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Research Papers

Analytical Determination of Rolling Friction Coefficient of Angular Bearings

[+] Author and Article Information
Kosmol Jan

Department of Machine Technology,
Silesian University of Technology,
Konarskiego 18a Street,
Gliwice 44-100, Poland
e-mail: jkosmol@polsl.pl

Manuscript received January 20, 2017; final manuscript received June 22, 2017; published online December 18, 2017. Assoc. Editor: Tony Schmitz.

J. Manuf. Sci. Eng 140(2), 021002 (Dec 18, 2017) (7 pages) Paper No: MANU-17-1032; doi: 10.1115/1.4037230 History: Received January 20, 2017; Revised June 22, 2017

This paper presents a proposal about a method for analytical rolling friction coefficient determination. Basis of the proposal was an assumption: rolling friction coefficient is proportional to the semi-axes of the contact ellipses. This paper shows how to compute the semi-axes of the contact ellipses. This paper shows example results of contact loads identification using finite element method (FEM) and example of experimental results of friction torque as motion resistance of an angular bearing. Comparison of analytical and experimental determination of rolling friction coefficient was examined too.

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References

Jiang, S. , and Mao, H. , 2010, “ Investigation of Variable Optimum Preload for Machine Tool Spindle,” Int. J. Mach. Tools Manuf., 50(1), pp. 19–28. [CrossRef]
Musiał, J. , and Styp-Rekowski, M. , 1999, “ Analytical and Experimental Method of Resistance Motion Coefficient Determination in Rolling Friction,” Problems of Unconventional Bearing Configuration, Łódź, Poland, pp. 59–65 (in Polish).
Styp-Rekowski, M. , 1999, “ Problems of Internal Resistance Motion in Angle Bearing,” Doctoral thesis, University Press of Technology and Agriculture Academy, Bydgoszcz, Poland (in Polish).
Harris, T. , and Kotzalas, M. , 2013, Rolling Bearing Analysis, Advanced Concepts of Bearing Technology, 5th ed., CRC Press, Boca Raton, FL, p. 352.
Chen, J.-S. , and Hwang, Y.-W. , 2006, “ Centrifugal Force Induced Dynamics of Motorized High-Speed Spindle,” Int. J. Adv. Manuf. Technol., 30, pp. 10–19. [CrossRef]
Antoine, F. , Abba, G. , and Molinari, A. , 2005, “ A New Proposal for Explicit Angle Calculation in Angular Contact Ball Bearing,” ASME J. Mech. Des., 128(2), pp. 468–478. [CrossRef]
Jędrzejewski, J. , and Kwaśny, W. , 2010, “ Modelling of Angular Contact Ball Bearings and Axial Displacements for High-Speed Spindles,” CIRP Ann. - Manuf. Technol., 59(1), pp. 377–383. [CrossRef]
Krzemiński-Freda, H. , 1985, Rolling Bearings, PWN, Warszawa, Poland, p. 233 (in Polish).
Harris, T. , and Kotzalas, M. , 2013, Rolling Bearing Analysis, Essential Concepts of Bearing Technology, 5th ed., CRC Press, Boca Raton, FL, p. 371.
Liao, N. T. , and Lin, J. F. , 2002, “ Ball Bearing Skidding Under Radial and Axial Loads,” Mech. Mach. Theory, 37(1), pp. 91–113. [CrossRef]
Kosmol, J. , 2016, Determination of Motion Resistances in High-Speed Spindle Angular Bearings (Monograph), Silesian University of Technology, Gliwice, Poland, p. 93.
Kania, M. , 2015, “ Experimental Research of Motion Resistances of Angular Bearing,” Masters thesis, Silesian University of Technology, Gliwice, Poland, p. 49 (in Polish).
Kosmol, J. , and Stawik, K. , 2016, “ Influence of Contact Stiffness and Friction Coefficient on Contact Loads in Angular Bearing,” Modelowanie Inżynierskie, 58(27), pp. 65–74 (in Polish).
Kosmol, J. , and Stawik, K. , 2016, “ Simulation Investigation of Rotational Speed Influence on Contact Loads in Angular Rolling Bearing,” Inżynieria Masz. (Wrocław), 1, pp. 28–42 (in Polish). [CrossRef]

Figures

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Fig. 1

Bearing load condition and selected kinematic parameters: M1(T), Ms*, Mother, Mv—friction torque, Fa, Fext(r), Fext(a)—thrust and external loads, Fc—centrifugal load, Mg—gyroscopic torque, Ms—spinning torque, Qi, Qo—contact loads, ω—angular speed of inner race, ωs—ball angular speed due to spinning, ωB—ball angular speed due to rolling motion, αi, αo—contact angels, i—inner raceway, o—outer raceway, a—axial, r—radial, s—spinning, and v—viscosity

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Fig. 2

Traditional model of rolling friction (a) and its application for the rolling bearing (b); fk—rolling friction coefficient, Tk—rolling friction load, Mk—rolling friction torque, M1(T)—friction torque due to rolling movement on raceways, i—inner raceway, and o—outer raceway

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Fig. 3

Hypothetical distribution of contact loads and contact deflection on the contact point of the ball and raceway: for a resting ball (a), ball in motion (b), and a contact ellipse obtained from FEM modeling (c): V—speed, fk—rolling friction coefficient, ξ—constant, Tk—rolling friction load, R—reaction, Q—normal load, Mk—rolling friction torque, N—resultant load, a—semimajor axis, and b—semiminor axis

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Fig. 4

Typical Hertzian model of contact zone: p(x,y), pmax—surface thrust in the geometrical center of the ellipse, a, b—semi-axes, δ—contact displacement, dT—elementary friction load, and D—diameter

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Fig. 5

Diagram assisting in determination of parameters a*, b* [11]: F(ξ)—curvature difference

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Fig. 6

Dependence of the rolling friction coefficient fk as a function of the contact load Q; fk(sym)—according to Eqs. (9) and (23) and fk(asym)—according to Eqs. (25) and (23)

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Fig. 7

Measurement concept for angular bearing friction torques: ns—rotational speed of the spindle

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Fig. 8

Impact of preload on the motion resistances (bearingwithout grease); Mexp—measured torque and Mother—hypothetic torque due to cage

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Fig. 9

Dependence between contact loads Qi and Qo and preload for the examined angular bearing type FAG B7013-E [11,13]

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Fig. 10

Relation between rolling friction coefficient fk and preload of examined angular bearing type FAG B7013-E; fk(best)—performance of friction coefficient for best tuned of coefficient ξ(9), and Model (23)sym and Model (25)asym—friction coefficient computed using models (23) and (25)

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Fig. 11

Example results of FEM modeling: (a) a trace of contact displacement for preload 1000 N, (b) distribution of normal stresses along the line a–a, and (c) distribution of normal stresses along the line b–b

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