Skip to main content
SpringerLink
Log in

Nonlinear dynamic analysis of defective rolling element bearing using Higuchi’s fractal dimension

  • Published:
Sādhanā Aims and scope Submit manuscript

Abstract

In the present study, localized surface defects are modelled in inner race and outer race, and nonlinear dynamic behaviour of the system has been observed and quantified using Higuchi’s fractal dimensions. The Hertzian contact among the rollers and races, clearance and nonlinear damping are considered as sources of nonlinearity. Dynamic responses show system behaviour as periodic, quasi-periodic and chaotic at different rotor speeds. The onset of chaotic motion is identified using Poincaré maps and Higuchi’s fractal dimensions. The system shows low peak-to-peak amplitude of vibration responses at higher speeds in the presence of defects. Results also indicate that Higuchi’s fractal dimensions can be effectively used as a diagnostic tool for health monitoring.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4

(Reconstructed from [43])

Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11

Similar content being viewed by others

Abbreviations

N :

number of observations in time series data

FD est :

estimated FD of the signal

FD th :

theoretical FD of the signal

k :

nonlinear stiffness, N/mm

L u :

useful length of the roller, mm

δ :

deformation of the roller at contact points, mm

Q :

contact force, N

W :

vertical radial load, N

\( F_{{\theta_{j} }} \) :

contact force at jth roller, N

θ j :

angular position of the jth roller, radian

R cr :

radial clearance, mm

X, Y :

state space variable

\( \mathop {P_{{d_{j} }} }\limits^{ \to } \) :

nonlinear dissipative force, N

U f :

unbalanced force, N

N r :

number of rollers

ω cg :

angular velocity of the cage, rad/s

ω ir :

angular velocity of the inner race, rad/s

δ s :

additional deflection due to defect (spall), mm

R ir :

radius of the inner race, mm

R or :

radius of the outer race, mm

dw :

defect width, mm

\( \phi_{dir} \) :

angle of defect corresponding to inner race, radian

\( \phi_{dor} \) :

angle of defect corresponding to outer race, radian

References

  1. Sharma A, Upadhyay N, Kankar P K and Amarnath M 2018 Nonlinear dynamic investigations on rolling element bearings: a review. Adv. Mech. Eng. 10: 1–15

    Article  Google Scholar 

  2. Gupta P K 1979 Dynamics of rolling element bearings-part I: cylindrical roller bearing analysis. J. Lubr. Technol. Trans. ASME 101: 293–304

    Article  Google Scholar 

  3. Gupta P K 1979 Dynamics of rolling element bearings-part II: cylindrical roller bearing results. J. Lubr. Technol. Trans. ASME 101: 305–311

    Article  Google Scholar 

  4. McFadden P D and Smith J D 1984 Model for the vibration produced by a single point defect in a rolling element bearing. J. Sound Vib. 96: 69–82

    Article  Google Scholar 

  5. McFadden P D and Smith J D 1985 The vibration produced by multiple point defects in a rolling element bearing. J. Sound Vib. 98: 263–273

    Article  Google Scholar 

  6. Tandon N and Choudhury A 1997 An analytical model for the prediction of the vibration response of rolling element bearings due to a localized defect. J. Sound Vib. 205: 275–292

    Article  Google Scholar 

  7. Tiwari M and Gupta K 2000 Dynamic response of an unbalanced rotor supported on ball bearings. J. Sound Vib. 238: 757–779

    Article  Google Scholar 

  8. Harsha S P 2005 Nonlinear dynamic analysis of an unbalanced rotor supported by roller bearing. Chaos Soliton Fract. 26: 47–66

    Article  Google Scholar 

  9. Lioulios A N and Antoniadis I A 2006 Effect of rotational speed fluctuations on the dynamic behaviour of rolling element bearings with radial clearances. Int. J. Mech. Sci. 48: 809–829

    Article  Google Scholar 

  10. Harsha S P, Nataraj C and Kankar P K 2006 The effect of ball waviness on nonlinear vibrations associated with rolling element bearings. Int. J. Acoust. Vib. 11: 56–66

    Google Scholar 

  11. Kankar P K, Sharma S C and Harsha S P 2011 Fault diagnosis of high speed rolling element bearings due to localized defects using response surface method. J. Dyn. Syst. Meas. Control Trans. ASME 133: 031007-1-14

    Article  Google Scholar 

  12. Arslan H and Aktürk N 2008 An investigation of rolling element vibrations caused by local defects. J. Tribol. Trans. ASME 130: 041101-1-12

    Article  Google Scholar 

  13. Ashtekar A, Sadeghi F and Stacke L 2008 A new approach to modeling surface defects in bearing dynamics simulations. J. Tribol. Trans. ASME 130: 041103-1-8

    Article  Google Scholar 

  14. Patil M S, Mathew J, Rajendrakumar P K and Desai S 2010 A theoretical model to predict the effect of localized defect on vibrations associated with ball bearing. Int. J. Mech. Sci. 52: 1193–1201

    Article  Google Scholar 

  15. Leblanc A, Nelias D and Defaye C 2009 Nonlinear dynamic analysis of cylindrical roller bearing with flexible rings. J. Sound Vib. 325: 145–160

    Article  Google Scholar 

  16. Sinou J J 2009 Non-linear dynamics and contacts of an unbalanced flexible rotor supported on ball bearings. Mech. Mach. Theory 44: 1713–1732

    Article  Google Scholar 

  17. Bai C, Zhang H and Xu Q 2010 Experimental and numerical studies on nonlinear dynamic behaviour of rotor system supported by ball bearings. J. Eng. Gas Turbines Power 132: 082502-1-5

    Article  Google Scholar 

  18. Patel V N, Tandon N and Pandey R K 2010 A dynamic model for vibration studies of deep groove ball bearings considering single and multiple defects in races. J. Tribol. Trans. ASME 132: 041101-1-10

    Article  Google Scholar 

  19. Ghafari S H, Abdel-Rahman E M, Golnaraghi F and Ismail F 2010 Vibrations of balanced fault-free ball bearings. J. Sound Vib. 329: 1332–1347

    Article  Google Scholar 

  20. Kankar P K, Sharma S C and Harsha S P 2012 Vibration based performance prediction of ball bearings caused by localized defects. Nonlinear Dyn. 69: 847–875

    Article  MathSciNet  Google Scholar 

  21. Shao Y, Liu J and Ye J 2014 A new method to model a localized surface defect in a cylindrical roller-bearing dynamic simulation. Proc. Inst. Mech. Eng. J. J. Eng. Tribol. 228: 140–159

    Article  Google Scholar 

  22. Sharma A, Amarnath M and Kankar P K 2014 Effect of varying the number of rollers on dynamics of a cylindrical roller bearing. In: Proceedings of the ASME 2014 International Design and Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE 2014), vol. 8: 26th Conference on Mechanical Vibration and Noise, Buffalo, New York, USA, August 17–20, Paper no. DETC2014-34824: V008T11A067, 8 pages

  23. Logan D and Mathew J 1996 Using the correlation dimension for vibration fault diagnosis of rolling element bearings-I: basic concepts. Mech. Syst. Signal Process. 10: 241–250

    Article  Google Scholar 

  24. Logan D and Mathew J 1996 Using the correlation dimension for vibration fault diagnosis of rolling element bearings-II: selection of experimental parameters. Mech. Syst. Signal Process. 10: 251–264

    Article  Google Scholar 

  25. Jiang J D, Chen J and Qu L S 1999 The application of correlation dimension in gearbox condition monitoring. J. Sound Vib. 223: 529–541

    Article  Google Scholar 

  26. Yang J, Zhang Y and Zhu Y 2007 Intelligent fault diagnosis of rolling element bearing based on SVMs and fractal dimension. Mech. Syst. Signal Process. 21: 2012–2024

    Article  Google Scholar 

  27. Ghafari S H, Golnaraghi F and Ismail F 2008 Effect of localized faults on chaotic vibration of rolling element bearings. Nonlinear Dyn. 53: 287–301

    Article  Google Scholar 

  28. Kappaganthu K and Nataraj C 2011 Nonlinear modeling and analysis of a rolling element bearing with a clearance. Commun. Nonlinear Sci. Numer. Simul. 16: 4134–4145

    Article  Google Scholar 

  29. Gupta T C, Gupta K and Sehgal D K 2011 Instability and chaos of a flexible rotor ball bearing system: an investigation on the influence of rotating imbalance and bearing clearance. J. Eng. Gas Turbines Power 133: 082501-11

    Article  Google Scholar 

  30. Sharma A, Amarnath M and Kankar P K 2015 Effect of unbalanced rotor on the dynamics of cylindrical roller bearings. In: Mechanisms and Machine Science, Proceedings of the 9th IFToMM International Conference on Rotor Dynamics, Milan, Italy, September 22–25, Switzerland: Springer International Publishing, vol. 21, pp. 1653–1663

    Chapter  Google Scholar 

  31. Sarkar N and Chaudhuri B B 1992 An efficient approach to estimate fractal dimension of textural images. Pattern Recogn. 25: 1035–1041

    Article  Google Scholar 

  32. Huang Q, Lorch J R and Dubes R C 1994 Can the fractal dimension of images be measured? Pattern Recogn. 27: 339–349

    Article  Google Scholar 

  33. Esteller R, Vachtsevanos G, Echauz J and Litt B 2001 A comparison of waveform fractal dimension algorithms. IEEE Trans. Circuits-I 48: 177–183

    Article  Google Scholar 

  34. Polychronaki G E, Ktonas P Y, Gatzonis S, Siatouni A, Asvestas P A, Tsekou H, Sakas D and Nikita K S 2010 Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection. J. Neural Eng. 7: 1–18

    Article  Google Scholar 

  35. Upadhyay R, Manglick A, Reddy D K, Padhy P K and Kankar P K 2015 Channel optimization and nonlinear feature extraction for electroencephalogram signals classification. Comput. Electr. Eng. 45: 222–234

    Article  Google Scholar 

  36. Mandelbrot B B 1967 How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156: 636–638

    Article  Google Scholar 

  37. Higuchi T 1988 Approach to an irregular time series on the basis of the fractal theory. Physica D 31: 277–283

    Article  MathSciNet  Google Scholar 

  38. Petrosian A 1995 Kolmogorov complexity of finite sequences and recognition of different preictal EEG patterns. In: Proceedings of the Eighth IEEE Symposium on Computer-Based Medical Systems, Lubbock, Texas, USA, June 9–10, pp. 212–217

  39. Katz M 1988 Fractals and the analysis of waveforms. Comput. Biol. Med. 18: 145–156

    Article  Google Scholar 

  40. Kantelhardt J W, Zschiegner S A, Koscielny-Bunde E, Havlin S, Bunde A and Stanley H E 2002 Multifractal detrended fluctuation analysis of nonstationary time series. Physica A 316: 87–114

    Article  Google Scholar 

  41. Chang-Jian C W 2010 Strong nonlinearity analysis for gear-bearing system under nonlinear suspension—bifurcation and chaos. Nonlinear Anal. Real World Appl. 11: 1760–1774

    Article  MathSciNet  Google Scholar 

  42. Brändlein J, Eschmann P, Hasbargen L and Weigand K 2000 Ball and roller bearings: theory, design and application, 3rd ed. Chichester: Wiley

    Google Scholar 

  43. Wensing J A 1998 On the dynamics of ball bearings. PhD Thesis, University of Twente, Enschede, The Netherlands

  44. Kuhnell B T 2004 Wear in rolling element bearings and gears—how age and contamination affect them. Mach. Lubr. 9. https://www.machinerylubrication.com/Read/664/wear-bearings-gears

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Dayalbagh Educational Institute, Agra, 282005, India

    Aditya Sharma

  2. Mechanical Engineering Discipline, PDPM Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, 482005, India

    M Amarnath

  3. Discipline of Mechanical Engineering, Indian Institute of Technology, Indore, 453552, India

    Pavan Kumar Kankar

Authors
  1. Aditya Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M Amarnath
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pavan Kumar Kankar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavan Kumar Kankar.

Appendix A

Appendix A

For a rolling element bearing, having rotating inner race, various corresponding frequencies are

$$ {\text{cage}}\;{\text{frequency}}\;(\omega_{c} ) = \frac{{\omega_{s} }}{2}\left[ {1 - \left( {\frac{d}{D}} \right)\cos \beta } \right]{\text{rad}}/{\text{s}} $$
(1)

outer race malfunction frequency or ball passage frequency

$$ (\omega_{bp} ) = \frac{{N_{r} \omega_{s} }}{2}\left[ {1 - \left( {\frac{d}{D}} \right)\cos \beta } \right]{\text{rad}}/{\text{s}} $$
(2)

inner race malfunction frequency or wave passage frequency

$$ (\omega_{wp} ) = \frac{{N_{r} \omega_{s} }}{2}\left[ {1 + \left( {\frac{d}{D}} \right)\cos \beta } \right]{\text{rad}}/{\text{s}} $$
(3)

and rolling element malfunction frequency or rolling element spin frequency or wave passage frequency of rolling element

$$ (\omega_{wpb} ) = \frac{{D\omega_{s} }}{d}\left[ {1 - \left( {\frac{{d^{2} }}{{D^{2} }}} \right)\cos^{2} \beta } \right]{\text{rad}}/{\text{s}} $$
(4)

where Nr is a number of rolling elements, \( \omega_{s} \)is shaft rotational frequency, d is rolling element diameter, D is pitch circle diameter and β is contact angle.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sharma, A., Amarnath, M. & Kankar, P.K. Nonlinear dynamic analysis of defective rolling element bearing using Higuchi’s fractal dimension. Sādhanā 44, 76 (2019). https://doi.org/10.1007/s12046-019-1060-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12046-019-1060-x

Keywords

Search

Navigation