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Analysis of static stiffness fluctuation in radially loaded ball and roller bearings

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Abstract

The radial stiffness of rolling bearings is the basis for analyzing the dynamic performance of bearing-rotor systems. The changes of rolling element position may cause continuous change of radial stiffness and relative displacement of inner and outer ring during the operation of the bearing-rotor systems. As a result, the vibration of bearing-rotor systems would be aggravated. In order to accurately study the influence of the changes of rolling element position on radial stiffness, two boundary positions of the inner ring supported by even or odd number of rolling elements are considered in this paper. A mathematical method for rolling bearings modeling and stiffness calculation is proposed based on Hertz elastic contact theory. Then, the differences of radial stiffness under two boundary positions and that of rotor center displacements are studied. The results show that there exist obvious fluctuation of radial stiffness and oscillation of rotor center during the operation of bearing-rotor systems. Moreover, the effect of bearing structure parameters including internal clearance, number of rolling elements on the fluctuation of radial stiffness and oscillation of rotor center has been systematically investigated for ball and roller bearings. Consequently, this paper not only proposes an effective method for the radial stiffness, load distribution and the displacement of rotor center calculation of radially loaded rolling bearings, but also provides a direction for the design of bearing-rotor systems to eliminate vibration.

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  • 13 January 2021

    Journal abbreviated title on top of the page has been corrected to “Arch Appl Mech”

References

  1. Wardle, F.P., Lacey, S.J., Poon, S.Y.: Dynamic and static characteristics of a wide speed range machine-tool spindle. Precis. Eng. 5(4), 175–183 (1983). https://doi.org/10.1016/0141-6359(83)90097-1

    Article  Google Scholar 

  2. Harris, T.A.: Rolling Bearing Analysis, vol. 4. Wiley, New York (2000)

    Google Scholar 

  3. Lim, T.C., Singh, R.: Vibration transmission through rolling element bearings, part I: bearing stiffness formulation. J. Sound Vib. 139(2), 179–199 (1990). https://doi.org/10.1016/0022-460X(90)90882-Z

    Article  Google Scholar 

  4. Lim, T.C., Singh, R.: Vibration transmission through rolling element bearings, part II: system studies. J. Sound Vib. 139(2), 201–225 (1990). https://doi.org/10.1016/0022-460X(90)90883-2

    Article  Google Scholar 

  5. Lazovic, T., Ristivojevic, M., Mitrovic, R.: Mathematical model of load distribution in rolling bearing. FME Trans. 36(4), 189–196 (2008)

    Google Scholar 

  6. Stribeck, R.: Ball bearings for various loads. Trans ASME 29, 420–463 (1907)

    Google Scholar 

  7. Palmgren, A.: Ball and roller bearing engineering. SKF Industries Inc, Philadelphia (1959)

    Google Scholar 

  8. Sjovall, H.: The load distribution within ball and roller bearings under given external radial and axial load. Teknisk Tidskrift, Mek Volume: 9 (1933)

  9. Houpert, L.: A uniform analytical approach for ball and roller bearings calculations. J. Tribol. 119(4), 851–858 (1997). https://doi.org/10.1115/1.2833896

    Article  Google Scholar 

  10. Ricci, M.C.: Internal loading distribution in statically loaded ball bearings subjected to an eccentric thrust load. Math. Prob. Eng. 2009, 1–36 (2009). https://doi.org/10.1155/2009/471804

    Article  MATH  Google Scholar 

  11. Tomović, R.: Calculation of the necessary level of external radial load for inner ring support on q rolling elements in a radial bearing with internal radial clearance. Int. J. Mech. Sci. 60(1), 23–33 (2012). https://doi.org/10.1016/j.ijmecsci.2012.04.002

    Article  Google Scholar 

  12. Ren, X. L., Zhai, J., Ren, G.: Calculation of radial load distribution on ball and roller bearings with positive, negative and zero clearance. Int. J. Mech. Sci. 131–132, 1–7 (2017). https://doi.org/10.1016/j.ijmecsci.2017.06.042

    Article  Google Scholar 

  13. Tomović, R.: Calculation of the boundary values of rolling bearing deflection in relation to the number of active rolling elements. Mech. Mach. Theory 47, 74–88 (2012). https://doi.org/10.1016/j.mechmachtheory.2011.08.006

    Article  Google Scholar 

  14. Jones, A.B.: A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions. ASME J. Basic Eng. 82(2), 309 (1960)

    Article  Google Scholar 

  15. Hernot, X., Sartor, M., Guillot, J.: Calculation of the stiffness matrix of angular contact ball bearings by using the analytical approach. J. Mech. Des. 122(1), 83–90 (2000). https://doi.org/10.1115/1.533548

    Article  Google Scholar 

  16. Cao, Y.Z., Altintas, Y.: A general method for the modeling of spindle-bearing systems. J. Mech. Des. 126(6), 1089–1104 (2004). https://doi.org/10.1115/1.1802311

    Article  Google Scholar 

  17. Gao, S.H., Long, X.H., Meng, G.: Nonlinear response and stability of a spindle system supported by ball bearings. Arch. Appl. Mech. 80(9), 1069–1081 (2009). https://doi.org/10.1007/s00419-009-0358-2

    Article  MATH  Google Scholar 

  18. Wang, N.F., Jiang, D.X., Behdinan, K.: Vibration response analysis of rubbing faults on a dual-rotor bearing system. Arch. Appl. Mech. 87(11), 1891–1907 (2017). https://doi.org/10.1007/s00419-017-1299-9

    Article  Google Scholar 

  19. Liew, H.-V., Lim, T.C.: Analysis of time-varying rolling element bearing characteristics. J. Sound Vib. 283(3–5), 1163–1179 (2005). https://doi.org/10.1016/j.jsv.2004.06.022

    Article  Google Scholar 

  20. Szumiński, P.: Determination of the stiffness of rolling kinematic pairs of manipulators. Mech. Mach. Theory 42(9), 1082–1102 (2007). https://doi.org/10.1016/j.mechmachtheory.2006.09.009

    Article  MATH  Google Scholar 

  21. Guo, Y., Parker, R.G.: Stiffness matrix calculation of rolling element bearings using a finite element/contact mechanics model. Mech. Mach. Theory 51, 32–45 (2012). https://doi.org/10.1016/j.mechmachtheory.2011.12.006

    Article  Google Scholar 

  22. Noel, D., Ritou, M., Furet, B., Le Loch, S.: Complete analytical expression of the stiffness matrix of angular contact ball bearings. J. Tribol. (2013). https://doi.org/10.1115/14024109

    Article  Google Scholar 

  23. Sheng, X., Li, B.Z., Wu, Z.P., Li, H.: Calculation of ball bearing speed-varying stiffness. Mech. Mach. Theory 81, 166–180 (2014). https://doi.org/10.1016/j.mechmachtheory.2014.07.003

    Article  Google Scholar 

  24. Zhang, J.H., Fang, B., Zhu, Y.S., Hong, J.: A comparative study and stiffness analysis of angular contact ball bearings under different preload mechanisms. Mech. Mach. Theory 115, 1–17 (2017). https://doi.org/10.1016/j.mechmachtheory.2017.03.012

    Article  Google Scholar 

  25. Fang, B., Zhang, J.H., Yan, K., Hong, J., Yu Wang, M.: A comprehensive study on the speed-varying stiffness of ball bearing under different load conditions. Mech. Mach. Theory 136, 1–13 (2019). https://doi.org/10.1016/j.mechmachtheory.2019.02.012

    Article  Google Scholar 

  26. Jafari, S.M., Rohani, R., Rahi, A.: Experimental and numerical study of an angular contact ball bearing vibration response with spall defect on the outer race. Arch. Appl. Mech. (2020). https://doi.org/10.1007/s00419-020-01733-z

    Article  Google Scholar 

  27. Cheng, H.C., Zhang, Y.M., Lu, W.J., Yang, Z.: Research on mechanical characteristics of fault-free bearings based on centrifugal force and gyroscopic moment. Arch. Appl. Mech. 90(10), 2157–2184 (2020). https://doi.org/10.1007/s00419-020-01714-2

    Article  Google Scholar 

  28. Petersen, D., Howard, C., Sawalhi, N., Moazen Ahmadi, A., Singh, S.: Analysis of bearing stiffness variations, contact forces and vibrations in radially loaded double row rolling element bearings with raceway defects. Mech. Syst. Sig. Process. 50–51, 139–160 (2015). https://doi.org/10.1016/j.ymssp.2014.04.014

    Article  Google Scholar 

  29. Petersen, D., Howard, C., Prime, Z.: Varying stiffness and load distributions in defective ball bearings: analytical formulation and application to defect size estimation. J. Sound Vib. 337, 284–300 (2015). https://doi.org/10.1016/j.jsv.2014.10.004

    Article  Google Scholar 

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Acknowledgements

This research was funded by the National Science Foundation of China (Grant No. 11872288 and No. 51575425) and the Shaanxi Provincial Natural Science Foundation (Grant No. 2019JM-219).

Funding

This research was funded by the National Science Foundation of China (Grant No. 11872288 and No. 51575425) and the Shaanxi Provincial Natural Science Foundation (Grant No. 2019JM-219).

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Correspondence to Lihua Yang.

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Han, Y., Yang, L. & Xu, T. Analysis of static stiffness fluctuation in radially loaded ball and roller bearings. Arch Appl Mech 91, 1757–1772 (2021). https://doi.org/10.1007/s00419-020-01853-6

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