Skip to main content
SpringerLink
Log in

Towards a complete numerical description of lubricant film dynamics in ball bearings

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

In this article, we propose a framework for a detailed finite element analysis of elastohydrodynamic lubrication in ball bearings. Our contribution to this field is twofold. First, we present a fully monolithic ALE method for the treatment of fluid–structure interaction. For the lubricant, we use the full Navier–Stokes equations in combination with a pressure-dependent viscosity law and include thermal effects. Second, we introduce a novel method for a fully implicit treatment of the evolution of the lubricants’ free surface using Nitsches method. This allows for arbitrarily large time steps independent of the spatial discretization. Despite the variety of numerical challenges present in this application, such as anisotropy and extreme values of pressure, our approach for the first time shows robustness up to high rotational speeds as required in industrial applications. We describe the numerical ingredients we use in detail and present numerical results that validate our approaches and demonstrate its capabilities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Almqvist T, Larsson R (2002) The Navier–Stokes approach for thermal EHL line contact solutions. Tribol Int 35:163–170

    Article  Google Scholar 

  2. Annavarapu C, Hautefeuille M, Dolbow JE (2012) A robust Nitsche’s formulation for interface problems. Comput Methods Appl Mech Eng 225–228:44–54

    Article  MathSciNet  Google Scholar 

  3. Apel T, Knopp T, Lube G (2008) Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem. Appl Numer Math 58(12):1830–1843

    Article  MATH  MathSciNet  Google Scholar 

  4. Bänsch E (2001) Finite element discretization of the Navier–Stokes equations with a free capillary surface. Numer Math 88(2): 203–235

    Google Scholar 

  5. Bayada G, Chambat M (1986) The transition between the Stokes equations and the Reynolds equation: a mathematical proof. Appl Math Optim 14(1):73–93

    Article  MATH  MathSciNet  Google Scholar 

  6. Becker R, Braack M (2001) A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4):173–199

    Article  MATH  MathSciNet  Google Scholar 

  7. Becker R, Braack M, Meidner D, Richter T, Vexler B (2005) The finite element toolkit Gascoigne. http://www.gascoigne.de. Accessed July 2013

  8. Belytschko T (1980) Fluid–structure interaction. Comput Struct 12:459–469

    Article  MATH  Google Scholar 

  9. Braack M (2008) A stabilized finite element scheme for the Navier–Stokes equations on quadrilateral anisotropic meshes. M2AN. Math Model Numer Anal 42(6):903–924

    Article  MATH  MathSciNet  Google Scholar 

  10. Braack M, Richter T (2006) Solutions of 3D Navier–Stokes benchmark problems with adaptive finite elements. Comput Fluids 35(4):372–392

    Article  MATH  Google Scholar 

  11. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335–354

    Article  MATH  MathSciNet  Google Scholar 

  12. Bruyere V, Fillot N, Morales-Espejel G, Vergne P (2012) Computational fluid dynamics and full elasticity model for sliding line thermal elastohydrodynamic contacts. Tribol Int 46(1):3–13

    Article  Google Scholar 

  13. Dettmer W, Perić D (2006) A computational framework for free surface fluid flows accounting for surface tension. Comput Methods Appl Mech Eng 195(23–24):3038–3071

    Article  MATH  Google Scholar 

  14. Dunne T, Rannacher R, Richter T (2010) Numerical simulation of fluid-structure interaction based on monolithic variational formulations. In: Galdi GP, Rannacher R (eds) Fundamental trends in fluid–structure interaction, Contemp. Chall. Math. Fluid Dyn. Appl., vol. 1. World Sci. Publ., Hackensack, pp 1–75

  15. Franta M, Málek J, Rajagopal KR (2005) On steady flows of fluids with pressure- and shear-dependent viscosities. Proc R Soc Lond Ser A Math Phys Eng Sci 461(2055):651–670

    Article  MATH  MathSciNet  Google Scholar 

  16. Girault V, Raviart PA (1979) Finite element methods for the Navier–Stokes equations. Springer, Berlin

  17. Gohar R (2001) Elastohydrodynamics, 2nd edn. World Scientific Publishing, Singapore

    Google Scholar 

  18. Hamrock B, Dowson D (1976) Isothermal elastohydrodynamic lubrication of point contacts. Tech. rep, NASA

  19. Hamrock B, Schmid S, Jacobson B (2004) Fundamentals of fluid film lubrication, 2nd edn. Marcel Decker, Inc., New York

    Book  Google Scholar 

  20. Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191(47–48):5537–5552

    Article  MATH  MathSciNet  Google Scholar 

  21. Hart D, Berzins M, Goodyer C, Jimack P (2011) Using adjoint error estimation techniques for elastohydrodynamic lubrication line contact problems. Int J Numer Methods Fluids 67:1559–1570

    Article  MATH  MathSciNet  Google Scholar 

  22. Hertz H (1882) Über die Berührung fester elastischer Körper. Journal für die reine und angewandte Mathematik 92:156–171

    MATH  Google Scholar 

  23. Hirt C, Amsden A, Cook J (1974) An Arbitrary Lagrangian–Eulerian computing method for all flow speeds. J Comput Phys 14:227–469

    Article  MATH  Google Scholar 

  24. Hron J, Turek S (2006) A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in Biomechanics. In: Bungartz JJ, Schäfer M (eds) Fluid–structure interaction. Springer, Berlin, pp 146–170

  25. Hughes TJR, Brooks AN (1982) Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible navier-stokes equation. Comput Methods Appl Mech Eng 32:199–259

    Article  MATH  MathSciNet  Google Scholar 

  26. Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59:85–99

    Google Scholar 

  27. Jubault I, Molimard J, Lubrecht A, Mansot J, Vergne P (2003) In situ pressure and film thickness measurements in rolling/sliding lubricated point contacts. Tribol Lett 15(4):421–429

    Article  Google Scholar 

  28. Knauf S (2011) Numerical simulations for ball bearings considering fluid–structure-interaction problems and non-newtonian fluids with a free boundary. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg.

  29. Kr̃upka I, Hartl M, Lis̃ka M (2005) The influence of contact pressure on central and minimal film thickness within ultrathin film lubricated contacts. J Tribol 127:890–892

    Article  Google Scholar 

  30. Nilsson B, Hansbo P (2011) A Stokes model with cavitation for the numerical simulation of hydrodynamic lubrication. Int J Numer Methods Fluids 67(12):2015–2025

    Article  MATH  MathSciNet  Google Scholar 

  31. Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. 36:9–15

    Article  MATH  MathSciNet  Google Scholar 

  32. Ohno N, Kuwano N, Hirano F (1995) Bulk modulus of solidified oil at high pressure as predominant factor affecting life of trust ball bearings. Tribol Trans 38:285–292

    Google Scholar 

  33. Rajagopal KR, Szeri AZ (2003) On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication. R Soc Lond Proc Ser A Math Phys Eng Sci 459(2039):2771–2786

    Article  MATH  MathSciNet  Google Scholar 

  34. Souli M, Zolesio J (2001) Arbitrary Lagrangian–Eulerian and free surface methods in fluid mechanics. Comput Methods Appl Mech Eng 191(3–5):451–466

    Article  MATH  Google Scholar 

  35. Tezduyar T, Mittal S, Ray S, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity–pressure elements. Comp Methods Appl Mech Eng 95:221–242

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the German Federal Ministry of Education and Research (BMBF) under contract no. 50 YB 0807 (VING II project), which is gratefully acknowledged. Furthermore, we would like to thank our industrial partner Rockwell Collins Germany GmbH for discussions on the physics of ball bearings and for providing the data used in our simulations.

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, Heidelberg University, INF 294, 69120 , Heidelberg, Germany

    Stefan Frei, Thomas Richter & Rolf Rannacher

  2. SeLasCo GmbH, Waldstückerring 44, 76756 , Bellheim, Germany

    Stefan Knauf

Authors
  1. Stefan Knauf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Frei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Richter
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rolf Rannacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Richter.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Knauf, S., Frei, S., Richter, T. et al. Towards a complete numerical description of lubricant film dynamics in ball bearings. Comput Mech 53, 239–255 (2014). https://doi.org/10.1007/s00466-013-0904-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-013-0904-1

Keywords

Search

Navigation