Thermal-mechanical fully coupled analysis of high-speed angular contact ball bearings


The thermal and mechanical properties of angular contact ball bearings are critical to their operating accuracy and service life. On the basis of the theory of dynamics and frictional heat generation, this work establishes a thermal-mechanical fully coupled model. The high-speed motion characteristics, temperature, and dynamic stiffness are analyzed, and the simulation results are compared with the theoretical calculation and experimental results. Then, the temperature and axial deformation of the bearings under different rotation speeds are analyzed, and the laws of mechanical and thermal characteristics of the bearings are obtained. Results show that an increase in the rotation speed of the inner ring causes the temperature of each component to increase, with the contact area of the balls showing the greatest increase in temperature. An increase in rotation speed also increases the axial deformation of the inner ring.

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M :


C :


K :


P :


C(T) :

Specific heat

K(T) :

Heat transfer

Q(T) :

Heat flow load

u :


T :


ω m :

Rotation angular velocity of ball

D b :

Diameter of ball

d m :

Pitch diameter

α i :

Contact angle of ball with inner raceway

α o :

Contact angle of ball with outer raceway

ω :

Angular velocity of inner ring

T :

Frictional stress

\(\dot y\) :

Slip rate

P f :

Friction energy dissipation rate

q :

Friction heat

f :

Proportion of frictional heat flowing into the first surface

η :

Coefficient of conversion of mechanical energy into thermal energy

k :

Thermal conductivity of lubricant

P r :

Prandtl number of lubricant

R e :

Reynolds number

u s :

Surface speed of cage

V 0 :

Kinematic viscosity of lubricating oil

T a :

Ambient temperature

k a :

Thermal conductivity of air

D o :

Outer diameter of bearing

u a :

Flow velocity of air

V a :

Kinematic viscosity of air


  1. [1]

    Q. Han, X. Li and F. L. Chu, Skidding behavior of cylindrical roller bearings under time-variable load conditions, International Journal of Mechanical Science, 135(1) (2018) 203–214.

    Article  Google Scholar 

  2. [2]

    Q. Han, Z. Ding, X. Xu, T. Wang and F. Chu, Stator current model for detecting rolling bearing faults in induction motors using magnetic equivalent circuits, Mechanical Systems and Signal Processing, 131(1) (2019) 554–575.

    Article  Google Scholar 

  3. [3]

    Q. Han, Z. Ding, Z. Qin, T. Wang, X. Xu and F. Chu, A triboelectric rolling ball bearing with self-powering and self-sensing capabilities, Nano Energy, 67(1) (2020) 104277.

    Article  Google Scholar 

  4. [4]

    N. Song, Analysis on Temperature Field and Thermo-stress Coupling of High Speed Angular Contact Ball Bearing with Spin Heat Generation, Jilin University, Jilin, China (2017).

    Google Scholar 

  5. [5]

    T. A. Harris and M. N. Kotzalas, Advanced Concepts of Bearing Technology, 5th Ed., CRC Press, Boca Raton, USA (2006).

    Google Scholar 

  6. [6]

    A. Tedric and M. N. K. Harris, Essential Concepts of Bearing Technology, 5th Ed. CRC Press, Boca Raton, USA (2007).

    Google Scholar 

  7. [7]

    A. Palmgren, Ball and Roller Bearing Engineering, 4th Ed., S. H. Burbank, Philadelphia (1959).

    Google Scholar 

  8. [8]

    Z. De-xing, C. Weifang and L. Miaomiao, An optimized thermal network model to estimate thermal performances on a pair of angular contact ball bearings under oil-air lubrication, Applied Thermal Engineering, 131(1) (2018) 328–339.

    Article  Google Scholar 

  9. [9]

    D. Zheng and W. Chen, Thermal performances on angular contact ball bearing of high-speed spindle considering structural constraints under oil-air lubrication, Tribology International, 109(1) (2017) 593–601.

    Article  Google Scholar 

  10. [10]

    D. Zheng, W. Chen and M. Li, An improved model on forecasting temperature rise of high-speed angular contact ball bearings considering structural constraints, Industrial Lubrication and Tribology, 70(1) (2018) 15–22.

    Article  Google Scholar 

  11. [11]

    J. Takabi and M. M. Khonsari, Experimental testing and thermal analysis of ball bearings, Tribology International, 60(1) (2013) 93–103.

    Article  Google Scholar 

  12. [12]

    P. He, F. Gao, Y. Li, W. Wu and D. Zhang, Study on thermomechanical coupling characteristics of angle contact ball bearing with fix-position preload, Industrial Lubrication and Tribology, 71(1) (2019) 795–802.

    Article  Google Scholar 

  13. [13]

    K. Yan, J. Hong, J. Zhang, W. Mi and W. Wu, Thermaldeformation coupling in thermal network for transient analysis of spindle-bearing system, International Journal of Thermal Sciences, 104(1) (2016) 1–12.

    Article  Google Scholar 

  14. [14]

    A. Zahedi and M. R. Movahhedy, Thermo-mechanical modeling of high speed spindles, Scientia Iranica, 19(2) (2012) 282–293.

    Article  Google Scholar 

  15. [15]

    H. Wang, Y. Cai and H. Wang, A dynamic thermal-mechanical model of the spindle-bearing system, Mechanical Sciences, 8(2) (2017) 277–288.

    Article  Google Scholar 

  16. [16]

    Z. Ye, L. Wang, G. Chen and D. Tang, Analysis of thermo-mechanical coupling of high-speed angular-contact ball bearings, Advances in Mechanical Engineering, 9(6) (2017) 1–14.

    Article  Google Scholar 

  17. [17]

    V. Than, C. Wang, T. Ngo and J. H. Huang, Estimating time-varying heat sources in a high speed spindle based on two measurement temperatures, International Journal of Thermal Sciences, 111(1) (2017) 50–65.

    Article  Google Scholar 

  18. [18]

    L. Cui, C. Cai and Q. Wang, Multi-objective optimization design algorithm of dynamic and thermal performances of high speed spindle bearing, 2016 IEEE Information Technology, Networking, Electronic and Automation Control Conference (2016) 1001–1004.

  19. [19]

    B. Yan, K. Yan, P. Zhang and Y. Zhu, Numerical simulation on the stress distribution of high speed angular contact rolling bearing, 2017 IEEE 7th Annual International Conference on CYBER Technology in Automation, Control, and Intelligent Systems (CYBER) (2017) 317–320.

  20. [20]

    B. Yan, K. Yan, T. Luo, Y. Zhu, B. Q. Li and J. Hong, Thermal coefficients modification of high speed ball bearing by multi-object optimization method, International Journal of Thermal Sciences, 137(1) (2019) 313–324.

    Article  Google Scholar 

  21. [21]

    C. Ma, X. Mei, J. Yang, L. Zhao and H. Shi, Thermal characteristics analysis and experimental study on the high-speed spindle system, The International Journal of Advanced Manufacturing Technology, 79(4) (2015) 469–489.

    Article  Google Scholar 

  22. [22]

    J. Huang, V. Than, T. Ngo and C. Wang, An inverse method for estimating heat sources in a high speed spindle, Applied Thermal Engineering, 105(1) (2016) 65–76.

    Article  Google Scholar 

  23. [23]

    P. Huang, Numerical Calculation Methods of Elastohydrodynamic Lubrication, 1st Ed., Tsinghua University Press, Beijing, China (2013).

    Google Scholar 

  24. [24]

    H. Xu, Numerical Simulation and Optimization Design of Heavy Load Rolling Bearings, 1st Ed., Tsinghua University Press, Beijing, China (2010).

    Google Scholar 

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This work is supported by the National Key Research and Development Program, China (2018YFB2000202).

Author information



Corresponding author

Correspondence to Xuanyu Sheng.

Additional information

Lanwen Wang is a Ph.D. student in Department of Mechanical Engineering, Tsinghua University, Beijing, China. He received his B.E. degree in Shandong University (2019). His research interests include friction, heat transfer and lubrication of bearing.

Xuanyu Sheng is an Associate Professor in Department of Mechanical Engineering, Tsinghua University, Beijing, China. He received his Ph.D. in Tsinghua University (1998). His research interests include friction properties analysis of materials, structural design, seismic analysis and simulation.

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Wang, L., Sheng, X. & Luo, J. Thermal-mechanical fully coupled analysis of high-speed angular contact ball bearings. J Mech Sci Technol 35, 669–678 (2021).

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  • Angular contact ball bearings
  • Dynamic stiffness
  • Thermal characteristics
  • Thermal-mechanical coupling