Skip to main content
SpringerLink
Log in

Pitchfork and Hopf bifurcations of geared systems with nonlinear suspension in permanent contact regime

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Gears are important mechanical parts with various industrial applications. Many researchers have investigated the complex nonlinear behavior of geared systems by studying the effect of time-varying mesh stiffness, clearance between the gears in mesh, radial clearance in the bearings, and bending of the supporting shafts. Most of these studies assume that the gear set operates under lightly loaded operational conditions, where the separation of the teeth in mesh occurs and the nonlinearity caused by the clearance between the gears in mesh has the major influence on the dynamic response of the system. Alternatively, in this work it is assumed that the transmitting load is great enough that gears in mesh do not separate, and consequently the clearance between the teeth does not participate in the dynamic response of the system. Then analytical and numerical techniques are used specifically to investigate the effect of the nonlinearity of the shafts on the dynamic behavior of the system. The results show that the nonlinear suspension has a significant influence on the creation of nontrivial equilibria and limit cycle within the parametrically unstable tongues which, for the right range of the parameters, can affect the rate of amplitude detonation and stabilization of the system.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Azimi, M.: Parametric frequency analysis of Mathieu–Duffing equation. Int. J. Bifurc. Chaos 31(12), 2150181 (2021). https://doi.org/10.1142/s0218127421501819

    Article  MathSciNet  MATH  Google Scholar 

  2. Azimi, M.: Parametric stability of geared systems with linear suspension in permanent contact regime. Nonlinear Dyn. (2021). https://doi.org/10.1007/s11071-021-06943-w

    Article  Google Scholar 

  3. Blankenship, G., Kahraman, A.: Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity. J. Sound Vib. 185(5), 743–765 (1995)

    Article  Google Scholar 

  4. Cao, Z., Chen, Z., Jiang, H.: Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness. Nonlinear Dyn. 99(2), 1227–1241 (2019). https://doi.org/10.1007/s11071-019-05348-0

    Article  Google Scholar 

  5. Cao, Z., Shao, Y., Rao, M., Yu, W.: Effects of the gear eccentricities on the dynamic performance of a planetary gear set. Nonlinear Dyn. 91(1), 1–15 (2018)

    Article  Google Scholar 

  6. Chang-Jian, C.W.: Nonlinear analysis for gear pair system supported by long journal bearings under nonlinear suspension. Mech. Mach. Theory 45(4), 569–583 (2010). https://doi.org/10.1016/j.mechmachtheory.2009.11.001

    Article  MATH  Google Scholar 

  7. Chang-Jian, C.W.: Nonlinear dynamic analysis for bevel-gear system under nonlinear suspension-bifurcation and chaos. Appl. Math. Modell. 35(7), 3225–3237 (2011). https://doi.org/10.1016/j.apm.2011.01.027

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Q., Zhou, J., Khushnood, A., Wu, Y., Zhang, Y.: Modelling and nonlinear dynamic behavior of a geared rotor-bearing system using tooth surface microscopic features based on fractal theory. AIP Adv. 9(1), 015201 (2019). https://doi.org/10.1063/1.5055907

    Article  Google Scholar 

  9. Cveticanin, L.: Free vibration of a Jeffcott rotor with pure cubic non-linear elastic property of the shaft. Mech. Mach. Theory 40(12), 1330–1344 (2005)

    Article  Google Scholar 

  10. Dadon, I., Koren, N., Klein, R., Bortman, J.: A realistic dynamic model for gear fault diagnosis. Eng. Fail. Anal. 84, 77–100 (2018). https://doi.org/10.1016/j.engfailanal.2017.10.012

    Article  Google Scholar 

  11. Farshidianfar, A., Saghafi, A.: Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems. Nonlinear Dyn. 75(4), 783–806 (2013). https://doi.org/10.1007/s11071-013-1104-4

    Article  MathSciNet  Google Scholar 

  12. Halse, C.K., Wilson, R.E., di Bernardo, M., Homer, M.E.: Coexisting solutions and bifurcations in mechanical oscillators with backlash. J. Sound Vib. 305(4–5), 854–885 (2007)

    Article  MathSciNet  Google Scholar 

  13. Huang, K., Cheng, Z., Xiong, Y., Han, G., Li, L.: Bifurcation and chaos analysis of a spur gear pair system with fractal gear backlash. Chaos Solitons Fractals 142, 110387 (2021). https://doi.org/10.1016/j.chaos.2020.110387

    Article  MathSciNet  Google Scholar 

  14. Huang, K., Yi, Y., Xiong, Y., Cheng, Z., Chen, H.: Nonlinear dynamics analysis of high contact ratio gears system with multiple clearances. J. Braz. Soc. Mech. Sci. Eng. (2020). https://doi.org/10.1007/s40430-020-2190-0

    Article  Google Scholar 

  15. Kumar, J.A., Sundar, D.S.: A numerical study on vibration control of a nonlinear Jeffcott rotor via Bouc-wen model. FME Trans. 47(1), 190–194 (2019)

    Article  Google Scholar 

  16. Li, Z., Peng, Z.: Nonlinear dynamic response of a multi-degree of freedom gear system dynamic model coupled with tooth surface characters: a case study on coal cutters. Nonlinear Dyn. 84(1), 271–286 (2015). https://doi.org/10.1007/s11071-015-2475-5

    Article  Google Scholar 

  17. Litak, G., Friswell, M.I.: Vibration in gear systems. Chaos Solitons Fractals 16(5), 795–800 (2003)

    Article  Google Scholar 

  18. Liu, J., Wang, S., Zhou, S., Wen, B.: Nonlinear behavior of a spur gear pair transmission system with backlash. J. Vibroeng. 16(8), 3922–3938 (2014)

    Google Scholar 

  19. Liu, J., Zhou, S., Wang, S.: Nonlinear dynamic characteristic of gear system with the eccentricity. J. Vibroeng. 17(5), 2187–2198 (2015)

    Google Scholar 

  20. Łuczko, J.: Chaotic vibrations in gear mesh systems. J. Theor. Appl. Mech. 46(4), 879–896 (2008)

    Google Scholar 

  21. Margielewicz, J., Gaska, D., Litak, G.: Modelling of the gear backlash. Nonlinear Dyn. 97(1), 355–368 (2019). https://doi.org/10.1007/s11071-019-04973-z

    Article  Google Scholar 

  22. Margielewicz, J., Gaska, D., Wojnar, G.: Numerical modelling of toothed gear dynamics. Sci. J. Silesian Univ. Technol. Ser. Transp. 97, 105–115 (2017). https://doi.org/10.20858/sjsutst.2017.97.10

    Article  Google Scholar 

  23. Mason, J.F., Piiroinen, P.T., Wilson, R.E., Homer, M.E.: Basins of attraction in nonsmooth models of gear rattle. Int. J. Bifurc. Chaos 19(01), 203–224 (2009). https://doi.org/10.1142/s021812740902283x

    Article  MathSciNet  Google Scholar 

  24. Morrison, T.M., Rand, R.H.: 2:1 resonance in the delayed nonlinear Mathieu equation. Nonlinear Dyn. 50(1–2), 341–352 (2007). https://doi.org/10.1007/s11071-006-9162-5

    Article  MathSciNet  MATH  Google Scholar 

  25. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Hoboken (1995). https://doi.org/10.1002/9783527617586

    Book  MATH  Google Scholar 

  26. Raghothama, A., Narayanan, S.: Bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method. J. Sound Vib. 226(3), 469–492 (1999)

    Article  Google Scholar 

  27. Rigaud, E., Sabot, J.: Effect of elasticity of shafts, bearings, casing and couplings on the critical rotational speeds of a gearbox. arXiv preprint arXiv:physics/0701038 (2007)

  28. Saghafi, A., Farshidianfar, A.: An analytical study of controlling chaotic dynamics in a spur gear system. Mech. Mach. Theory 96, 179–191 (2016). https://doi.org/10.1016/j.mechmachtheory.2015.10.002

    Article  Google Scholar 

  29. Shabaneh, N., Zu, J.W.: Nonlinear dynamic analysis of a rotor shaft system with viscoelastically supported bearings. J. Vib. Acoust. 125(3), 290–298 (2003)

    Article  Google Scholar 

  30. Shen, Y., Yang, S., Liu, X.: Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method. Int. J. Mech. Sci. 48(11), 1256–1263 (2006)

    Article  Google Scholar 

  31. Strogatz, S.H.: Nonlinear Dynamics and Chaos. CRC Press, Boca Raton (2018). https://doi.org/10.1201/9780429492563

    Book  Google Scholar 

  32. Theodossiades, S., Natsiavas, S.: Non-linear dynamics of gear-pair systems with periodic stiffness and backlash. J. Sound Vib. 229(2), 287–310 (2000)

    Article  Google Scholar 

  33. Theodossiades, S., Natsiavas, S.: Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash. Chaos Solitons Fractals 12(13), 2427–2440 (2001)

    Article  Google Scholar 

  34. Wang, J., Li, R., Peng, X.: Survey of nonlinear vibration of gear transmission systems. Appl. Mech. Rev. 56(3), 309–329 (2003)

    Article  Google Scholar 

  35. Wang, J., Wang, H., Guo, L.: Analysis of effect of random perturbation on dynamic response of gear transmission system. Chaos Solitons Fractals 68, 78–88 (2014). https://doi.org/10.1016/j.chaos.2014.08.004

    Article  MATH  Google Scholar 

  36. Wang, J., Zhang, W., Long, M., Liu, D.: Study on nonlinear bifurcation characteristics of multistage planetary gear transmission for wind power increasing gearbox. IOP Conf. Ser. Mater. Sci. Eng. 382, 042007 (2018). https://doi.org/10.1088/1757-899x/382/4/042007

    Article  Google Scholar 

  37. Warminski, J.: Regular and chaotic vibrations of a parametrically and self-excited system under internal resonance condition. Meccanica 40(2), 181–202 (2005). https://doi.org/10.1007/s11012-005-3306-4

    Article  MathSciNet  MATH  Google Scholar 

  38. Warmiński, J., Litak, G., Szabelski, K.: Synchronisation and chaos in a parametrically and self-excited system with two degrees of freedom. Nonlinear Dyn. 22(2), 125–143 (2000)

    Article  Google Scholar 

  39. Wojnarowski, J., Margielewicz, J.: Nonlinear dynamics of single-stage gear transmission. In: IFToMM World Congress on Mechanism and Machine Science, pp. 1007–1016. Springer (2019)

  40. Xia, Y., Wan, Y., Chen, T.: Investigation on bifurcation and chaos control for a spur pair gear system with and without nonlinear suspension. In: 2018 37th Chinese Control Conference (CCC). IEEE (2018). https://doi.org/10.23919/chicc.2018.8484012

  41. Xiong, Y., Huang, K., Xu, F., Yi, Y., Sang, M., Zhai, H.: Research on the influence of backlash on mesh stiffness and the nonlinear dynamics of spur gears. Appl. Sci. 9(5), 1029 (2019). https://doi.org/10.3390/app9051029

    Article  Google Scholar 

  42. Zhou, S., Liu, J., Li, C., Wen, B.: Nonlinear behavior of a spur gear pair transmission system with backlash. J. Vibroeng. 16(8), 3850–3861 (2014)

    Google Scholar 

Download references

Funding

This study is not funded by any organization or company.

Author information

Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, 85721, USA

    Mohsen Azimi

Authors
  1. Mohsen Azimi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Optional

Corresponding author

Correspondence to Mohsen Azimi.

Ethics declarations

Conflict of interest

The author declare that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Consent to participate

The author warrants that the work has not been published before in any form except as a preprint, that the work is not being concurrently submitted to and is not under consideration by another publisher, that the persons listed above are listed in the proper order, and that no author entitled to credit has been omitted.

Consent for publication

The author hereby transfers to the publisher the copyright of the Work. As a result, the publisher shall have the exclusive and unlimited right to publish the work throughout the World.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Azimi, M. Pitchfork and Hopf bifurcations of geared systems with nonlinear suspension in permanent contact regime. Nonlinear Dyn 107, 3339–3363 (2022). https://doi.org/10.1007/s11071-021-07110-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-07110-x

Keywords

Search

Navigation