Research Papers: Contact Mechanics

Four-Point Contact Ball Bearing Model With Deformable Rings

[+] Author and Article Information
Samy Lacroix

e-mail: samy.lacroix@insa-lyon.fr

Daniel Nélias

e-mail: daniel.nelias@insa-lyon.fr
Université de Lyon
UMR CNRS 5259,
INSA Lyon,
Villeurbanne, 69621, France

Alexandre Leblanc

Assistant Professor
Université Artois,
Béthune, 62400, France
e-mail: alexandre.leblanc@univ-artois.fr

Contributed by the Tribology Division of ASME for publication in the Journal of Tribology. Manuscript received November 27, 2012; final manuscript received March 10, 2013; published online May 2, 2013. Assoc. Editor: James R. Barber.

J. Tribol 135(3), 031402 (May 02, 2013) (8 pages) Paper No: TRIB-12-1218; doi: 10.1115/1.4024103 History: Received November 27, 2012; Revised March 10, 2013

In many applications, such as four-point contact slewing bearings or main shaft angular contact ball bearings, the rings and housings are so thin that the assumption of rigid rings does not hold anymore. In this paper, several methods are proposed to account for the flexibility of rings in a quasi-static ball bearing numerical model. The modeling approach consists of coupling a semianalytical approach and a finite element (FE) model to describe the deformation of the rings and housings. The manner in which this weak coupling is made differs depending on how the structural deformation of the ring and housing assemblies is injected into the set of nonlinear geometrical and equilibrium equations in order to solve them. These methods enable us to account for ring ovalization, ring twist, and raceway opening (including change of conformity) since a tulip deformation mode of the ring groove is observed for high contact angles. Either the torus fitting technique or mean displacement computation are used to determine these geometrical parameters. A comparison between the different approaches allows us to study, in particular, the impact of raceway conformity change. The loads used in this investigation are chosen in order that the maximum contact pressure (the Hertz pressure) at the ball-raceway interface remains below 2000 MPa, without any contact ellipse truncation. For the ball bearing example considered here, relative differences of up to 30% on the axial displacement, 10% on the maximum contact pressure, and 10% on the contact angle are observed by comparing rigid and deformable rings for a typical loading representative of the one encountered in operation. Despite the local change of conformity, which becomes significant at high contact angles and for thin ball bearing flanges, it is shown that this hardly affects the internal load distribution. The paper ends with a discussion on how the ring and housing flexibility may affect the loading envelope when the truncation of the contact ellipse is an issue.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Young, W., and Budynas, R., 2002, Roark's Formulas for Stress and Strain, McGraw-Hill, New York.
Harris, T. A., 2001, Rolling Bearing Analysis, 4th ed., Wiley New York.
Cavallaro, G., Nelias, D., and Bon, F., 2005, “Analysis of High-Speed Intershaft Cylindrical Roller Bearing With Flexible Rings,” Tribol. Trans., 48(2), pp. 154–164. [CrossRef]
Yao, T., Chi, Y., and Huang, Y., 2012, “Research on Flexibility of Bearing Rings for Multibody Contact Dynamics of Rolling Bearings,” Procedia Eng., 31, pp. 586–594. [CrossRef]
Leblanc, A., Nelias, D., and Defaye, C., 2009. “Nonlinear Dynamic Analysis of Cylindrical Roller Bearing With Flexible Rings,” J. Sound Vib., 325(1–2), pp. 145–160. [CrossRef]
Daidié, A., Chaib, Z., and Ghosn, A., 2008, “3D Simplified Finite Elements Analysis of Load and Contact Angle in a Slewing Ball Bearing,” ASME J. Mech. Des., 130, p. 082601. [CrossRef]
Smolnicki, T., and Rusiński, E., 2006, “Superelement-Based Modeling of Load Distribution in Large-Size Slewing Bearings,” ASME J. Mech. Des., 129(10), pp. 459–463. [CrossRef]
Bourdon, A., Rigal, J. F., and Play, D., 1999. “Static Rolling Bearing Models in a C.A.D. Environment for the Study of Complex Mechanisms—Part 1: Rolling Bearing Model,” ASME J. Tribol., 121, pp. 205–214. [CrossRef]
Bourdon, A., Rigal, J. F., and Play, D., 1999, “Static Rolling Bearing Models in a C.A.D. Environment for the Study of Complex Mechanisms—Part 2: Complete Assembly Model,” ASME J. Tribol., 121, pp. 215–223. [CrossRef]
Chen, G., and Wen, J., 2012, “Load Performance of Large-Scale Rolling Bearings With Supporting Structure in Wind Turbines,” ASME J. Tribol., 134(4), p. 041105. [CrossRef]
Olave, M., Sagartzazu, X., Damian, J., and Serna, A., 2010, “Design of Four Contact-Point Slewing Bearing With a New Load Distribution Procedure to Account for Structural Stiffness,” ASME J. Mech. Des., 132(2), p. 021006. [CrossRef]
Leblanc, A., and Nelias, D., 2007, “Ball Motion and Sliding Friction in a Four-Contact-Point Ball Bearing,” ASME J. Tribol., 129(4), pp. 801–808. [CrossRef]
Leblanc, A., and Nelias, D., 2008, “Analysis of Ball Bearings With 2, 3 or 4 Contact Points,” Tribol. Trans., 51(3), pp. 372–380. [CrossRef]
Jones, A. B., 1959, “Ball Motion and Sliding Friction in Ball Bearings,” ASME J. Basic Eng., 81(1), pp. 1–12.
Hamrock, B. J., and Anderson, W. J., 1973, “Analysis of an Arched Outer-Race Ball Bearing Considering Centrifugal Forces,” ASME J. Lubr. Technol., 95(3), pp. 265–271. [CrossRef]
Hamrock, B., 1975, “Ball Motion and Sliding Friction in an Arched Outer Race Ball Bearing,” ASME J. Lubr. Technol., 97(2), pp. 202–210. [CrossRef]
Nelias, D., Sainsot, P., and Flamand, L., 1994, “Power Loss of Gearbox Ball Bearing Under Axial and Radial Loads,” Tribol. Trans., 37(1), pp. 83–90. [CrossRef]
Nelias, D. and Yoshioka, T., 1998, “Location of an Acoustic Emission Source in a Radially Loaded Deep Groove Ball-Bearing,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 212(J1), pp. 33–45. [CrossRef]
Johnson, K. L., 1987, Contact Mechanics, Cambridge University Press, Cambridge, England.
Lukács, G., Martin, R., and Marshall, D., 1998, “Faithful Least-Squares Fitting of Spheres, Cylinders, Cones and Tori for Reliable Segmentation,” ECCV’98, Proceedings of the 5th European Conference on Computer Vision, Springer-Verlag, Berlin, Vol. 1, pp. 671–686.
Shakarji, C. M., 1998, “Least-Squares Fitting Algorithms of the NIST Algorithm Testing System,” J. Res. Natl. Inst. Stand. Technol., 103(6), pp. 633–641. [CrossRef]


Grahic Jump Location
Fig. 1

Simplified algorithm-initialization: first estimation of internal parameters, - relevance: numerical check for N-R stability, - convergence: on geometry and N-R variables

Grahic Jump Location
Fig. 2

Position of the three loading points equivalent to the normal force at the ball-ring contact. The gray dashed line represents the deformed shape of the ring, including the contact deformation.

Grahic Jump Location
Fig. 3

Total displacement along the curvilinear abscissa of the raceway for several loadings: one punctual force, three equivalent punctual forces, and the Hertz pressure distribution

Grahic Jump Location
Fig. 4

Geometry of the point selection area for the torus fitting

Grahic Jump Location
Fig. 5

Contact point displacement

Grahic Jump Location
Fig. 6

Contact point displacement for Q = 2000 N and αc = 34 deg

Grahic Jump Location
Fig. 7

Ball bearing cross-section and boundary conditions

Grahic Jump Location
Fig. 8

Normal load on the inner right raceway. Comparison between rigid rings, torus fitting with disk selection (Torus F), simplified torus fitting with disk selection (Torus S), mean displacement with disk selection (Mean D), and mean displacement with sector selection (Mean DS)—Case 3.

Grahic Jump Location
Fig. 9

Radial displacement of the curvature center for the inner and outer rings. Comparison between rigid rings, torus fitting with disk selection (Torus F), simplified torus fitting with disk selection (Torus S), mean displacement with disk selection (Mean D), and mean displacement with sector selection (Mean DS)—Case 3.

Grahic Jump Location
Fig. 10

Contact angle, contact ellipse position (dashed lines), and contact pressure on the inner and outer rings—Case 4

Grahic Jump Location
Fig. 11

Maximum top position of the contact ellipse versus moment Mz for two radial forces Fy and an axial force Fx = 20 kN—Case 5



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