Research Papers: Elastohydrodynamic Lubrication

A Thermal Elastohydrodynamic Lubrication Model for Crowned Rollers and Its Application on Apex Seal–Housing Interfaces

[+] Author and Article Information
Zhong Liu, David Pickens III, Tao He, Xin Zhang, Yuchuan Liu, Q. Jane Wang

Department of Mechanical Engineering,
Northwestern University,
2145 Sheridan Road,
Evanston, IL 60208

Takayuki Nishino

Powertrain Division,
Mazda Motor Corporation,
3-1 Shinchi, Fuchu-cho, Aki-gun,
Hiroshima 730-8670, Japan

1Present address: GM Powertrain, Pontiac, MI 48340.

2Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received September 30, 2018; final manuscript received December 26, 2018; published online February 13, 2019. Assoc. Editor: Liming Chang.

J. Tribol 141(4), 041501 (Feb 13, 2019) (14 pages) Paper No: TRIB-18-1407; doi: 10.1115/1.4042503 History: Received September 30, 2018; Revised December 26, 2018

This paper presents a thermal elastohydrodynamic lubrication (EHL) model for analyzing crowned roller lubrication performances under the influence of frictional heating. In this thermal EHL model, the Reynolds equation is solved to obtain the film thickness and pressure results while the energy equation and temperature integration equation are evaluated for the temperature rise in the lubricant and at the surfaces. The discrete convolution fast Fourier transform (DC-FFT) method is utilized to calculate the influence coefficients for both the elastic deformation and the temperature integration equations. The influences of the slide-to-roll ratio (SRR), load, crowning radius, and roller length on the roller lubrication and temperature rise are investigated. The results indicate that the thermal effect becomes significant for the cases with high SRRs or heavy loads. The proposed thermal EHL model is used to study the thermal-tribology behavior of an apex seal–housing interface in a rotary engine, and to assist the design of the apex seal crown geometry. A simplified crown design equation is obtained from the analysis results, validated through comparison with the optimal results calculated using the current crowned-roller thermo-EHL (TEHL) model.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Cheng, H. S. , and Sternlicht, B. , 1964, “A Numerical Solution for the Pressure, Temperature, and Film Thickness Between Two Infinitely Long, Lubricated Rolling and Sliding Cylinders, Under Heavy Loads,” ASME J. Basic Eng., 87(3), pp. 695–704. [CrossRef]
Dowson, D. , and Whitaker, A. V. , 1966, “A Numerical Procedure for the Solution of the Elastohydrodynamic Problem of Rolling and Sliding Contacts Lubricated by a Newtonian Fluid,” Proc. Inst. Mech. Eng., 180, pp. 119–134.
Yang, P. R. , and Wen, S. H. , 1992, “The Behavior of Non-Newtonian Thermal EHL Film in Line Contacts at Dynamic Loads,” ASME J. Tribol., 114(1), pp. 81–85. [CrossRef]
Hsiao, H. S. , and Hamrock, B. J. , 1994, “Non-Newtonian and Thermal Effects on Film Generation and Traction Reduction in EHL Line Contact Conjunctions,” ASME J. Tribol., 116(3), pp. 559–568. [CrossRef]
Hsiao, H. S. , and Hamrock, B. J. , 1994, “Temperature Distribution and Thermal-Degradation of the Lubricant in EHL Line Contact Conjunctions,” ASME J. Tribol., 116(4), pp. 794–803. [CrossRef]
Yang, P. , Wang, J. , and Kaneta, M. , 2006, “Thermal and Non-Newtonian Numerical Analyses for Starved EHL Line Contacts,” ASME J. Tribol., 128(2), pp. 282–290. [CrossRef]
Almqvist, T. , and Larsson, R. , 2002, “The Navier-Stokes Approach for Thermal EHL Line Contact Solutions,” Tribol. Int., 35(3), pp. 163–170. [CrossRef]
Almqvist, T. , and Larsson, R. , 2008, “Thermal Transient Rough EHL Line Contact Simulations by Aid of Computational Fluid Dynamics,” Tribol. Int., 41(8), pp. 683–693. [CrossRef]
Kumar, P. , and Khonsari, M. M. , 2008, “Traction in EHL Line Contacts Using Free-Volume Pressure-Viscosity Relationship With Thermal and Shear-Thinning Effects,” ASME J. Tribol., 131(1), p. 011503. [CrossRef]
Zhang, B. B. , Wang, J. , Omasta, M. , and Kaneta, M. , 2015, “Effect of Fluid Rheology on the Thermal EHL Under ZEV in Line Contact,” Tribol. Int., 87, pp. 40–49. [CrossRef]
Ochoa, E. D. , Otero, J. E. , Lopez, A. S. , Tanarro, E. C. , and Lopez, B. D. , 2018, “Film Thickness Formula for Thermal EHL Line Contact Considering a New Reynolds-Carreau Equation,” Tribol. Lett., 31, p. 66.
Bruggemann, H. , and Kollmann, F. G. , 1982, “A Numerical Solution of the Thermal Elastohydrodynamic Lubrication in an Elliptical Contact,” ASME J. Lubr. Technol., 104(3), pp. 392–400. [CrossRef]
Zhu, D. , and Wen, S. Z. , 1984, “A Full Numerical-Solution for the Thermoelastohydrodynamic Problem in Elliptical Contacts,” ASME J. Tribol., 106(2), pp. 246–254. [CrossRef]
Ma, M. T. , 1997, “An Expedient Approach to the Non-Newtonian Thermal EHL in Heavily Loaded Point Contacts,” Wear, 206(1–2), pp. 100–112. [CrossRef]
Liu, X. L. , and Yang, P. R. , 2009, “Effects of Solid Body Temperature on the Non-Newtonian Thermal EHL Behavior in Point Contacts,” Advanced Tribology, Springer, Berlin, pp. 169–170.
Liu, X. L. , Jiang, M. , Yang, P. R. , and Kaneta, M. , 2005, “Non-Newtonian Thermal Analyses of Point EHL Contacts Using the Eyring Model,” ASME J. Tribol., 127(1), pp. 70–81. [CrossRef]
Kaneta, M. , and Yang, P. R. , 2003, “Effects of Thermal Conductivity of Contacting Surfaces on Point EHL Contacts,” ASME J. Tribol., 125(4), pp. 731–738. [CrossRef]
Zhang, J. J. , Yang, P. R. , Wang, J. , and Guo, F. , 2015, “The Effect of an Oscillatory Entrainment Velocity on the Film Thickness in Thermal EHL Point Contact,” Tribol. Int., 90, pp. 519–532. [CrossRef]
Kaneta, M. , Cui, J. L. , Yang, P. R. , Krupka, I. , and Hartl, M. , 2016, “Influence of Thermal Conductivity of Contact Bodies on Perturbed Film Caused by a Ridge and Groove in Point EHL Contacts,” Tribol. Int., 100, pp. 84–98. [CrossRef]
Kaneta, M. , and Yang, P. R. , 2003, “Formation Mechanism of Steady Multi-Dimples in Thermal EHL Point Contacts,” ASME J. Tribol., 125(2), pp. 241–251. [CrossRef]
Yang, P. R. , Cui, J. L. , Jin, Z. M. , and Dowson, D. , 2008, “Influence of Two-Sided Surface Waviness on the EHL Behavior of Rolling/Sliding Point Contacts Under Thermal and Non-Newtonian Conditions,” ASME J. Tribol., 130(4), p. 041502. [CrossRef]
Zhang, B. B. , Wang, J. , Omasta, M. , and Kaneta, M. , 2016, “Variation of Surface Dimple in Point Contact Thermal EHL Under ZEV Condition,” Tribol. Int., 94, pp. 383–394. [CrossRef]
Kuroda, S. , and Arai, K. , 1985, “Elastohydrodynamic Lubrication Between Two Rollers,” JSME, 28(241), pp. 1367–1372. [CrossRef]
Mostofi, A. , and Gohar, R. , 1983, “Elastohydrodynamic Lubrication of Finite Line Contacts,” ASME J. Lubr. Technol., 105(4), pp. 598–604. [CrossRef]
Park, T. J. , and Kim, K. W. , 1998, “Elastohydrodynamic Lubrication of a Finite Line Contact,” Wear, 223(1–2), pp. 102–109. [CrossRef]
Zhu, D. , Wang, J. X. , Ren, N. , and Wang, Q. J. , 2012, “Mixed Elastohydrodynamic Lubrication in Finite Roller Contacts Involving Realistic Geometry and Surface Roughness,” ASME J. Tribol., 134(1), p. 011504. [CrossRef]
He, T. , Wang, J. X. , Wang, Z. J. , and Zhu, D. , 2015, “Simulation of Plasto-Elastohydrodynamic Lubrication in Line Contacts of Infinite and Finite Length,” ASME J. Tribol., 137(4), p. 041505. [CrossRef]
Park, T. J. , and Kim, K. W. , 1998, “A Numerical Analysis of the Elastohydrodynamic Lubrication of Elliptical Contacts,” Wear, 136, pp. 299–312.
Najjari, M. , and Guilbault, R. , 2014, “Edge Contact Effect on Thermal Elastohydrodynamic Lubrication of Finite Contact Lines,” Tribol. Int., 71, pp. 50–61. [CrossRef]
Wang, X. P. , Liu, Y. C. , and Zhu, D. , 2017, “Numerical Solution of Mixed Thermal Elastohydrodynamic Lubrication in Point Contacts With Three-Dimensional Surface Roughness,” ASME J. Tribol., 139(1), p. 011501. [CrossRef]
Kim, H. J. , Ehret, P. , Dowson, D. , and Taylor, C. M. , 2001, “Thermal Elastohydrodynamic Analysis of Circular Contacts—Part 1: Newtonian Model,” Proc. Inst. Mech. Eng., 215(3), pp. 339–352. [CrossRef]
He, T. , Zhu, D. , Wang, J. X. , and Wang, Q. J. , 2017, “Experimental and Numerical Investigations of the Stribeck Curves for Lubricated Counterformal Contacts,” ASME J. Tribol., 139(2), p. 021505. [CrossRef]
Nikas, G. K. , 2017, “Miscalculation of Film Thickness, Friction and Contact Efficiency by Ignoring Tangential Tractions in Elastohydrodynamic Contacts,” Tribol. Int., 110(6), pp. 252–263. [CrossRef]
Chen, W. W. , and Wang, Q. J. , 2008, “A Numerical Model for the Point Contact of Dissimilar Materials Considering Tangential Tractions,” Mech. Mater., 40(11), pp. 936–948. [CrossRef]
Liu, X. L. , Ma, M. , Yang, P. , and Guo, F. , 2018, “A New Method for Eyring Shear-Thinning Models in Elliptical Contacts Thermal Elastohydrodynamic Lubrication,” ASME J. Tribol., 140(5), p. 051503. [CrossRef]
Liu, S. B. , Wang, Q. , and Liu, G. , 2000, “A Versatile Method of Discrete Convolution and FFT (DC-FFT) for Contact Analyses,” Wear, 243(1–2), pp. 101–111. [CrossRef]
Liu, S. B. , and Wang, Q. , 2002, “Studying Contact Stress Fields Caused by Surface Tractions With a Discrete Convolution and Fast Fourier Transform Algorithm,” ASME J. Tribol., 124(1), pp. 36–45. [CrossRef]
Roelands, C. J. A. , Vlutger, J. C. , and Watermann, H. I. , 1963, “The Viscosity Temperature Pressure Relationship of Lubricating Oils and Its Correlation With Chemical Constitution,” ASME J. Basic Eng., 85(4), pp. 601–607. [CrossRef]
Yang, P. R. , and Wen, S. Z. , 1990, “A Generalized Reynolds Equation for Non-Newtonian Thermal Elastohydrodynamio Lubrication,” ASME J. Tribol., 112, pp. 631–636. [CrossRef]
Liu, Y. C. , Wang, H. , Wang, W. Z. , Hu, Y. Z. , and Zhu, D. , 2002, “Methods Comparison in Computation of Temperature Rise on Frictional Interfaces,” Tribol. Int., 35(8), pp. 549–560. [CrossRef]
Liu, Y. C. , Wang, Q. J. , Zhu, D. , and Shi, F. , 2008, “A Generalized Thermal EHL Model for Point Contact Problems,” ASME Paper No. IJTC2008-71120.
Bos, J. , and Moes, H. , 1995, “Frictional Heating of Tribological Contacts,” ASME J. Tribol., 117(1), pp. 171–177. [CrossRef]
Chen, W. W. , and Wang, Q. J. , 2008, “Thermomechanical Analysis of Elastoplastic Bodies in a Sliding Spherical Contact and the Effects of Sliding Speed, Heat Partition, and Thermal Softening,” ASME J. Tribol., 130(4), p. 041402. [CrossRef]
Ai, X. L. , 1993, “Numerical Analyses of Elastohydrodynamically Lubricated Line and Point Contacts With Rough Surfaces by Using Semi-System and Multigrid Methods,” Northwestern University, Evanston, IL.
Zhu, D. , and Hu, Y. Z. , 1999, “The Study of Transition From Full Film Elastohydrodynamic to Mixed and Boundary Lubrication,” STLE/ASME HS Cheng Tribology Surveillance, pp. 150–156.
Hamrock, B. J. , and Dowson, D. , 1976, “Isothermal Elastohydrodynamic Lubrication of Point Contacts—Part I: Theoretical Formulation,” ASME J. Lubr. Technol., 98(2), pp. 223–228. [CrossRef]
Brewe, D. E. , and Hamrock, B. J. , 1977, “Simplified Solution for Elliptical-Contact Deformation Between Two Elastic Solids,” ASME J. Lubr. Technol., 99(4), pp. 485–487. [CrossRef]


Grahic Jump Location
Fig. 1

Crowned roller contact geometry and the solution domain in the lubricant

Grahic Jump Location
Fig. 2

Geometry and cross-sectional views of a roller

Grahic Jump Location
Fig. 3

Flowchart for TEHL crowned roller contacts

Grahic Jump Location
Fig. 4

Comparison of dimensionless minimum film thicknesses and temperature rises obtained from the present study and Refs. [25] and [29] under different meshes and two different load conditions: (a) light load 47.6 kN/m and (b) high load 190 kN/m

Grahic Jump Location
Fig. 5

Three-dimensional temperature rise distribution in the middle layer of the lubricant: L =2.37, w =190 kN/m, u =0.8 m/s, and SRR = 0.3

Grahic Jump Location
Fig. 6

Temperature rise distribution in the xoz cross section in a crowned TEHL: w =500 kN/m, u =0.625 m/s, SRR = 0.25, and Ry = 254 mm

Grahic Jump Location
Fig. 7

Three-dimensional temperature rise distributions (left column) and contours of temperature rise (right column) in (a) the upper surface, (b) the middle layer surface, and (c) the lower surface in crowned roller TEHL. w =500 kN/m, u=0.625 m/s, SRR = 0.25, and Ry = 254 mm.

Grahic Jump Location
Fig. 8

Distributions of the centerline pressure and temperature rise in a crowned roller TEHL. w =500 kN/m, u=0.625 m/s, SRR = 0.25, and Ry = 254 mm.

Grahic Jump Location
Fig. 9

Comparisons of centerline pressure and film thickness distributions for EHL and TEHL crowned roller interfaces (a) along the motion direction and (b) along the length direction. w =500 kN/m, u=0.625 m/s, SRR = 0.25, and Ry = 254 mm.

Grahic Jump Location
Fig. 10

(a) Variations of maximum temperature rises and maximum pressures. (b) Variations of central and minimum film thicknesses for SRR = 0–2 under w =500 kN/m, u=0.625 m/s, and Ry = 254 mm.

Grahic Jump Location
Fig. 11

Comparisons of (a) centerline pressure and film thickness distributions. (b) Centerline temperature distributions for the cases of SRR = 0–2 under w =500 kN/m, u=0.625 m/s, and Ry= 254 mm.

Grahic Jump Location
Fig. 12

(a) Maximum pressure and maximum temperature rise. (b) Central and minimum film thickness for the cases of w =250–1000 kN/m under u=0.625 m/s and SRR = 0.25.

Grahic Jump Location
Fig. 13

(a) Centerline pressure, (b) temperature rise, and (c) film thickness distributions for the cases of w =250–1000 kN/m under u=0.625 m/s and SRR = 0.25

Grahic Jump Location
Fig. 14

(a) Maximum pressure and maximum temperature rise and (b) central and minimum film thickness for rollers with different crown radii Ry= 62–10,000 mm under w =500 kN/m, u=0.625 m/s, SRR = 0.25, and Rx= 19.05 mm. Note that the minimum film thickness drops to 0 due to the edges.

Grahic Jump Location
Fig. 15

(a) Dimensionless pressure, (b) film thickness, and (c) middle layer temperature rise distributions for the cases of different crowning radii Ry= 62–10,000 mm under conditions of w =500 kN/m, u=0.625 m/s, SRR = 0.25, and Rx= 19.05 mm

Grahic Jump Location
Fig. 16

(a) Pressure and maximum temperature rise; (b) central and minimum film thickness for straight rollers of different lengths L =2–20 mm under w =500 kN/m, u=0.625 m/s, SRR = 0.25, Rx= 19.05 mm, and Ry= 2.5×1011 m (representing a finite-length cylinder)

Grahic Jump Location
Fig. 17

Schematic of a typical crowned apex seal of a rotary engine. Here, the crown height is related to the crown radius, and a higher height means a smaller crown radius.

Grahic Jump Location
Fig. 18

Comparisons of the EHL and TEHL results for the apex seal–housing interface under the conditions listed in Table 3 (a) along the motion direction and (b) along the length direction

Grahic Jump Location
Fig. 19

(a) Pressure distributions, (b) upper layer temperature rise distributions, (c) central and minimum film thickness, and (d) maximum pressure and maximum temperature rise at the interfaces with the apex seals of different crown heights

Grahic Jump Location
Fig. 20

F(Ry) plotted against Ry for the cases of λ = 0.8–1 under w =500 kN/m, E′ = 220 GPa, Rx = 19.05 mm, and L =5 mm

Grahic Jump Location
Fig. 21

Optimal crown radius Ry calculated based on simulation results and F(Ry) = 0 for rollers of different roller lengths L =1–6 mm under λ = 0.8–1, w =500 kN/m, E′ = 220 GPa, and Rx = 19.05 mm



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In