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Research Papers: Applications

Load–Displacement Relationship of a Ball Bearing With Axial, Radial, and Angular Displacements for Both the Inner and Outer Rings

[+] Author and Article Information
Hiroyuki Ohta

Department of Mechanical Engineering,
Graduate School of Engineering,
Nagaoka University of Technology,
1603-1 Kamitomika,
Nagaoka, Niigata 940-2188, Japan
e-mail: ohta@mech.nagaokaut.ac.jp

Tomoya Sakaguchi

CAE Department,
Automotive Business HQ,
NTN Corporation,
1578 Higashi-Kaizuka,
Iwata, Shizuoka 438-8510, Japan
e-mail: tomoya_sakaguchi@ntn.co.jp

Masaharu Uchiumi

Japan Aerospace Exploration Agency,
1 Koganesawa, Kimigaya,
Kakuda, Miyagi 981-1525, Japan
e-mail: uchiumi.masaharu@jaxa.jp

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received November 20, 2015; final manuscript received February 17, 2016; published online July 20, 2016. Assoc. Editor: Xiaolan Ai.

J. Tribol 139(1), 011103 (Jul 20, 2016) (8 pages) Paper No: TRIB-15-1417; doi: 10.1115/1.4033136 History: Received November 20, 2015; Revised February 17, 2016

This paper deals with the load–displacement relationship of a ball bearing with axial, radial, and angular displacements for both the inner and outer rings. First, the expressions for the load–displacement relationship of ball bearings with any number of balls under the combined axial, radial, and moment loads were presented by using a system in which both the inner and outer rings are allowed to move in the axial, radial, and angular directions. Second, the presented expressions were compared with Jones' expressions (which are typical conventional expressions for the load–displacement relationship), then the range of application of Jones's expressions were elucidated. Third, the relative axial displacement, the relative radial displacement, and the relative angular displacement of a miniature ball bearing type 692 under the combined axial, radial, and moment loads were calculated. Finally, it was shown that the relative angular displacement in the case with no inner ring angular displacement is Ri/Ro times the relative angular displacement in the case with no outer ring angular displacement, in which Ri and Ro are the radii of the inner and outer race curvature center loci.

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References

Beatty, R. F. , and Rowan, B. F. , 1982, “ Determination of Ball Bearing Dynamic Stiffness,” Workshop, Rotor Dynamic Instability Problems in High Performance Turbomachinery, NASA Conference Publication 2250, pp. 98–104.
Okayasu, A. , Ohta, T. , Azuma, T. , Fujita, T. , and Aoki, H. , 1990, “ Vibration Problem in the LE-7 LH2 Turbo Pump,” AIAA Paper No. 90-2250.
Butner, M. F. , Murphy, B. T. , and Akian, R. A. , 1991, “ Influence of Mounting Compliance and Operating Conditions on the Radial Stiffness of Ball Bearings,” ASME Rotating Mach. Veh. Dyn., 35, pp. 155–162.
Bugra, H. , Ertas, B. H. , and Vance, J. M. , 2004, “ Effect of Static and Dynamic Misalignment on Ball Bearing Radial Stiffness,” J. Propul. Power, 20(4), pp. 634–647. [CrossRef]
Jones, A. B. , 1946, Analysis of Stresses and Deflections, Vols. 1 and 2, New Departure Division, GMC, Bristol, CT.
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Houpert, L. , 2015, “ Load-Displacement Relationships for Ball and Spherical Roller Bearings,” ASME J. Tribol., 137(2), p. 021102. [CrossRef]
Aoki, S. , 1961, “ Study on the Radial Ball Bearing,” Report of the Government Mechanical Laboratory, pp. 14–23, Report No. 44.
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Hertz, H. , 1895, Gesammelte Werke, Band 1, Johann Ambrosius Barth, Leipzig, Germany.

Figures

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Fig. 1

Coordinate system and ball position

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Fig. 2

Ball-race contact [5]

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Fig. 3

Displacements of the inner and outer rings due to combined axial, radial, and moment loads

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Fig. 4

Loci of race curvature centers before displacements (at reference position)

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Fig. 5

Loci of race curvature centers after displacements

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Fig. 6

Ri/Ro of 76 commercial ball bearings with bore diameters 1.5–130 mm

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Fig. 7

Solutions of (hi − ho), (ki − ko), and (Riαi − Roαo) for the calculation example

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Fig. 8

Relative angular displacement (αi − αo) for the calculation example

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Fig. 9

(αi − αo)αi=0/(αi − αo)αo = 0 for the calculation example

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