Research Papers: Applications

Rolling-Element Bearing Heat Transfer—Part II: Housing, Shaft, and Bearing Raceway Partial Differential Equation Solutions

[+] Author and Article Information
William M. Hannon

The Timken Company,
North Canton, OH 44720-5450
e-mail: william.hannon@timken.com

The Timken Bearing Syber Analysis Program calculated torque. This program calculates global and local deflection and bearing life, as well as the local rolling element contact stress, film thickness, torque, and power losses. The output of Syber becomes the input to this heat transfer model.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received July 29, 2014; final manuscript received January 6, 2015; published online March 25, 2015. Assoc. Editor: Mihai Arghir.

J. Tribol 137(3), 031103 (Jul 01, 2015) (11 pages) Paper No: TRIB-14-1189; doi: 10.1115/1.4029733 History: Received July 29, 2014; Revised January 06, 2015; Online March 25, 2015

Part I of this three-part series presented a heat transfer rolling-element bearing model. The model is composed of solid conduction partial differential equations (PDEs), control volume formulation for lubricant temperatures, and heat partitioning. The model applies to systems with a shaft, housing, numerous bearings, gears, and various methods of lubrication. Part II, this work, presents a solution to the thermal conduction equations. The raceways are three-dimensional (3D), the shaft and housing models are two-dimensional (2D) and lumped in the third direction. This generalized method applies to ball, cylindrical, spherical, and tapered rolling-element bearings. Semi-analytic solutions are obtained by imposing integral transforms. This approach accounts for the axial and circumferential variations in the bearing load zone and rib heating, as well as the ability to link many bearings and gears within an assembly. The housing and shaft equations are radially lumped. The lumped fluxes account for internal and external convection and radiation, as well as conduction fluxes from contiguous bearings and gears. These equations are solved using a Fourier transform. The 3D bearing raceway solution uses a Fourier transform and a modified Hankel transform. Part III of this series presents additional results and experimental validation.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Harris, T., and Kotzalas, M. N., 2007, Advanced Concepts of Bearing Technology, 4th ed., CRC, Boca Raton, FL, Vol. 2, pp. 191–208.
Hannon, W. M., 2015, “Rolling-Element Bearing Heat Transfer—Part I: Analytic Model,” ASME J. Tribol., 137(3), p. 031101. [CrossRef]
Brown, J. R., and Forster, N. H., 2003, “Carbon-Phenolic Cages for High-Speed Bearings,” Report No. AFRL-PR-WP-TR-2003-2033.
Tarawneh, C. M., Fuentes, A. A., Kypuros, J. A., Navarro, L. A., Vaipan, A. G., and Wilson, B. M., 2012, “Thermal Modeling of a Railroad Tapered-Roller Bearing Using Finite Element Analysis,” ASME J. Therm. Sci. Eng. Appl., 4(3), pp. 1636–1645. [CrossRef]
Wright, B., 2012, “Thermal Behavior of Work Rolls in the Hot Mill Rolling Process,” Ph.D. thesis, Cardiff University, Cardiff, UK.
Hannon, W. M., and Braun, M. J., 2008, The Generalized Universal Reynolds Equation for Variable Property Fluid-Film Lubrication and Variable Geometry Self-Acting Bearings, VDM Verlag Dr. Mueller, Germany.
Pantakar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Company, New York.
Bairi, A. A., 2004, “Three Dimensional Stationary Thermal Behavior of a Ball Bearing,” Int. J. Therm. Sci., 43(6), pp. 561–568. [CrossRef]
Ozisik, M. N., 1981, Boundary Heat Conduction, Dover Publications, Inc., New York.
Hannon, W. M., 2015, “Rolling-Element Bearing Heat Transfer—Part III: Experimental Validation,” ASME J. Tribol., 137(3), p. 031103. [CrossRef]
Abramowitz, M., and Stegun, I. A., 1972, Handbook of Mathematical Function, 10th printing, Dover Publishing, Inc., New York.
McLachlan, N. W., 1941, Bessel Functions for Engineers, Oxford University, London.


Grahic Jump Location
Fig. 1

Rolling-element bearing geometry and dimension definitions

Grahic Jump Location
Fig. 2

Grease-lubricated two bearing single shaft and housing example

Grahic Jump Location
Fig. 3

Bearing one raceway temperature distribution

Grahic Jump Location
Fig. 4

Housing and shaft temperature distribution. (a) Housing Temperature and (b) Shaft Temperature.

Grahic Jump Location
Fig. 5

Graphical determination of the radial eigenvalues




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