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Research Papers: Applications

Load-Displacement Relationships for Ball and Spherical Roller Bearings

[+] Author and Article Information
L. Houpert

TIMKEN Europe,
B.P. 60089, Colmar,
Cedex 68002, France
e-mail: luc.houpert@timken.com

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received September 5, 2014; final manuscript received November 4, 2014; published online December 12, 2014. Assoc. Editor: Dong Zhu.

J. Tribol 137(2), 021102 (Apr 01, 2015) (17 pages) Paper No: TRIB-14-1216; doi: 10.1115/1.4029042 History: Received September 05, 2014; Revised November 04, 2014; Online December 12, 2014

Analytical relationships for calculating three rolling element bearing loads (Fx, Fy, and Fz) and two tilting moments (My and Mz) as a function of three relative race translations (dx, dy, and dz) and two relative race tilting angles (dθy and dθz) have been given in a previous paper. The previous approach was suggested for any rolling element bearing type, although it has been recognized that the assumption of a constant rolling element-race contact angle is not well supported by deep groove ball bearings (DGBB) or angular contact ball bearings (ACBB). The new approach described in this paper addresses the latter weaknesses by accounting for the variation of the contact angle on the most loaded ball and also shows that misalignment effects on spherical roller bearing (SRB) loads are negligible. Comparisons between the simplified approach (option 1) and the “enhanced” numerical approach (option 2, which requires a summation of the load components on each ball with the appropriate contact angle included) is made, showing a good correlation as long as the relative misalignment remains reasonable or occurs in the plane corresponding to maximum radial displacement. Option 2 can, however, be recommended since it is easy to program and quite accurate at any misalignment level. Other pros and cons of both options are described. As in the previous paper, a full coupling between all displacements and forces, as well as roller and raceway crown radii, are considered, meaning that Hertzian point contact stiffness is used in roller bearings at low load with a smooth transition toward Hertzian line contact as the load increases. This approach is particularly recommended for programming the rolling element bearing behavior in any finite element analysis or multibody system dynamic tool, since only two nodes are considered: one for the inner race (IR) center, usually connected to a shaft, and another node for the outer race (OR) center, connected to the housing.

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References

Houpert, L., 1997, “A Uniform Analytical Approach for Ball and Roller Bearings,” ASME J. Tribol., 119(4), pp. 851–857. [CrossRef]
Houpert, L., 2014, “An Enhanced Study of the Load-Displacement Relationships for Rolling Element Bearings,” ASME J. Tribol., 136(1), p. 011105. [CrossRef]
Houpert, L., 2001, “An Engineering Approach to Hertzian Contact Elasticity– Part I,” ASME J. Tribol., 123(3), pp. 582–588. [CrossRef]
Houpert, L., 2001, “An Engineering Approach to Non-Hertzian Contact Elasticity–Part II,” ASME J. Tribol., 123(3), pp. 589–594. [CrossRef]
Eschmann, P., Hasbargen, L., and Weigand, K., 1978, Ball and Roller Bearings, Theory, Design and Application, 2nd ed., R.Oldenbourg, ed., Verlag, München, Germany.
Harris, T. A., 1991, Rolling Bearing Analysis, 3rd ed., Wiley-Interscience Publication, Wiley, New York.
Hoeprich, M. R., 1986, “Numerical Procedure for Designing Rolling Element Contact Geometry as a Function of Load Cycle,” SAE Technical Paper Series No. 850764. [CrossRef]
Houpert, L., 1995, “Prediction of Bearing, Gear and Housing Performances,” Proceedings of the Rolling Bearing Practice Today Seminar, Institution of Mechanical Engineers, London, UK.
Jones, A. B., 1960, “A General Theory for Elastically Constrained Ball and Radial Roller Bearings Under Arbitrary Load and Speed Conditions,” ASME J. Basic Eng., 82(2), pp. 309–320. [CrossRef]
Sjöval, S., 1933, “The Load Distribution Within Ball and Roller Bearings Under Given External Radial and Axial Load,” Tek. Tidskr., Mek., p. 9.
Tripp, J., 1985, “Hertzian Contact in Two and Three Dimensions,” NASA Technical Paper No. 2473.
Cretu, S., 1996, “Initial Plastic Deformation of Cylindrical Roller Generatrix Stress Distribution Analysis and Fatigue Life Tests,” Acta Tribol., 4(1–2), pp. 1–6.
Houpert, L., and Merckling, J., 1998, “A Successful Transition From Physically Measured to Numerically Simulated Bearings, Shafts, Gears and Housing Deflections in a Transmission,” Proceedings of the Conference GPC’98 Global PowerTrain Congress New Powertrain Materials and Processes, Detroit, MI, Vol. 4, pp. 131–137.
Hauswald, T., and Houpert, L., 2000, “Numerical and Experimental Simulations of Performances of Bearing System, Shaft and Housing; Account for Global and Local Deformations,” Proceeding of the Conference ‘SIA Seminar Fiabilité experimentale, Paris, France.
Houpert, L., 1999, “Numerical and Analytical Calculations in Ball Bearings,” Proceedings of the 8th European Space Mechanism and Tribology Symposium, Toulouse, France.
Houpert, L., 2010, “CAGEDYN: A Contribution to Roller Bearing Dynamic Calculations; Part I: Basic Tribology Concepts,” STLE Tribol. Trans., 53(1), pp. 1–9. [CrossRef]
Houpert, L., 2010, “CAGEDYN: A Contribution to Roller Bearing Dynamic Calculations; Part II: Description of the Numerical Tool and its Outputs,” STLE Tribol. Trans., 53(1), pp. 10–21. [CrossRef]
Houpert, L., 2010, “CAGEDYN: A Contribution to Roller Bearing Dynamic Calculations; Part III: Experimental Validation,” STLE Tribol. Trans., 53(6), pp. 848–859. [CrossRef]

Figures

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Fig. 1

Basic TRB geometry, relative race displacements, and total roller race geometrical interference

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Fig. 2

Description of a DGBB geometry and displacements (not to scale)

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Fig. 3

The new coordinate frame used for estimating ψr using Ref. [2]

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Fig. 4

Error relative to the exact numerical integrals

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Fig. 5

DGBB pure radial load case (dx = 0, dy = −0.05 mm)

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Fig. 6

DGBB general case without misalignment

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Fig. 7

General case with misalignment

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Fig. 8

DGBB case with a large misalignment in another plane

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Fig. 9

DGBB case with a large misalignment in the same plane as the radial displacement

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Fig. 10

DGBB large axial displacement case

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Fig. 11

Low load, narrow load zone case

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Fig. 12

Low load, narrow load zone case in another radial plane

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Fig. 13

Radial load variation versus applied direction

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Fig. 14

Basic SRB geometry

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Fig. 15

Graphical presentation of Δα = α − αn

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