roller and an infiniteplane. The infinite plane is called solid a and the rollersolid b. Assume that the tangential velocity of surface ais always greater than that of surface b. The same xy

coordinate system as shown in Fig. 1 is applied for thefilm, solid a and solid b. For the film, the z-coordinateis perpendicular to the xy plane and its direction is fromsurface a to surface b. In the same direction as z, the za-coordinate is used for solid a and the zb-coordinate isused for solid b.

The symbols used in this section are defined in the

139X. Liu, P. Yang / Tribology International 35 (2002) 137–144

Nomenclature. The mathematical model is composed ofthe following equations.

2.1. Reynolds equation

Based on the work of Yang and Wen [5], the gen-eralized Reynolds equation for a Newtonian steady-statefinite line contact can be written as

∂∂x�(r /h)eh3

∂p∂x� �

∂∂y�(r /h)eh3

∂p∂y� (1)

� 12ue

∂∂x

(r∗h)

where (r /h)e � 12(her�e /h�e�r�e), r∗ � [r�ehe(ub�

ua) � reua] /ue, re � (1 /h)�h

0

r dz, r�e �

(1 /h2)�h

0

r�z

0

(1 /h) dz� dz, r�e �

(1 /h3)�h

0

r�z

0

(z� /h) dz� dz, 1 /he �

(1 /h)�h

0

(1 /h) dz, 1 /h�e � (1 /h2)�h

0

(z /h) dz.

In solving Eq. (1), the boundary and cavitation con-ditions must be satisfied; these conditions can be writ-ten as

�p(xin,y) � p(xout,y) � p(x,yout) � 0

p(x,y)�0 (xin � x � xout, 0 � y � yout)

2.2. Film thickness equation

For the finite line contact shown in Fig. 1, the filmthickness equation can be written as

h(x,y) � h00 �x2

2Rx

�(y�l /2)2

2Ry

f� (2)

�2pE��� p(x�,y�)

�(x�x�)2 � (y�y�)2dx� dy�

where f� is a symbolic function, f�=1 if yl/2 and f�=0if yl/2.

2.3. Viscosity relationship

In this analysis the viscosity of oil is allowed to varyin all directions. The viscosity–pressure–temperaturerelationship proposed by Roelands [7] is adopted. Thisrelationship can be expressed in SI units as

h � h0 exp�(ln h0 � 9.67) × ��1 � (1 (3)

� 5.1 × 10�9p)z0� t�138t0�138��s0��

In correspondence with the more popular relationship

h � h0 exp[ap�b(t�t0)]

the dimensionless constants z0 and s0 in Eq. (3) can becalculated as follows:

z0 � a / [5.1 × 10�9(ln h0 � 9.67)]

s0 � b(t0�138) / (ln h0 � 9.67)

2.4. Density relationship

The Dowson–Higginson’s relationship [8] of density–pressure–temperature is employed, i.e.

r � r0[1 � C1p / (1 � C2p)�C3(t�t0)] (4)

where C1 � 0.6 × 10�9 Pa�1, C2 � 1.7 × 10�9 Pa�1,C3 � 0.00065 K�1.

2.5. Load equation

The load equation is given by

��p dx dy � w (5)

2.6. Energy equation of the oil film

The temperature within the oil film can be obtainedfrom the energy equation for the flowing oil. Withoutthe heat conduction in the x- and y-directions, this equ-ation can be written as

c�ru∂t∂x

� rv∂t∂y

�q∂t∂z� � k

∂2t∂z2�

tr

∂r∂t�u

∂p∂x

� v∂p∂y� (6)

� h��∂u∂z�2

� �∂v∂z�2�

where

q �∂∂x�z

0

ru dz� �∂∂y�z

0

rv dz�

The inlet boundary condition of Eq. (6) is given byt(xin,y,z)=t0 (if u(xin,y,z)�0), while for u(xin,y,z) � 0 noboundary condition is required.

2.7. Energy equations of solids

Similar to Eq. (6), the energy equations of solids aand b are given by

�caraua ∂t /∂x � ka ∂2t /∂z2a

cbrbub ∂t /∂x � kb ∂2t /∂z2b

(7)

The temperature domains of solids a and b are con-sidered to be infinite layers with finite thickness d. In

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Table 1Properties of the lubricant and solids

Ambient viscosity of lubricant, h0, N s/m2 0.08Ambient density of lubricant, r0, kg/m3 870Density of solids a and b, ra,b, kg/m3 7850Specific heat of lubricant, c, J/kg K 2000Specific heat of solids a and b, ca,b, J/kg K 470Thermal conductivity of lubricant, k, W/m K 0.14Thermal conductivity of solids a and b, ka,b, W/m K 46Viscosity–pressure coefficient, a, m2/N 2.2×10�8

Viscosity–temperature coefficient, b, K�1 0.042Material parameter, G, aE�, dimensionless 5000Maximum Hertzian contact pressure, pH, GPa 0.5 (W=3.04×10�5)Ambient temperature, t0, K 313 (40 °C)Radius of cylindrical roller, Rx, mm 12.7Length of the straight part of roller, l, mm 12.7Length of roller, L, mm 14.7

this paper, it is assumed that a fixed ambient temperatureis at the depth of d=3.15bH in the contacting bodies,which is regarded as sufficiently far from the surface.For solid a, the boundary conditions of Eq. (7) are givenby t(xin,y,za)=t0,t(x,y,�d)=t0, and for solid b byt(xin,y,zb)=t0,t(x,y,d)=t0. On the two oil–solid interfaces,the following heat flux continuity conditions must be sat-isfied:

�ka(∂t /∂za)|za=0 � k(∂t /∂z)|z=0

kb(∂t /∂zb)|zb=0 � k(∂t /∂z)|z=h

2.8. Frictional coefficient

The frictional coefficient on surface a is defined as

ma � �1w���h∂u

∂z�z=0

dx dy (8)

3. Numerical method

In this paper, the algorithm developed by Guo andYang [9] for the point contact thermal EHL problem isextended to the finite line contact problem, and hasproved to be an efficient method.

In order to facilitate the numerical analysis, equations(1)–(8) together with the boundary conditions should bewritten in dimensionless forms. The dimensionless quan-tities are defined as: G � aE�, h̄ � hRx /b2

H, P � p /pH,T � t/t0, Ue � ueh0/E�Rx, U � u/ue, V � v/ue, W � w/(E�RxL), X � x /bH, Y � y /bH, Z � z /h, Za,b � za,b /bH,h̄ � h /h0, r̄ � r /r0. The numerical solution is achi-eved by a pressure–temperature iteration between theReynolds equation (1) and the energy equations (6) and(7) with their boundary conditions. The pressure field issolved with a multilevel method, which essentially fol-lows Venner [10]. The elastic deformation is calculated

with the multilevel multi-integration method, which isstrictly the same as that developed by Brandt and Lub-recht [11]. In solving the energy equations, a sequentialcolumn sweeping technique is employed.

The calculation is carried out only in half of the wholedomain, because of the symmetry with respect to the x-axis. The real calculating domain for a finite line contactis xin=�4.5bH, xout=4.5bH, yout=0.5L, d=3.15bH. Five lev-els of grids are used for the calculation of pressure andfilm thickness. The numbers of nodes on the finest levelare 257 along both the x-direction and the y-direction.The number of nodes employed for the solution of thethree-dimensional energy equations is 11 across the film,and 6 in both za and zb directions within solids a andb, respectively.

4. Results

The properties of the lubricant and solids used in theanalysis are shown in Table 1. The lubricant propertiescoincide with those of the oil Shell Turbo 33, which isa typical lubricant for EHL problems. The properties forboth solids coincide with those of steel. For all theresults the film thickness is given by H=105×h/Rx.

The thermal results obtained for the finite line contactof Ue=6.0×10�11, Ry=4 mm and x=1.5 are illustrated inFig. 2. In Figs 2(a) and (c) the surfaces of pressure P andfilm thickness H are plotted over the calculating domain.Within the film the highest temperature occurs in thelayer of Z=0.6, therefore in Fig. 2(e) the surface of tem-perature T is plotted over this layer. Corresponding tothe results shown in Figs 2(a), (c) and (e), contour mapsof P, H and T are shown over the end region in Figs2(b), (d) and (f), respectively. Because of side-leakage,the lubricating characteristics between the middle andthe end of the roller are significantly different. That is,the lubricating situation in the end region is much worsethan that in the middle region. Note that along the direc-

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Fig. 2. Thermal EHL results of the finite line contact for Ue=6.0×10�11, Ry=4 mm and x=1.5. (a) Surface of pressure P; (b) pressure contours inthe end region; (c) surface of film thickness H; (d) film thickness contours in the end region; (e) surface of temperature T at Z=0.6; (f) temperaturecontours at Z=0.6 in the end region.

tion of entrainment, in the end region the maximumpressure and the highest film temperature are muchhigher than in the middle region, but the minimum filmthickness is much lower.

Figure 3 shows comparisons between different sol-utions for Ue=6.0×10�11 and x=1.5. For the finite longroller the results are given on the cross-section of Y=0and solutions are obtained for Ry=4 mm. Figures 3(a)and (b) show that the isothermal finite solution is verydifferent from the thermal solution. The isothermalpressure spike is clearly higher than the thermal spike,and the isothermal minimum film thickness is thereforelarger than the thermal one. Figure 3 also shows the dif-ference between infinitely long and finite thermal sol-

utions on the cross-section of Y=0, the difference isreasonably small.

The effects of the velocity parameter Ue on the ther-mal EHL performance of the finite line contact arepresented in Fig. 4. In Figs 4(a), (b) and (c), variationsin the minimum film thickness, maximum pressure andthe highest temperature are plotted against the velocityparameter for the fixed slide/roll ratio of x=1.5. The twocurves in each of these figures represent the results inthe middle part of the roller and in the end region,respectively. We can see that the minimum film thick-ness in the end region is much lower than that in theplane of Y=0, and on the contrary, the maximum pressureand the highest temperature in the end region are much

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Fig. 3. Comparisons between different solutions for Ue=6.0×10�11 and x=1.5. For the finite long roller the results are on the cross-section of Y=0with the end radius of Ry=4 mm. (a) Pressure profiles; (b) film thickness profiles; (c) temperature profiles at Z=0.6.

Fig. 4. Effect of velocity parameter on the minimum film thickness, maximum pressure, the highest temperature and the frictional coefficient forRy=4 mm and x=1.5. (a) Minimum film thickness; (b) maximum pressure; (c) the highest film temperature at Z=0.6; (d) frictional coefficient ofsurface a.

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higher than those in the middle. From Fig. 4(a) we knowthat at low speed the minimum film thickness at the endof roller is dangerously small, while in the middle of theroller the minimum film thickness possesses a reason-ably large value. This suggests that under large slidingconditions the engineering design based on the infinitelyline contact TEHL theory may be very dangerous. Theminimum film thickness and the highest temperatureincrease with the increasing velocity parameter, whilethe maximum pressure slowly decreases when Ue

exceeds a certain value. Figure 4(d) shows the variationin the frictional coefficient calculated on surface a,which also decreases as the velocity parameter increases.Note that the values of ma are obtained with Eq. (8).

The effect of the slide/roll ratio x on the minimumfilm thickness, the maximum pressure, the highest tem-perature, and the frictional coefficient of surface a isshown in Fig. 5 for Ue=6.0×10�11 and Ry=4 mm. Similar

Fig. 5. Effect of slide/roll ratio on the minimum film thickness, maximum pressure, the highest temperature and the frictional coefficient forUe=6.0×10�11 and Ry=4 mm. (a) Minimum film thickness; (b) maximum pressure; (c) the highest film temperature at Z=0.6; (d) the highesttemperature of the film, surfaces a and b; (e) frictional coefficient of surface a.

to Figs 4(a)–(c), Figs 5(a)–(c) compare the film featuresbetween the middle and the end parts of the roller. FromFigs 5(a) and (b) we know that as the slide/roll ratioincreases, the minimum film thickness becomes lower,the maximum pressure first becomes higher and thengets lower. From Fig. 5(c) we know that at large x, avery high temperature can be generated by viscous heatdissipation. Under pure rolling conditions (x=0), how-ever, the temperature rise is very small so that the iso-thermal solution can be used to replace the thermal sol-ution. For more information, Fig. 5(d) shows the highesttemperatures on both solid surfaces. It is very interestingto note that, above a certain value of x, the highest tem-perature on surface a becomes independent of theslide/roll ratio, suggesting that some kind of heat balancecan be reached between the heat generation and heattransfer. The variation in the frictional coefficient withthe slide/roll ratio x is given in Fig. 5(e); the frictional

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Fig. 6. Effect of the profile radius of the roller end on the film thickness, pressure, temperature and frictional coefficient for Ue=6.0×10�11 andx=1.5. (a) Minimum film thickness; (b) maximum pressure; (c) the highest film temperature at Z=0.6; (d) frictional coefficient of surface a.

coefficient first increases and then decreases. Such atendency has been reported many times in the TEHLliterature.

For Ue=6.0×10�11 and x=1.5, Fig. 6 shows variationsin the minimum film thickness, the maximum pressure,the highest film temperature, and the frictional coef-ficient versus the profile radius Ry. Note that when Ry isincreased, both the length of the roller, L, and the lengthof its straight part, l, are fixed. In the end region theminimum film thickness, the maximum pressure and thehighest film temperature are drastically changed by Ry;however, in the middle part of the roller these featuresare almost unchanged. The tendencies of the maximumpressure and the highest temperature are similar to eachother while Ry increases. When the value of Ry isbetween 1 mm and 2 mm, the minimum film thicknessdecreases to a small value. The effect of radius Ry onthe frictional coefficient is weak, almost negligible.

In addition, it is very possible that the fluctuations inPmax as shown in Fig. 4(b), Fig. 5(b) and Fig. 6(b) arecaused by the inadequate nodes on the finest grid level.When the multilevel technique is applied to the EHLpressure computation, the height of the spike is usuallysensitive to the mesh structure.

5. Conclusions

A numerical solution to the thermal EHL of a finiteline contact problem is presented for an axially profiledroller contacting with a flat plane. The results indicate

that the lubricating behavior in the end region cannot bepredicted by the corresponding infinitely long solution,and with a large slide/roll ratio the thermal solution can-not be replaced by the corresponding isothermal sol-ution.

References

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