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An efficient numerical analysis of starved thermohydrodynamically lubricated rolling line contacts
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Tribology International 41 (2008) 940–946 www.elsevier.com/locate/triboint
An efficient numerical analysis of starved thermohydrodynamically lubricated rolling line contacts N. Anandan, R.K. Pandey, C.R. Jagga Industrial Tribology Machine Dynamics and Maintenance Engineering Centre (ITMMEC), IIT Delhi, New Delhi 110016, India Received 12 December 2006; accepted 2 October 2007 Available online 19 November 2007
Abstract Effect of starvation in thermohydrodynamically lubricated high rolling speed line contacts has been investigated numerically by using an efficient numerical method in which temperature variations across the lubricant film is approximated by the second-order of Legendre polynomial. Mechanism of starvation at the contact has been set by creating gradual reduction in the length of the computational domain from the inlet side. In the solution, the lubricant has been assumed to be a Newtonian fluid. Minimum film thickness and rolling traction coefficient under fully flooded and starved conditions have been computed in this work. The rolling traction coefficient, minimum film thickness, and maximum mid film temperature rise in the starved line contact are found to be lesser than the fully flooded contact condition. r 2007 Elsevier Ltd. All rights reserved. Keywords: Thermohydrodynamic lubrication; Starvation effect; Rolling traction; Minimum film thickness; High rolling speed
1. Introduction The contacts between rollers and races in cylindrical/ needle roller bearings are well characterized as line contacts. Such contacts generally operate in the elastohydrodynamic lubrication (EHL) regime under heavily loaded conditions. However, in many high-speed applications, cylindrical/needle roller bearings are generally run under lightly loaded conditions. The line contacts formed between the roller ends and races are subjected to starvation due to lubricant thinning and enhanced centrifugal effects (which throws out lubricant from the contact) at the elevated rolling speeds. Therefore, investigation of thermohydrodynamic lubrication (THL) of concentrated line contacts at high rolling speeds in the presence of starvation is of great practical significance. In the past, Lauder [1] and Dalmaz and Godet [2,3] have studied hydrodynamic lubrication of non-conformal contacts at low rolling speeds under fully flooded conditions. Corresponding author.
E-mail addresses: [email protected] (N. Anandan), [email protected] (R.K. Pandey). 0301-679X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2007.10.001
These authors have investigated variation of minimum film thickness with operating parameters in their corresponding works. Influence of starvation in concentrated contacts have been investigated by Wedeven et al. [4], Kingsbury [5], Hargreaves and Higginson [6], Brewe and Hamrock [7], Goksem and Hargreaves [8], and Ghosh et al. [9] in their respective works. Based on the experimental work on cylindrical roller bearings, reduction in traction, minimum film thickness and temperature rise with increase in the degree of starvation has been reported in Ref. [6]. Similarly, in Ref. [8], a theoretical study of starvation in inlet zone of an EHL line contact has been described by incorporating viscous shear heating. The authors [8] have amply demonstrated reduction in the minimum film thickness and rolling traction through their analysis. Moreover, empirical relations for the prediction of minimum film thickness and rolling traction in fully flooded and starved conditions have been presented in Ref. [8] for EHL line contact. Prasad et al. [10] and Ghosh and Gupta [11] have analyzed the thermal effects in fully flooded hydrodynamic lubrication of rigid rollers for non-Newtonian and Newtonian lubricants, respectively. The authors of both Refs. [10,11] have reported significant effect of
ARTICLE IN PRESS N. Anandan et al. / Tribology International 41 (2008) 940–946
Nomenclature b Cp fx,T fN,T F h h0 hx,T hN,T k _ m Pk p R R1,R2 T T0
half Hertzian width (m) specific heat of lubricant (J/kg K) traction coefficient with inlet boundary at X traction coefficient with inlet boundary at infinity friction force (N) film thickness (m) central film thickness (m) thermal film thickness with inlet boundary at X (m) thermal film thickness with inlet boundary at infinity (m) thermal conductivity of lubricant (W/m K) lineal mass flux (kg/m s) Legendre polynomial, kth order [P0 ¼ 1; P1 ¼ z; P2 ¼ (3z21)/2; P3 ¼ (5z33z)/2] pressure (Pa) equivalent radius of curvature (m), R ¼ (R1+R2)/R1R2 radii of rollers (m) temperature (K) inlet lubricant temperature (K)
viscous shear heating on the film thickness, rolling traction and load carrying capacity. Based on the literature review, it has been observed that in the regime of hydrodynamic lubrication of concentrated line contacts, one of the most important aspects that have not been properly investigated is the lubricant starvation. In the past, most theoretical and experimental works have been carried out on the fully flooded problems of concentrated line/point contacts. Thus, dearth of published articles on thermal hydrodynamic lubrication of starved line contact at high rolling speeds can be found in spite of its practical significance. Therefore, in this theoretical study, the authors have carried out starvation analysis of thermohydrodynamically lubricated line contacts by using a computationally efficient numerical method developed by Elrod and Brewe [12] and further used by Refs. [13–15]. It is pertinent to mention here that this analysis incorporates only those starvation cases where beginning of inlet boundaries are always greater than half Hertzian width of the contact. In this work, results are being presented for film thickness and traction coefficient for starved lubrication of lightly loaded line contacts. It is found from this analysis that the effect of starvation on the minimum film thickness and rolling traction coefficient is significant at high rolling speeds. 2. Governing equations The coupled solutions of Reynolds and energy equations with temperature dependence of lubricant property (viscosity)
TL TU T¯ k u uL uU W x xi xe X z z Z0 g x¯ k x r w f
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lower rolling surface temperature (K) upper rolling surface temperature (K) Legendre coefficient of temperature at kth location rolling velocity in x-direction (m/s) velocity of lower rolling surface (m/s) velocity of upper rolling surface (m/s) load carrying capacity (N/m) coordinate in rolling direction (m) inlet of the domain exit of the domain non-dimensional coordinate (x/R) coordinate perpendicular to rolling direction (m) dimensionless coordinate transverse to film (z ¼ 2z/h) lubricant viscosity at ambient conditions (Pa s) temperature viscosity coefficient of lubricant (K1) Legendre coefficient of fluidity at kth location fluidity (1/Z), (Pa s)1 lubricant density (kg/m3) thermal diffusivity (m2/s), k/(rcp) dissipation factor, Z(@u/@z)2
have been achieved for THL of starved line contacts. In order to reduce the complexity of the computations, second-order Legendre polynomial temperature profile across the film thickness has been assumed in the energy equation. Present analysis treats the bounding solids (rollers) as rigid owing to their lightly loaded conditions. The coordinate system for this line contact geometry has been defined in Fig. 1. 2.1. Energy equation Considering non-inertial laminar lubricating films without dilatational viscosity, the energy equation for the lubrication of line contacts can be written as [12,13]: 2 @T @ @T @u rC p u ¼ k (1) þZ @x @z @z @z The variation of lubricant viscosity (Z) with temperature has been simulated using the following viscosity relation [13]: Z ¼ Z0 egðTT 0 Þ
(2)
Eq. (2) does not have pressure term in it since the effect of pressure on viscosity is negligible for hydrodynamic lubrication problems due to their light loads. In the energy equation (1), the terms due to conduction along the lubricant film, i.e. (@/@x)[k(@T/@x)] and convection across the film [rCpu(@T/@z)] are omitted. The magnitudes of these terms are usually very small in comparison to the convection along the film and conduction
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Viscosity of lubricating oil varies in the longitudinal as well as in the transverse direction to the film. In this analysis, viscosity has been expressed in the form of fluidity (reciprocal of viscosity (x ¼ 1/Z), where Z is modeled using Eq. (2)) for convenience of mathematical derivation. The expression for the fluidity variation across the film thickness is expressed by Legendre polynomial of order 2 and written as follows: xðzÞ ¼
3 X
x¯ k Pk ðzÞ
(4)
k¼0
For Legendre polynomial of order 2, the Lobatto locations, zk, and weight factors, wi, are provided in Table 1. The temperature variation across the film is also represented by a Legendre polynomial of order 2 written as: TðzÞ ¼
3 X
T¯ k Pk ðzÞ
(5)
k¼0
The Legendre coefficients of temperature, T¯ k , are evaluated as: Z 1 2 TðzÞPk ðzÞ dz ¼ T¯ k 2k þ1 1 or 2k þ 1 T¯ k ¼ 2 Fig. 1. Geometry of line contact. (a) Rigid rollers in contact; (b) equivalent rigid roller and rigid plane; (c) Lobatto points.
across the film [13]. The in-plane conduction terms in energy equation have been neglected. These terms are small compared with the cross-film conduction terms [13]. These terms are also small compared with the convection terms as the Peclet number is very high in this analysis. Therefore, surface temperatures (TU and TL) of both solids (both rollers) have been assumed constant and set equal to inlet temperature of lubricant (T0). The expression for velocity appearing in energy equation (1) is obtained by double integration of momentum equation [@p/@x ¼ (@/@z)(Z(@u/@z))], across the film thickness using boundary conditions (z ¼ 1, u ¼ uL and z ¼ +1, u ¼ uU). Thus, velocity expression for lubricant flow is: Z z Z z u ¼ uL þ A x dz þ B xz dz (3) 1
1
where R1
A¼
uU uL B 1 xz dz ; R1 1 x dz
2 h rp; B¼ 2
and z ¼
2z h
3 X
wi T i Pk ðzi Þ
(6)
i¼0
In order to avoid confusion related to non adoption of higher order Legendre polynomials in this analysis, it is relevant to mention here that higher orders (i.e. more than 4 Lobatto points) of Legendre polynomials do not bring perceptible difference between the results [13]. Eq. (6) can be used to obtain four equations for Legendre coefficients of temperature, i.e. T¯ 0 ; T¯ 1 ; T¯ 2 ; and T¯ 3 . In the expressions for T¯ 0 ; T¯ 1 ; T¯ 2 ; and T¯ 3 , surface temperatures (of rollers) TU and TL are considered constant and set equal to the ambient temperature (T0). From the expressions for T¯ 0 ; T¯ 1 ; T¯ 2 ; and T¯ 3 (obtained from Eq. (6)), the following two equations have been obtained: ðT U þ T L Þ ¯ T2 ¼ (7) T¯ 0 2 Table 1 Lobatto locations and weight factor Location, zi
Weight factor, wi
1 pffiffiffi 1= 5 pffiffiffi 1= 5 1
1/6 5/6 5/6 1/6
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7 ðT U T L Þ ¯ T3 ¼ T¯ 1 3 2
(8)
In this problem, there are two unknown temperatures (at Lobatto points in the film as shown in Fig. 1), which have to be computed by using energy equation (1). In order to compute temperatures at locations ‘‘+1’’ and ‘‘1’’, two equations involving T+1 and T1 are required. Therefore, zeroth and first moment of energy equation (1) have been taken across the film thickness as follows: Zeroth moment: Z 1 Z 1 h @T 2w @T @T 1 h u dz ¼ f dz þ h @z U @z L rC p 1 2 1 2 @x (9)
The Legendre coefficients of fluidity ðx¯ k Þ appearing in Eqs. (11)–(13) are evaluated at Lobatto locations as: Z 1 2 ¯ x xðzÞPk ðzÞ dz ¼ 2k þ1 k 1 or 2k þ 1 x¯ k ¼ 2
3 X
wi xi Pk ðzi Þ
@T @T þ TU þ TL @z U @z L Z 1 2 1 h fz dz ð10Þ þ rC p 1 4
h2 @T z dz ¼ w u @x 1 4 1
(14)
i¼0
_ Divergence of lineal mass flux ðr ðm=rÞ ¼ 0Þ leads to generalized thermal Reynolds equation as follows: r x¯ p h3 rp ¼ 6ðuL þ uU Þrh 2r
First moment: Z
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x¯ 1 hðuL uU Þ, x¯ 0
(15)
2 where x¯ p ¼ x¯ 0 þ 0:4x¯ 2 ðx¯ 1 =3x¯ 0 Þ
2.3. Miscellaneous equations The expression for film thickness can be written as:
where ‘U’ and ‘L’ stands for upper and lower roller surfaces, respectively. With the aid of Legendre series for the fluidity and temperature (Eqs. (4) and (5)), the integrals appearing in the zeroth (Eq. (9)) and first moments (Eq. (10)) of the energy equation are evaluated. The resulting simplified equations are as follows: Zeroth moment of energy equation yields:
x2 (16) 2R Load carrying capacity per unit length of rigid roller is computed as: Z xe W¼ p dx (17)
@T¯ 0 @T¯ 1 @T¯ 2 @T¯ 3 c1 þ c2 þ c3 þ c4 @x @x @x @x 2 2 2 @T @T 1 2 c5 ¼w þ h @z U @z L rC p h
Whereas the friction force per unit length of roller is obtained as below: Z xe @u F¼ Z dx (18) @z h=2 xi
ð11Þ
First moment of energy equation yields:
h ¼ h0 þ
xi
and rolling traction coefficient is obtained as follows:
@T¯ 0 @T¯ 1 @T¯ 2 @T¯ 3 þ c7 þ c8 þ c9 c6 @x @x @x @x 2 2 2 @T @T 1 2 ¼w þ TU þ TL þ c10 h @z U @z L rC p h
f ¼
F W
(19)
3. Computational procedure
ð12Þ Coefficients appearing in Eqs. (11) and (12) are given in Appendix. 2.2. Generalized Reynolds equation The derivation of generalized Reynolds equation has been obtained by taking divergence of lineal mass flux across the film thickness in the line contact (Fig. 1). The R þh=2 _ ¼ h=2 ru dzÞ, is obtained lineal mass flux expression, ðm by the integration of velocity terms (Eq. (3)) across the film thickness and the simplified expression obtained is as follows: _ h h¯ h ¯ 2¯ m x þ x B ¼ ðuU þ uL Þ x1 A (13) 2 3 3 0 5 2 r
In this analysis, the upper and lower surface temperatures of bounding solids (rollers) have been assumed constant throughout (justification for the same has been provided after Eq. (2)) and set equal to ambient temperature (T0). The solution to the present thermohydrodynamically lubricated line contact problem begins with the known pressure distribution within the contact as obtained from solution of Eq. (20) for isothermal films, which assumes that the temperature in the entire fluid film is equal to the inlet oil temperature. @ h3 @p ðuL þ uU Þ @h ¼ . (20) @x 12Z @x 2 @x Computational methodology involves coupled solution of , energy equations (11) and (12) and generalized thermal
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i n
For temperature: P P ð T i Þ T i n n1
P o0:001 Ti
(22)
n
where n represents number of iterations. By reducing the inlet distance, starvation is created and its effects on film thickness and rolling traction have been studied in THL conjunction of line contact.
4. Results and discussion Input data for lubricant properties and roller geometry are given in Table 2. The line contact problem analyzed here assumes the rollers to be infinitely long and therefore, the pressure gradient along the length of the rollers is taken as zero. Moreover, the roller surface temperatures are assumed constant in this analysis and equal to the inlet oil temperature, i.e. 311.11 K. It is pertinent to mention here that in the full film EHL solution, surface temperature rise is calculated by using appropriate boundary conditions which take into account of the conduction through a semiinfinite body. However, in the present analysis, it did not result in any significant change while keeping the solid surface temperature constant. Results for the performance parameters of the starved hydrodynamically lubricated line contact are presented for various values of inlet starvation parameters for a range of operating speeds and loads in Fig. 2 through Fig. 5. Table 2 Input data for lubricant and rollers Inlet viscosity of the lubricant (Pa s) Inlet temperature of lubricant (K) Pressure–viscosity coefficient of the lubricant (Pa1) Temperature viscosity coefficient of oil (K1) Thermal diffusivity of the oil (m2/s) rCp of the lubricant (J/m3 K) Radius of equivalent roller on plane (m) Thermal conductivity of the oil (W/m K)
0.13885 311.11 2.30 108 0.045 7.306 108 1.7577 106 1.11125 102 0.1284
4.1. Influence of starvation on film thickness Variations of starvation film thickness parameters for 15, 25, 35, and 50 m/s rolling speeds at load 15,000 N/m has been plotted in Fig. 2. At high degree of starvation, drastic reduction in film thickness parameter have been found due to increase in viscous shear heating. Fig. 3 shows variation of starvation film thickness parameter for four loads (15,000, 17,000, 19,000, and 21,000 N/m) at rolling speed 35 m/s. It can be seen from Figs. 2 and 3 that a unique relationship between starvation film thickness parameter and inlet boundary distance exist for the entire range of rolling speeds and loads taken in the analysis. These figures show that the film thickness parameter gradually reduces from its fully flooded (i.e. |X|=1.0) value and it goes sharply down at |X|=0.15. It is essential to mention here that this analysis does not incorporate starvation cases where |X|ob. 4.2. Influence of starvation on rolling traction Figs. 4 and 5 show the variation of starvation rolling traction parameter (fx,T/fN,T) with inlet distance (|X|) at 15,000 N/m load for 15, 25, 35, and 50 m/s rolling speeds Starvation film thickness parameter
Reynolds equation (15) by using appropriate boundary conditions. In computation, wherever reverse flow arises in the domain, upwind differencing has been resorted. The Legendre coefficients of temperatures T¯ 0 ; T¯ 1 ; T¯ 2 and T¯ 3 are obtained by solving Eqs. (6), (11) and (12). Pressure (thermal) is computed iteratively using Eq. (15) after incorporating thermal effects in viscosity (using Eq. (2)). The solution of governing equations have been made converged by satisfying the criterion (Eqs. (21) and (22)) at the non-dimensional mesh size Dx=0.002. The mesh size has been arrived at by performing grid independent test. For pressure: P
P p n pi n1 P i o0:001 (21) p
1.0 0.8 0.6 15 m/s 25 m/s 35 m/s 50 m/s
0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 2. Variation of starvation film thickness parameter (hx,T/hN,T) with inlet distance (X) at W ¼ 15,000 N/m.
1.0 Starvation film thickness parameter
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0.8 0.6 15000 N/m 17000 N/m 19000 N/m 21000 N/m
0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 3. Variation of starvation film thickness parameter (hx,T/hN,T) with inlet distance (X) at u ¼ 35 m/s.
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and 15,000, 17,000, 19,000 and 21,000 N/m load at 35 m/s rolling speed, respectively. The plots for starved rolling traction parameter indicate a gradual reduction in its magnitude with the increase in the starvation level. The trend of starvation traction parameter reduction is similar to that of film thickness parameter reduction, which can be seen from the presented figures. In THL, when the film thickness has 85% of its fully flooded value due to presence of starvation, the corresponding traction value has only fallen to 82% of its fully flooded value unlike EHL where traction falls steeply in comparison to film thickness [8]. Present analysis reveals that in the case of THL, both minimum film thickness and rolling traction fall with almost the same gradients in the presence of starvation. From the figures, it appears that minimum film thickness and traction variations in THL with degree of starvation are entirely different than that of thermo-elastohydrodynamic lubrication. Variations of non-dimensional minimum film thickness and maximum mid-film temperature for an operating condition (W ¼ 15,000 N/m and U ¼ 15 m/s) are provided in Table 3 at various levels of starvation. With the increase in the level of starvation, both non-dimensional minimum film thickness and maximum mid-film temperature reduce.
Starvation rolling traction parameter
1.0 0.8 0.6
15 m/s 25 m/s 35 m/s 50 m/s
0.4 0.2
Table 3 Variation of minimum film thickness and maximum mid-film temperature with starvation [W ¼ 15,000 N/m and u ¼ 15 m/s] Load (N/m)
Speed (m/s)
Inlet level |X|
Nondimensional minimum film thickness (h/R)
Maximum mid-film temperature (K)
15,000
15
1.0 0.75 0.50 0.25 0.15 0.05
4.21E04 3.70E04 3.15E04 2.55E04 2.29E04 2.00E04
316.685 314.070 313.771 313.726 312.261 311.817
The reduction in rolling traction with starvation has been noticed. Thus, in the presence of continuous film at the contact, starvation is beneficial. 5. Conclusions An efficient numerical method is used to study the effects of lubricant starvation and viscous shear heating on minimum film thickness and rolling traction under the conditions of hydrodynamically lubricated line contact. Varying the inlet distance of the domain (|X| ¼ 0.15–1.0), starvation has been created. The analysis is valid for high speed rolling line contact at lightly loaded conditions for which elastic deformation of mating surfaces are negligible. Starvation has a significant effect on minimum film thickness and rolling traction. For a range of operating conditions, the benefits of starvation in terms of reduced rolling traction and reduced temperature rise can be exploited in the presence of a continuous minimum film thickness at the contact. Acknowledgments
0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 4. Variation of starvation traction parameter (fx,T/fN,T) with inlet distance (X) at 15,000 N/m.
Starvation rolling traction parameter
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The computational facilities provided by IIT Delhi for this work and the leave sanctioned under QIP to the first author by Pondicherry Engineering College are gratefully acknowledged. Appendix
1.0 0.8
x0 ¼ xðT 0 Þ 15000 N/m
0.6
x1 ¼ xðT 1 Þ
17000 N/m
x2 ¼ xðT 2 Þ x3 ¼ xðT 3 Þ
19000 N/m
0.4
21000 N/m
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 5. Variation of starvation traction parameter (fx,T/fN,T) with inlet distance (X) at u ¼ 35 m/s.
2 x¯ x¯ p ¼ x¯ 0 þ 0:4x¯ 2 1 3x0 2¯ 2¯ 2¯ 4 ¯ ¯ c1 ¼ 2uL þ 2x0 x1 A þ x0 þ x1 x2 B 3 3 3 15 2¯ 2 ¯ 2 ¯ 2 ¯ x x AB þ x x B c2 ¼ 3 0 15 2 15 1 35 3
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2 ¯ 2 ¯ 2 ¯ 2 ¯ x x Aþ x þ x B c3 ¼ 15 1 35 3 15 0 105 2 2 ¯ 4 ¯ 2 ¯ x Aþ x þ x B c4 ¼ 35 2 105 1 315 3 2¯ 4 4¯ x0 þ x¯ 2 B2 þ x1 AB c5 ¼ 2x¯ 0 A2 þ 3 15 3 2¯ 2 2 ¯ 2 x0 x¯ 2 A þ x1 x¯ 3 B c6 ¼ 3 15 15 35 2 2 4 ¯ x A c7 ¼ uL þ 2x¯ 0 x¯ 1 3 15 105 3 2 2 8 ¯ x2 B þ x¯ 0 þ x¯ 1 15 9 105 2¯ 2 2 ¯ 2 x0 x¯ 2 A þ x1 x¯ 3 B c8 ¼ 3 15 15 35 2 ¯ 2 ¯ 2 ¯ 2 ¯ x1 x3 A þ x0 x2 B c9 ¼ 35 315 35 105 2¯ 2 2¯ 4 ¯ 2¯ 4 x1 þ x3 B2 þ x0 þ x¯ 2 2AB c10 ¼ x1 A þ 3 5 35 3 15
References [1] Lauder W. Hydrodynamic lubrication of proximate cylindrical surfaces of large relative curvature. Proc Inst Mech Eng 1965; 180(3B):64–9. [2] Dalmaz G, Godet M. An apparatus for the simultaneous measurement of load, traction, and film thickness in lubricated sliding point contacts. Tribology 1972;5(3):111–7.
[3] Dalmaz G, Godet M. Traction, load, and film thickness in lightly loaded lubricated point contacts. J Mech Eng Sci 1973;15(6):400–9. [4] Wedeven LD, Evans D, Cameron AC. Optical analysis of ball bearing starvation. ASME J Lubr Technol 1971;93:349–63. [5] Kingsbury E. Cross flow in a starved EHD contact. ASLE Trans 1973;16:276–80. [6] Hargreaves RA, Higginson GR. Some effects of lubricant starvation in cylindrical roller bearings. ASME Trans J Lubr Technol 1976;98: 66–72. [7] Brewe DE, Hamrock BJ. Analysis of starvation effects on hydrodynamic lubrication in non-conforming contacts. ASME Trans J Lubr Technol 1982;104:410–7. [8] Goksem PG, Hargreaves RA. The effect of viscous shear heating on both film thickness and rolling traction in an EHL line contact—Part II: Starved conditions. ASME Trans J Lubr Technol 1978;100(2): 353–8. [9] Ghosh MK, Hamrock BJ, Brewe DE. Starvation effects on the hydrodynamic lubrication of rigid non-conformal contacts in combined rolling and normal motion. ASLE Trans 1986;30:91–9. [10] Prasad D, Shukla JB, Singh P, Sinha P, Chhabra RP. Thermal effects in lubrication of asymmetrical rollers. Tribol Int 1991;24(4):239–46. [11] Ghosh MK, Gupta K. Thermal effect in hydrodynamic lubrication of line contacts—piezoviscous effect neglected. Int J Mech Sci 1998; 40(6):603–16. [12] Elrod HG, Brewe DE. Thermohydrodynamic analysis for laminar lubricating films. NASA technical memorandum no. 88845; 1986. [13] Elrod HG. Efficient numerical method for computation of the thermo hydrodynamics of laminar lubricating films. ASME Trans J Tribol 1991;113:506–11. [14] Ghosh MK, Pandey RK. Thermal elastohydrodynamic lubrication of heavily loaded line contacts—an efficient inlet zone analysis. ASME Trans J Tribol 1998;120:119–25. [15] Pandey RK, Ghosh MK. Temperature rise due to sliding in rolling/ sliding elastohydrodynamic lubrication of line contact: an efficient numerical analysis for contact zone temperatures. Tribol Int 1998;31(12):745–52.
Tribology International 41 (2008) 940–946 www.elsevier.com/locate/triboint
An efficient numerical analysis of starved thermohydrodynamically lubricated rolling line contacts N. Anandan, R.K. Pandey, C.R. Jagga Industrial Tribology Machine Dynamics and Maintenance Engineering Centre (ITMMEC), IIT Delhi, New Delhi 110016, India Received 12 December 2006; accepted 2 October 2007 Available online 19 November 2007
Abstract Effect of starvation in thermohydrodynamically lubricated high rolling speed line contacts has been investigated numerically by using an efficient numerical method in which temperature variations across the lubricant film is approximated by the second-order of Legendre polynomial. Mechanism of starvation at the contact has been set by creating gradual reduction in the length of the computational domain from the inlet side. In the solution, the lubricant has been assumed to be a Newtonian fluid. Minimum film thickness and rolling traction coefficient under fully flooded and starved conditions have been computed in this work. The rolling traction coefficient, minimum film thickness, and maximum mid film temperature rise in the starved line contact are found to be lesser than the fully flooded contact condition. r 2007 Elsevier Ltd. All rights reserved. Keywords: Thermohydrodynamic lubrication; Starvation effect; Rolling traction; Minimum film thickness; High rolling speed
1. Introduction The contacts between rollers and races in cylindrical/ needle roller bearings are well characterized as line contacts. Such contacts generally operate in the elastohydrodynamic lubrication (EHL) regime under heavily loaded conditions. However, in many high-speed applications, cylindrical/needle roller bearings are generally run under lightly loaded conditions. The line contacts formed between the roller ends and races are subjected to starvation due to lubricant thinning and enhanced centrifugal effects (which throws out lubricant from the contact) at the elevated rolling speeds. Therefore, investigation of thermohydrodynamic lubrication (THL) of concentrated line contacts at high rolling speeds in the presence of starvation is of great practical significance. In the past, Lauder [1] and Dalmaz and Godet [2,3] have studied hydrodynamic lubrication of non-conformal contacts at low rolling speeds under fully flooded conditions. Corresponding author.
E-mail addresses: [email protected] (N. Anandan), [email protected] (R.K. Pandey). 0301-679X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2007.10.001
These authors have investigated variation of minimum film thickness with operating parameters in their corresponding works. Influence of starvation in concentrated contacts have been investigated by Wedeven et al. [4], Kingsbury [5], Hargreaves and Higginson [6], Brewe and Hamrock [7], Goksem and Hargreaves [8], and Ghosh et al. [9] in their respective works. Based on the experimental work on cylindrical roller bearings, reduction in traction, minimum film thickness and temperature rise with increase in the degree of starvation has been reported in Ref. [6]. Similarly, in Ref. [8], a theoretical study of starvation in inlet zone of an EHL line contact has been described by incorporating viscous shear heating. The authors [8] have amply demonstrated reduction in the minimum film thickness and rolling traction through their analysis. Moreover, empirical relations for the prediction of minimum film thickness and rolling traction in fully flooded and starved conditions have been presented in Ref. [8] for EHL line contact. Prasad et al. [10] and Ghosh and Gupta [11] have analyzed the thermal effects in fully flooded hydrodynamic lubrication of rigid rollers for non-Newtonian and Newtonian lubricants, respectively. The authors of both Refs. [10,11] have reported significant effect of
ARTICLE IN PRESS N. Anandan et al. / Tribology International 41 (2008) 940–946
Nomenclature b Cp fx,T fN,T F h h0 hx,T hN,T k _ m Pk p R R1,R2 T T0
half Hertzian width (m) specific heat of lubricant (J/kg K) traction coefficient with inlet boundary at X traction coefficient with inlet boundary at infinity friction force (N) film thickness (m) central film thickness (m) thermal film thickness with inlet boundary at X (m) thermal film thickness with inlet boundary at infinity (m) thermal conductivity of lubricant (W/m K) lineal mass flux (kg/m s) Legendre polynomial, kth order [P0 ¼ 1; P1 ¼ z; P2 ¼ (3z21)/2; P3 ¼ (5z33z)/2] pressure (Pa) equivalent radius of curvature (m), R ¼ (R1+R2)/R1R2 radii of rollers (m) temperature (K) inlet lubricant temperature (K)
viscous shear heating on the film thickness, rolling traction and load carrying capacity. Based on the literature review, it has been observed that in the regime of hydrodynamic lubrication of concentrated line contacts, one of the most important aspects that have not been properly investigated is the lubricant starvation. In the past, most theoretical and experimental works have been carried out on the fully flooded problems of concentrated line/point contacts. Thus, dearth of published articles on thermal hydrodynamic lubrication of starved line contact at high rolling speeds can be found in spite of its practical significance. Therefore, in this theoretical study, the authors have carried out starvation analysis of thermohydrodynamically lubricated line contacts by using a computationally efficient numerical method developed by Elrod and Brewe [12] and further used by Refs. [13–15]. It is pertinent to mention here that this analysis incorporates only those starvation cases where beginning of inlet boundaries are always greater than half Hertzian width of the contact. In this work, results are being presented for film thickness and traction coefficient for starved lubrication of lightly loaded line contacts. It is found from this analysis that the effect of starvation on the minimum film thickness and rolling traction coefficient is significant at high rolling speeds. 2. Governing equations The coupled solutions of Reynolds and energy equations with temperature dependence of lubricant property (viscosity)
TL TU T¯ k u uL uU W x xi xe X z z Z0 g x¯ k x r w f
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lower rolling surface temperature (K) upper rolling surface temperature (K) Legendre coefficient of temperature at kth location rolling velocity in x-direction (m/s) velocity of lower rolling surface (m/s) velocity of upper rolling surface (m/s) load carrying capacity (N/m) coordinate in rolling direction (m) inlet of the domain exit of the domain non-dimensional coordinate (x/R) coordinate perpendicular to rolling direction (m) dimensionless coordinate transverse to film (z ¼ 2z/h) lubricant viscosity at ambient conditions (Pa s) temperature viscosity coefficient of lubricant (K1) Legendre coefficient of fluidity at kth location fluidity (1/Z), (Pa s)1 lubricant density (kg/m3) thermal diffusivity (m2/s), k/(rcp) dissipation factor, Z(@u/@z)2
have been achieved for THL of starved line contacts. In order to reduce the complexity of the computations, second-order Legendre polynomial temperature profile across the film thickness has been assumed in the energy equation. Present analysis treats the bounding solids (rollers) as rigid owing to their lightly loaded conditions. The coordinate system for this line contact geometry has been defined in Fig. 1. 2.1. Energy equation Considering non-inertial laminar lubricating films without dilatational viscosity, the energy equation for the lubrication of line contacts can be written as [12,13]: 2 @T @ @T @u rC p u ¼ k (1) þZ @x @z @z @z The variation of lubricant viscosity (Z) with temperature has been simulated using the following viscosity relation [13]: Z ¼ Z0 egðTT 0 Þ
(2)
Eq. (2) does not have pressure term in it since the effect of pressure on viscosity is negligible for hydrodynamic lubrication problems due to their light loads. In the energy equation (1), the terms due to conduction along the lubricant film, i.e. (@/@x)[k(@T/@x)] and convection across the film [rCpu(@T/@z)] are omitted. The magnitudes of these terms are usually very small in comparison to the convection along the film and conduction
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Viscosity of lubricating oil varies in the longitudinal as well as in the transverse direction to the film. In this analysis, viscosity has been expressed in the form of fluidity (reciprocal of viscosity (x ¼ 1/Z), where Z is modeled using Eq. (2)) for convenience of mathematical derivation. The expression for the fluidity variation across the film thickness is expressed by Legendre polynomial of order 2 and written as follows: xðzÞ ¼
3 X
x¯ k Pk ðzÞ
(4)
k¼0
For Legendre polynomial of order 2, the Lobatto locations, zk, and weight factors, wi, are provided in Table 1. The temperature variation across the film is also represented by a Legendre polynomial of order 2 written as: TðzÞ ¼
3 X
T¯ k Pk ðzÞ
(5)
k¼0
The Legendre coefficients of temperature, T¯ k , are evaluated as: Z 1 2 TðzÞPk ðzÞ dz ¼ T¯ k 2k þ1 1 or 2k þ 1 T¯ k ¼ 2 Fig. 1. Geometry of line contact. (a) Rigid rollers in contact; (b) equivalent rigid roller and rigid plane; (c) Lobatto points.
across the film [13]. The in-plane conduction terms in energy equation have been neglected. These terms are small compared with the cross-film conduction terms [13]. These terms are also small compared with the convection terms as the Peclet number is very high in this analysis. Therefore, surface temperatures (TU and TL) of both solids (both rollers) have been assumed constant and set equal to inlet temperature of lubricant (T0). The expression for velocity appearing in energy equation (1) is obtained by double integration of momentum equation [@p/@x ¼ (@/@z)(Z(@u/@z))], across the film thickness using boundary conditions (z ¼ 1, u ¼ uL and z ¼ +1, u ¼ uU). Thus, velocity expression for lubricant flow is: Z z Z z u ¼ uL þ A x dz þ B xz dz (3) 1
1
where R1
A¼
uU uL B 1 xz dz ; R1 1 x dz
2 h rp; B¼ 2
and z ¼
2z h
3 X
wi T i Pk ðzi Þ
(6)
i¼0
In order to avoid confusion related to non adoption of higher order Legendre polynomials in this analysis, it is relevant to mention here that higher orders (i.e. more than 4 Lobatto points) of Legendre polynomials do not bring perceptible difference between the results [13]. Eq. (6) can be used to obtain four equations for Legendre coefficients of temperature, i.e. T¯ 0 ; T¯ 1 ; T¯ 2 ; and T¯ 3 . In the expressions for T¯ 0 ; T¯ 1 ; T¯ 2 ; and T¯ 3 , surface temperatures (of rollers) TU and TL are considered constant and set equal to the ambient temperature (T0). From the expressions for T¯ 0 ; T¯ 1 ; T¯ 2 ; and T¯ 3 (obtained from Eq. (6)), the following two equations have been obtained: ðT U þ T L Þ ¯ T2 ¼ (7) T¯ 0 2 Table 1 Lobatto locations and weight factor Location, zi
Weight factor, wi
1 pffiffiffi 1= 5 pffiffiffi 1= 5 1
1/6 5/6 5/6 1/6
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7 ðT U T L Þ ¯ T3 ¼ T¯ 1 3 2
(8)
In this problem, there are two unknown temperatures (at Lobatto points in the film as shown in Fig. 1), which have to be computed by using energy equation (1). In order to compute temperatures at locations ‘‘+1’’ and ‘‘1’’, two equations involving T+1 and T1 are required. Therefore, zeroth and first moment of energy equation (1) have been taken across the film thickness as follows: Zeroth moment: Z 1 Z 1 h @T 2w @T @T 1 h u dz ¼ f dz þ h @z U @z L rC p 1 2 1 2 @x (9)
The Legendre coefficients of fluidity ðx¯ k Þ appearing in Eqs. (11)–(13) are evaluated at Lobatto locations as: Z 1 2 ¯ x xðzÞPk ðzÞ dz ¼ 2k þ1 k 1 or 2k þ 1 x¯ k ¼ 2
3 X
wi xi Pk ðzi Þ
@T @T þ TU þ TL @z U @z L Z 1 2 1 h fz dz ð10Þ þ rC p 1 4
h2 @T z dz ¼ w u @x 1 4 1
(14)
i¼0
_ Divergence of lineal mass flux ðr ðm=rÞ ¼ 0Þ leads to generalized thermal Reynolds equation as follows: r x¯ p h3 rp ¼ 6ðuL þ uU Þrh 2r
First moment: Z
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x¯ 1 hðuL uU Þ, x¯ 0
(15)
2 where x¯ p ¼ x¯ 0 þ 0:4x¯ 2 ðx¯ 1 =3x¯ 0 Þ
2.3. Miscellaneous equations The expression for film thickness can be written as:
where ‘U’ and ‘L’ stands for upper and lower roller surfaces, respectively. With the aid of Legendre series for the fluidity and temperature (Eqs. (4) and (5)), the integrals appearing in the zeroth (Eq. (9)) and first moments (Eq. (10)) of the energy equation are evaluated. The resulting simplified equations are as follows: Zeroth moment of energy equation yields:
x2 (16) 2R Load carrying capacity per unit length of rigid roller is computed as: Z xe W¼ p dx (17)
@T¯ 0 @T¯ 1 @T¯ 2 @T¯ 3 c1 þ c2 þ c3 þ c4 @x @x @x @x 2 2 2 @T @T 1 2 c5 ¼w þ h @z U @z L rC p h
Whereas the friction force per unit length of roller is obtained as below: Z xe @u F¼ Z dx (18) @z h=2 xi
ð11Þ
First moment of energy equation yields:
h ¼ h0 þ
xi
and rolling traction coefficient is obtained as follows:
@T¯ 0 @T¯ 1 @T¯ 2 @T¯ 3 þ c7 þ c8 þ c9 c6 @x @x @x @x 2 2 2 @T @T 1 2 ¼w þ TU þ TL þ c10 h @z U @z L rC p h
f ¼
F W
(19)
3. Computational procedure
ð12Þ Coefficients appearing in Eqs. (11) and (12) are given in Appendix. 2.2. Generalized Reynolds equation The derivation of generalized Reynolds equation has been obtained by taking divergence of lineal mass flux across the film thickness in the line contact (Fig. 1). The R þh=2 _ ¼ h=2 ru dzÞ, is obtained lineal mass flux expression, ðm by the integration of velocity terms (Eq. (3)) across the film thickness and the simplified expression obtained is as follows: _ h h¯ h ¯ 2¯ m x þ x B ¼ ðuU þ uL Þ x1 A (13) 2 3 3 0 5 2 r
In this analysis, the upper and lower surface temperatures of bounding solids (rollers) have been assumed constant throughout (justification for the same has been provided after Eq. (2)) and set equal to ambient temperature (T0). The solution to the present thermohydrodynamically lubricated line contact problem begins with the known pressure distribution within the contact as obtained from solution of Eq. (20) for isothermal films, which assumes that the temperature in the entire fluid film is equal to the inlet oil temperature. @ h3 @p ðuL þ uU Þ @h ¼ . (20) @x 12Z @x 2 @x Computational methodology involves coupled solution of , energy equations (11) and (12) and generalized thermal
ARTICLE IN PRESS N. Anandan et al. / Tribology International 41 (2008) 940–946
i n
For temperature: P P ð T i Þ T i n n1
P o0:001 Ti
(22)
n
where n represents number of iterations. By reducing the inlet distance, starvation is created and its effects on film thickness and rolling traction have been studied in THL conjunction of line contact.
4. Results and discussion Input data for lubricant properties and roller geometry are given in Table 2. The line contact problem analyzed here assumes the rollers to be infinitely long and therefore, the pressure gradient along the length of the rollers is taken as zero. Moreover, the roller surface temperatures are assumed constant in this analysis and equal to the inlet oil temperature, i.e. 311.11 K. It is pertinent to mention here that in the full film EHL solution, surface temperature rise is calculated by using appropriate boundary conditions which take into account of the conduction through a semiinfinite body. However, in the present analysis, it did not result in any significant change while keeping the solid surface temperature constant. Results for the performance parameters of the starved hydrodynamically lubricated line contact are presented for various values of inlet starvation parameters for a range of operating speeds and loads in Fig. 2 through Fig. 5. Table 2 Input data for lubricant and rollers Inlet viscosity of the lubricant (Pa s) Inlet temperature of lubricant (K) Pressure–viscosity coefficient of the lubricant (Pa1) Temperature viscosity coefficient of oil (K1) Thermal diffusivity of the oil (m2/s) rCp of the lubricant (J/m3 K) Radius of equivalent roller on plane (m) Thermal conductivity of the oil (W/m K)
0.13885 311.11 2.30 108 0.045 7.306 108 1.7577 106 1.11125 102 0.1284
4.1. Influence of starvation on film thickness Variations of starvation film thickness parameters for 15, 25, 35, and 50 m/s rolling speeds at load 15,000 N/m has been plotted in Fig. 2. At high degree of starvation, drastic reduction in film thickness parameter have been found due to increase in viscous shear heating. Fig. 3 shows variation of starvation film thickness parameter for four loads (15,000, 17,000, 19,000, and 21,000 N/m) at rolling speed 35 m/s. It can be seen from Figs. 2 and 3 that a unique relationship between starvation film thickness parameter and inlet boundary distance exist for the entire range of rolling speeds and loads taken in the analysis. These figures show that the film thickness parameter gradually reduces from its fully flooded (i.e. |X|=1.0) value and it goes sharply down at |X|=0.15. It is essential to mention here that this analysis does not incorporate starvation cases where |X|ob. 4.2. Influence of starvation on rolling traction Figs. 4 and 5 show the variation of starvation rolling traction parameter (fx,T/fN,T) with inlet distance (|X|) at 15,000 N/m load for 15, 25, 35, and 50 m/s rolling speeds Starvation film thickness parameter
Reynolds equation (15) by using appropriate boundary conditions. In computation, wherever reverse flow arises in the domain, upwind differencing has been resorted. The Legendre coefficients of temperatures T¯ 0 ; T¯ 1 ; T¯ 2 and T¯ 3 are obtained by solving Eqs. (6), (11) and (12). Pressure (thermal) is computed iteratively using Eq. (15) after incorporating thermal effects in viscosity (using Eq. (2)). The solution of governing equations have been made converged by satisfying the criterion (Eqs. (21) and (22)) at the non-dimensional mesh size Dx=0.002. The mesh size has been arrived at by performing grid independent test. For pressure: P
P p n pi n1 P i o0:001 (21) p
1.0 0.8 0.6 15 m/s 25 m/s 35 m/s 50 m/s
0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 2. Variation of starvation film thickness parameter (hx,T/hN,T) with inlet distance (X) at W ¼ 15,000 N/m.
1.0 Starvation film thickness parameter
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0.8 0.6 15000 N/m 17000 N/m 19000 N/m 21000 N/m
0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 3. Variation of starvation film thickness parameter (hx,T/hN,T) with inlet distance (X) at u ¼ 35 m/s.
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and 15,000, 17,000, 19,000 and 21,000 N/m load at 35 m/s rolling speed, respectively. The plots for starved rolling traction parameter indicate a gradual reduction in its magnitude with the increase in the starvation level. The trend of starvation traction parameter reduction is similar to that of film thickness parameter reduction, which can be seen from the presented figures. In THL, when the film thickness has 85% of its fully flooded value due to presence of starvation, the corresponding traction value has only fallen to 82% of its fully flooded value unlike EHL where traction falls steeply in comparison to film thickness [8]. Present analysis reveals that in the case of THL, both minimum film thickness and rolling traction fall with almost the same gradients in the presence of starvation. From the figures, it appears that minimum film thickness and traction variations in THL with degree of starvation are entirely different than that of thermo-elastohydrodynamic lubrication. Variations of non-dimensional minimum film thickness and maximum mid-film temperature for an operating condition (W ¼ 15,000 N/m and U ¼ 15 m/s) are provided in Table 3 at various levels of starvation. With the increase in the level of starvation, both non-dimensional minimum film thickness and maximum mid-film temperature reduce.
Starvation rolling traction parameter
1.0 0.8 0.6
15 m/s 25 m/s 35 m/s 50 m/s
0.4 0.2
Table 3 Variation of minimum film thickness and maximum mid-film temperature with starvation [W ¼ 15,000 N/m and u ¼ 15 m/s] Load (N/m)
Speed (m/s)
Inlet level |X|
Nondimensional minimum film thickness (h/R)
Maximum mid-film temperature (K)
15,000
15
1.0 0.75 0.50 0.25 0.15 0.05
4.21E04 3.70E04 3.15E04 2.55E04 2.29E04 2.00E04
316.685 314.070 313.771 313.726 312.261 311.817
The reduction in rolling traction with starvation has been noticed. Thus, in the presence of continuous film at the contact, starvation is beneficial. 5. Conclusions An efficient numerical method is used to study the effects of lubricant starvation and viscous shear heating on minimum film thickness and rolling traction under the conditions of hydrodynamically lubricated line contact. Varying the inlet distance of the domain (|X| ¼ 0.15–1.0), starvation has been created. The analysis is valid for high speed rolling line contact at lightly loaded conditions for which elastic deformation of mating surfaces are negligible. Starvation has a significant effect on minimum film thickness and rolling traction. For a range of operating conditions, the benefits of starvation in terms of reduced rolling traction and reduced temperature rise can be exploited in the presence of a continuous minimum film thickness at the contact. Acknowledgments
0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 4. Variation of starvation traction parameter (fx,T/fN,T) with inlet distance (X) at 15,000 N/m.
Starvation rolling traction parameter
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The computational facilities provided by IIT Delhi for this work and the leave sanctioned under QIP to the first author by Pondicherry Engineering College are gratefully acknowledged. Appendix
1.0 0.8
x0 ¼ xðT 0 Þ 15000 N/m
0.6
x1 ¼ xðT 1 Þ
17000 N/m
x2 ¼ xðT 2 Þ x3 ¼ xðT 3 Þ
19000 N/m
0.4
21000 N/m
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
X Fig. 5. Variation of starvation traction parameter (fx,T/fN,T) with inlet distance (X) at u ¼ 35 m/s.
2 x¯ x¯ p ¼ x¯ 0 þ 0:4x¯ 2 1 3x0 2¯ 2¯ 2¯ 4 ¯ ¯ c1 ¼ 2uL þ 2x0 x1 A þ x0 þ x1 x2 B 3 3 3 15 2¯ 2 ¯ 2 ¯ 2 ¯ x x AB þ x x B c2 ¼ 3 0 15 2 15 1 35 3
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2 ¯ 2 ¯ 2 ¯ 2 ¯ x x Aþ x þ x B c3 ¼ 15 1 35 3 15 0 105 2 2 ¯ 4 ¯ 2 ¯ x Aþ x þ x B c4 ¼ 35 2 105 1 315 3 2¯ 4 4¯ x0 þ x¯ 2 B2 þ x1 AB c5 ¼ 2x¯ 0 A2 þ 3 15 3 2¯ 2 2 ¯ 2 x0 x¯ 2 A þ x1 x¯ 3 B c6 ¼ 3 15 15 35 2 2 4 ¯ x A c7 ¼ uL þ 2x¯ 0 x¯ 1 3 15 105 3 2 2 8 ¯ x2 B þ x¯ 0 þ x¯ 1 15 9 105 2¯ 2 2 ¯ 2 x0 x¯ 2 A þ x1 x¯ 3 B c8 ¼ 3 15 15 35 2 ¯ 2 ¯ 2 ¯ 2 ¯ x1 x3 A þ x0 x2 B c9 ¼ 35 315 35 105 2¯ 2 2¯ 4 ¯ 2¯ 4 x1 þ x3 B2 þ x0 þ x¯ 2 2AB c10 ¼ x1 A þ 3 5 35 3 15
References [1] Lauder W. Hydrodynamic lubrication of proximate cylindrical surfaces of large relative curvature. Proc Inst Mech Eng 1965; 180(3B):64–9. [2] Dalmaz G, Godet M. An apparatus for the simultaneous measurement of load, traction, and film thickness in lubricated sliding point contacts. Tribology 1972;5(3):111–7.
[3] Dalmaz G, Godet M. Traction, load, and film thickness in lightly loaded lubricated point contacts. J Mech Eng Sci 1973;15(6):400–9. [4] Wedeven LD, Evans D, Cameron AC. Optical analysis of ball bearing starvation. ASME J Lubr Technol 1971;93:349–63. [5] Kingsbury E. Cross flow in a starved EHD contact. ASLE Trans 1973;16:276–80. [6] Hargreaves RA, Higginson GR. Some effects of lubricant starvation in cylindrical roller bearings. ASME Trans J Lubr Technol 1976;98: 66–72. [7] Brewe DE, Hamrock BJ. Analysis of starvation effects on hydrodynamic lubrication in non-conforming contacts. ASME Trans J Lubr Technol 1982;104:410–7. [8] Goksem PG, Hargreaves RA. The effect of viscous shear heating on both film thickness and rolling traction in an EHL line contact—Part II: Starved conditions. ASME Trans J Lubr Technol 1978;100(2): 353–8. [9] Ghosh MK, Hamrock BJ, Brewe DE. Starvation effects on the hydrodynamic lubrication of rigid non-conformal contacts in combined rolling and normal motion. ASLE Trans 1986;30:91–9. [10] Prasad D, Shukla JB, Singh P, Sinha P, Chhabra RP. Thermal effects in lubrication of asymmetrical rollers. Tribol Int 1991;24(4):239–46. [11] Ghosh MK, Gupta K. Thermal effect in hydrodynamic lubrication of line contacts—piezoviscous effect neglected. Int J Mech Sci 1998; 40(6):603–16. [12] Elrod HG, Brewe DE. Thermohydrodynamic analysis for laminar lubricating films. NASA technical memorandum no. 88845; 1986. [13] Elrod HG. Efficient numerical method for computation of the thermo hydrodynamics of laminar lubricating films. ASME Trans J Tribol 1991;113:506–11. [14] Ghosh MK, Pandey RK. Thermal elastohydrodynamic lubrication of heavily loaded line contacts—an efficient inlet zone analysis. ASME Trans J Tribol 1998;120:119–25. [15] Pandey RK, Ghosh MK. Temperature rise due to sliding in rolling/ sliding elastohydrodynamic lubrication of line contact: an efficient numerical analysis for contact zone temperatures. Tribol Int 1998;31(12):745–52.