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Microstress cycle and contact fatigue of spiral bevel gears by rolling-sliding of asperity contact

  • Wei Cao
  • Si Ren
  • Wei PuEmail author
  • Ke Xiao
Open Access
Research Article


The rolling contact fatigue (RCF) model is commonly used to predict the contact fatigue life when the sliding is insignificant in contact surfaces. However, many studies reveal that the sliding, compared to the rolling state, can lead to a considerable reduction of the fatigue life and an excessive increase of the pitting area, which result from the microscopic stress cycle growth caused by the sliding of the asperity contact. This suggests that fatigue life in the rolling-sliding condition can be overestimated based only on the RCF model. The rubbing surfaces of spiral bevel gears are subject to typical rolling-sliding motion. This paper aims to study the mechanism of the micro stress cycle along the meshing path and provide a reasonable method for predicting the fatigue life in spiral bevel gears. The microscopic stress cycle equation is derived with the consideration of gear meshing parameters. The combination of the RCF model and asperity stress cycle is developed to calculate the fatigue life in spiral bevel gears. We find that the contact fatigue life decreases significantly compared with that obtained from the RCF model. There is strong evidence that the microscopic stress cycle is remarkably increased by the rolling-sliding motion of the asperity contact, which is consistent with the experimental data in previous literature. In addition, the fatigue life under different assembling misalignments are investigated and the results demonstrate the important role of misalignments on fatigue life.


rolling/sliding contact fatigue stress cycle spiral bevel gear mixed elasto-hydrodynamic lubrication assembling misalignment 



This study is funded by National Science Foundation of China (No. 51875369) and General Projects of Basic Science and Frontier Technology Research of Chongqing (Nos. cstc2016jcyjA0511, cstc2018jcyjAX0451). Wei PU would like to thank Fundamental Research Funds for the Central Universities (No. YJ201752).


  1. [1]
    Sadeghi F, Jalalahmadi B. Probabilistic life prediction models for rolling contact fatigue. In Encyclopedia of Tribology. Wang Q J, Chung Y W, Eds. Boston: Springer, 2013.Google Scholar
  2. [2]
    Bujold M P, Zhu D, Epstein D, Wang Q, Keer L M. Investigation of sliding/rolling contact fatigue life with both two disk experiments and computer model based prediction. In STLE 2004 Annual Meeting, Toronto, 2004.Google Scholar
  3. [3]
    Ramalho A, Esteves M, Marta P. Friction and wear behaviour of rolling-sliding steel contacts. Wear302(1-2): 1468–1480 (2013)CrossRefGoogle Scholar
  4. [4]
    Lee D H, Seo J W, Kwon S J. Numerical analysis of the effect of slip ratio on the fatigue crack initiation life in rolling contact. Adv Mater Res891-892: 1791–1796 (2014)CrossRefGoogle Scholar
  5. [5]
    Seo J W, Jun H K, Kwon S J, Lee D H. Rolling contact fatigue and wear of two different rail steels under rolling-sliding contact. Int J Fatigue83: 184–194 (2016)CrossRefGoogle Scholar
  6. [6]
    Oksanen V, Valtonen K, Andersson P, Vaajoki A, Laukkanen A, Holmberg K, Kuokkala V T. Comparison of laboratory rolling-sliding wear tests with in-service wear of nodular cast iron rollers against wire ropes. Wear340-341: 73–81 (2015)CrossRefGoogle Scholar
  7. [7]
    Pu W, Zhu D, Wang J X, Wang Q J. Rolling-sliding contact fatigue of surfaces with sinusoidal roughness. Int J Fatigue90: 57–68 (2016)CrossRefGoogle Scholar
  8. [8]
    Cao W, Pu W, Wang J X, Xiao K. Effect of contact path on the mixed lubrication performance, friction and contact fatigue in spiral bevel gears. Tribol Int123: 359–371 (2018)CrossRefGoogle Scholar
  9. [9]
    Lundberg G, Palmgren A. Dynamic capacity of rolling bearings. Acta Polytechnica Mech Eng Ser I Roy Swed Acad Eng Sci1(3): 7 (1947)Google Scholar
  10. [10]
    Weibull W A. A statistical theory of the strength of materials. Proc Roy Swed Inst Eng Res151: 5–45 (1939)Google Scholar
  11. [11]
    Ioannides E, Harris T A. A new fatigue life model for rolling bearings. J Tribol107(3): 367–377 (1985)CrossRefGoogle Scholar
  12. [12]
    Zaretsky E Y. Fatigue criterion to system design, life, and reliability. J Propul Power3(1): 76–83 (1987)CrossRefGoogle Scholar
  13. [13]
    Tallian T E. A data-fitted rolling bearing life prediction model-Part I: Mathematical model. Tribol Trans39(2): 249–258 (1996)CrossRefGoogle Scholar
  14. [14]
    Tripp J H, Ioannides E. Effects of surface roughness on rolling bearing life. In Proceedings of Japan International Tribology Conference, Nagoya, 1990: 797–802.Google Scholar
  15. [15]
    Ai X L. Effect of three-dimensional random surface roughness on fatigue life of a lubricated contact. J Tribol120(2): 159–164 (1998)CrossRefGoogle Scholar
  16. [16]
    Epstein D, Yu T H, Wang Q J, Keer L M, Cheng H S, Liu S, Harris S J, Gangopadhyay A. An efficient method of analyzing the effect of roughness on fatigue life in mixed-EHL contact. Tribol Trans46(2): 273–281 (2003)CrossRefGoogle Scholar
  17. [17]
    Zhu D, Ren N, Wang Q J. Pitting life prediction based on a 3D line contact mixed EHL analysis and subsurface von Mises stress calculation. J Tribol131(4): 041501 (2009)Google Scholar
  18. [18]
    Xu G, Sadeghi F. Thermal EHL analysis of circular contacts with measured surface roughness. J Tribol118(3): 473–482 (1996)CrossRefGoogle Scholar
  19. [19]
    Zhu D, Ai X L. Point contact EHL based on optically measured three-dimensional rough surfaces. J Tribol119(3): 375–384 (1997)CrossRefGoogle Scholar
  20. [20]
    Jiang X F, Hua D Y, Cheng H S, Ai X L, Lee S C. A mixed elastohydrodynamic lubrication model with asperity contact. J Tribol121(3): 481–491 (1999)CrossRefGoogle Scholar
  21. [21]
    Shi F H, Salant R F. A mixed soft elastohydrodynamic lubrication model with interasperity cavitation and surface shear deformation. J Tribol122(1): 308–316 (2000)CrossRefGoogle Scholar
  22. [22]
    Hu Y Z, Zhu D. A full numerical solution to the mixed lubrication in point contacts. J Tribol122(1): 1–9 (2000)CrossRefGoogle Scholar
  23. [23]
    Holmes M J A, Qiao H, Evans H P, Snidle R W. Surface contact and damage in micro-EHL. Tribol Interface Eng Ser48: 605–616 (2005)CrossRefGoogle Scholar
  24. [24]
    Bayada G, Martin S, Vázquez C. Micro-roughness effects in (elasto)hydrodynamic lubrication including a mass-flow preserving cavitation model. Tribol Int39(12): 1707–1718 (2006)CrossRefGoogle Scholar
  25. [25]
    Zhu D. On some aspects of numerical solutions of thin-film and mixed elastohydrodynamic lubrication. Proc Inst Mech Eng, Part J: J Eng Tribol221(5): 561–579 (2007)CrossRefGoogle Scholar
  26. [26]
    Zhu D, Liu Y C, Wang Q. On the numerical accuracy of rough surface EHL solution. Tribol Trans57(4): 570–580 (2014)CrossRefGoogle Scholar
  27. [27]
    Greco A, Martini A, Liu Y C, Lin C, Wang Q J. Rolling contact fatigue performance of vibro-mechanical textured surfaces. Tribol Trans53(4): 610–620 (2010)CrossRefGoogle Scholar
  28. [28]
    Li S, Kahraman A. A fatigue model for contacts under mixed elastohydrodynamic lubrication condition. Int J Fatigue33(3): 427–436 (2011)CrossRefGoogle Scholar
  29. [29]
    Li S, Anisetti A. A tribo-dynamic contact fatigue model for spur gear pairs. Int J Fatigue98: 81–91 (2017)CrossRefGoogle Scholar
  30. [30]
    Pu W, Wang J X, Zhang Y, Zhu D. A theoretical analysis of the mixed elastohydrodynamic lubrication in elliptical contacts with an arbitrary entrainment angle. J Tribol136(4): 041505 (2014)Google Scholar
  31. [31]
    Pu W, Wang J X, Zhu D. Friction and flash temperature prediction of mixed lubrication in elliptical contacts with arbitrary velocity vector. Tribol Int99: 38–46 (2016)CrossRefGoogle Scholar
  32. [32]
    Pu W, Wang J X, Yang R S, Zhu D. Mixed elastohydrodynamic lubrication with three-dimensional machined roughness in spiral bevel and hypoid gears. J Tribol137(4): 041503 (2015)Google Scholar
  33. [33]
    Ural A, Heber G, Wawrzynek P A, Ingraffea A R, Lewicki D G, Neto J B C. Three-dimensional, parallel, finite element simulation of fatigue crack growth in a spiral bevel pinion gear. Eng Fract Mech72(8): 1148–1170 (2005)CrossRefGoogle Scholar
  34. [34]
    Ural A, Wawrzynek P A, Ingraffea A R, Lewicki D G. Simulating fatigue crack growth in spiral bevel gears using computational fracture mechanics. In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, 2003: 195–199.Google Scholar
  35. [35]
    Asi O. Experimental surface contact fatigue failure analysis of a hypoid pinion used in differential system of a truck. Int J Heavy Vehicle Syst18(1): 104–114 (2011)CrossRefGoogle Scholar
  36. [36]
    Xi L, Wang S W. Experimental investigation of bending fatigue life of driving pinion tooth of hypoid bevels. J Test Eval44(6): 20140378 (2016)Google Scholar
  37. [37]
    Goldfarb V, Barmina N. Theory and Practice of Gearing and Transmissions: In Honor of Professor Faydor L. Litvin. Cham (Germany): Springer, 2016.CrossRefGoogle Scholar
  38. [38]
    Álvarez Á, Calleja A, Arizmendi M, González H, de Lacalle L N L. Spiral bevel gears face roughness prediction produced by CNC end milling centers. Materials11(8): 1301 (2018)CrossRefGoogle Scholar
  39. [39]
    Litvin F L, Fuentes A, Hayasaka K. Design, manufacture, stress analysis, and experimental tests of low-noise high endurance spiral bevel gears. Mech Machine Theory41(1): 83–118 (2006)zbMATHCrossRefGoogle Scholar
  40. [40]
    Fan Q. Enhanced algorithms of contact simulation for hypoid gear drives produced by face-milling and face-hobbing processes. J Mech Des129(1): 31–37 (2007)CrossRefGoogle Scholar
  41. [41]
    Cheng W, Cheng H S, Keer L M. Experimental investigation on rolling/sliding contact fatigue crack initiation with artificial defects. Tribol Trans37(1): 1–12 (1994)CrossRefGoogle Scholar
  42. [42]
    Wang Y Z, Chen Y Y, Zhou G M, Lv Q J, Zhang Z Z, Tang W, Liu Y. Roughness model for tooth surfaces of spiral bevel gears under grinding. Mech Machine Theory104: 17–30 (2016)CrossRefGoogle Scholar
  43. [43]
    Zhu D, Cheng H S. An analysis and computational procedure for EHL film thickness, friction and flash temperature in line and point contacts. Tribol Trans32(3): 364–370 (1989)CrossRefGoogle Scholar
  44. [44]
    Bair S, Winer W O. A rheological model for elastohydrodynamic contacts based on primary laboratory data. J Lub Tech101(3): 258–264 (1979)CrossRefGoogle Scholar
  45. [45]
    Johnson K L. Contact Mechanics. Cambridge (UK): Cambridge University Press, 1985.zbMATHCrossRefGoogle Scholar
  46. [46]
    Rabaso P, Gauthier T, Diaby M, Ville F. Rolling contact fatigue: Experimental study of the influence of sliding, load, and material properties on the resistance to micropitting of steel discs. Tribol Trans56(2): 203–214 (2013)CrossRefGoogle Scholar
  47. [47]
    Govindarajan N, Gnanamoorthy R. Rolling/sliding contact fatigue life prediction of sintered and hardened steels. Wear262(1-2): 70–78 (2007)CrossRefGoogle Scholar
  48. [48]
    Gao C K, Qi X M, Snidle R W, Evans H P. Effect of film thickness ratio on gearing contact fatigue life. In IUTAM Symposium on Elastohydrodynamics and Microelastohydrodynamics. Snidle R W, Evans H P, Eds. Dordrecht: Springer, 2006: 423–434.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Construction MachineryChang’an UniversityXi’anChina
  2. 2.School of Aeronautics and AstronauticsSichuan UniversityChengduChina
  3. 3.Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.College of Mechanical EngineeringChongqing UniversityChongqingChina

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