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Study on safety characteristics of the spur gear pair considering time-varying backlash in the established multi-level safety domains

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Abstract

Multi-state meshing characteristics induced by backlash and contact ratio in the gear pair are an unsolved puzzle about its relationship to safety characteristics of the gear transmission system. A nonlinear dynamics model of spur gear pair considering time-varying backlash, multi-state meshing, contact ratio and other time-varying parameters is improved. The time-varying backlash comprehensively determined by static backlash, tooth elastic deformation, thermal deformation and oil film thickness is calculated to identify the multi-state meshing of the system. The safety characteristics of the system are classified into nine safety levels according to the influence of meshing states on operation stability. Nine safety domains corresponding to the safety levels are established, respectively. An algorithm is proposed to calculate the safe basin of the gear transmission system in established safety domains. The evolution of safety characteristics in established safety domains is investigated from a local and global perspective by safe basin, nonlinear analysis methods, bifurcation and safety dendrograms and statistical methods. The results show that the value of backlash is affected by meshing states and working conditions of the system. Time-varying backlash is important in identifying multi-state meshing and establishing multi-level safety domains. The safety levels of system responses can be judged by established multi-level safety domains, and the safety characteristics of the system are affected by system parameters, motions and bifurcations. It provides a theoretical basis for the prediction of meshing states and the design of system parameters.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

References

  1. Chen, S., Tang, J., Luo, C., et al.: Nonlinear dynamic characteristics of geared rotor bearing systems with dynamic backlash and friction. Mech. Mach. Theory 46, 466–478 (2011)

    Article  Google Scholar 

  2. Wang, G., Chen, L., Yu, L., et al.: Research on the dynamic transmission error of a spur gear pair with eccentricities by finite element method. Mech. Mach. Theory 109, 1–13 (2017)

    Article  Google Scholar 

  3. Liu, H., Zhang, C., Xiang, C.: Tooth profile modification based on lateral-torsional-rocking coupled nonlinear dynamic model of gear system. Mech. Mach. Theory 105, 606–619 (2016)

    Article  Google Scholar 

  4. Shi, J., Gou, X., Zhu, L.: Calculation of time-varying backlash for an involute spur gear pair. Mech. Mach. Theory 152, 103956 (2020)

    Article  Google Scholar 

  5. Shi, J., Gou, X., Zhu, L.: Generation mechanism and evolution of five-state meshing behavior of a spur gear system considering gear-tooth time-varying contact characteristics. Nonlinear Dynamic 106(3), 2035–2060 (2021). https://doi.org/10.1007/s11071-021-06891-5

    Article  Google Scholar 

  6. Shi, J., Gou, X., Zhu, L.: Modeling and analysis of a spur gear pair considering multi-state mesh with time-varying parameters and backlash. Mech. Mach. Theory 134, 582–603 (2019)

    Article  Google Scholar 

  7. Luo, Y., Baddour, N., Liang, M.: Effects of gear center distance variation on time varying mesh stiffness of a spur gear pair. Eng. Fail. Anal. 75, 37–53 (2017)

    Article  Google Scholar 

  8. Dai, H., Long, X., Chen, F., et al.: An improved analytical model for gear mesh stiffness calculation. Mech. Mach. Theory 159, 104262 (2021)

    Article  Google Scholar 

  9. Xiong, Y., Huang, K., Xu, F., et al.: Research on the influence of backlash on mesh stiffness and the nonlinear dynamics of spur gears. Appl. Sci. 5, 1029 (2019)

    Article  Google Scholar 

  10. Chen, Z., Zhou, Z., Zhai, W., et al.: Improved analytical calculation model of spur gear mesh excitations with tooth profile deviations. Mech. Mach. Theory 149, 103838 (2020)

    Article  Google Scholar 

  11. Chen, Z., Ning, J., Wang, K.: An improved dynamic model of spur gear transmission considering coupling effect between gear neighboring teeth. Nonlinear Dyn. 106(1), 339–357 (2021)

    Article  Google Scholar 

  12. Chen, Z., Shao, Y.: Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack. Mech. Mach. Theory 62, 63–74 (2013)

    Article  Google Scholar 

  13. Chen, Z., Zhang, J., Zhai, W., et al.: Improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack. Eng. Fail. Anal. 82, 72–81 (2017)

    Article  Google Scholar 

  14. Ma, H., Zeng, J., Feng, R., et al.: An improved analytical method for mesh stiffness calculation of spur gears with tip relief. Mech. Mach. Theory 98, 64–80 (2016)

    Article  Google Scholar 

  15. Ma, H., Feng, M., Li, Z., et al.: Time-varying mesh characteristics of a spur gear pair considering the tip-fillet and friction. Meccanica 52(7), 1695–1709 (2017)

    Article  MathSciNet  Google Scholar 

  16. Liang, X., Zhang, H., Liu, L.: The influence of tooth pitting on the mesh stiffness of a pair of external spur gears. Mech. Mach. Theory 106, 1–15 (2016)

    Article  Google Scholar 

  17. Long, Z., Lee, S., Kim, J.: Estimation of survival probability for a ship in beam seas using the safe basin. Ocean Eng. 37, 418–424 (2010)

    Article  Google Scholar 

  18. Yang, S.: Parameter-induced fractal erosion of the safe basin in a softening duffing oscillator. J. Vibroeng. 18, 3329–3336 (2016)

    Article  Google Scholar 

  19. Zhu, Z., Li, X., Xu, J.: Stochastic nonlinear dynamic characteristics and safe basin of Li-doped graphene impacted by hydrogen atoms. Int. J. Hydrogen Energy 40, 12889–12896 (2015)

    Article  Google Scholar 

  20. Rega, G., Lenci, S.: Identifying, evaluating, and controlling dynamical integrity measures in non-linear mechanical oscillators. Nonlinear Anal. 63, 902–914 (2005)

    Article  MathSciNet  Google Scholar 

  21. Belardinelli, P., Sajadi, B., Lenci, S., et al.: Global dynamics and integrity of a micro-plate pressure sensor. Commun. Nonlinear Sci. Numer. Simul. 69, 432–444 (2019)

    Article  Google Scholar 

  22. Shang, H., Xu, J.: Delayed feedbacks to control the fractal erosion of safe basins in a parame- trically excited system. Chaos Solitions Fractals 41, 1880–1896 (2009)

    Article  Google Scholar 

  23. Shi, J., Gou, X., Zhu, L.: Bifurcation and erosion of safe basin for a spur gear system. Int. J. Bifurcation and Chaos 28, 1830048 (2018)

    Article  MathSciNet  Google Scholar 

  24. Erdem, U.: Examination of the stability of trawlers in beam seas by using safe basins. Ocean Eng. 38, 1908–1915 (2011)

    Article  Google Scholar 

  25. Zhu, L., Li, Z., Gou, X., et al.: Analysis of safety characteristics by nonlinear dynamics and safety basin methods for the spur gear pair in the established teeth contact safety domain. Mech. Syst. Signal Process. 158, 107718 (2021)

    Article  Google Scholar 

  26. Zhu, L., Li, Z., Gou, X., et al.: Evolutionary mechanism of safety performance for spur gear pair based on meshing safety domain. Nonlinear Dynamic 104, 215–239 (2021)

    Article  Google Scholar 

  27. Sánchez, B., Pleguezuelos, M., Pedrero, I.: Enhanced model of load distribution along the line of contact for non-standard involute external gears. Meccanica 48, 527–543 (2013)

    Article  Google Scholar 

  28. Pedrero, J., Pleguezuelos, M., Artés, M.: Load distribution model along the line of contact for involute external gears. Mech. Mach. Theory 45, 780–794 (2010)

    Article  Google Scholar 

  29. Tian, X.: Dynamic Simulation for system response of gearbox including localized gear faults, pp. 22–30. University of Alberta, Alberta (2004)

    Google Scholar 

  30. Sainsot, P., Velex, P., Duverger, O.: Contribution of gear body to tooth deflections a new bidi- mensional analytical formula. J. Mech. Des. 126, 748–752 (2004)

    Article  Google Scholar 

  31. Masjedi, M., Khonsari, M.: Film thickness and asperity load formulas for line-contact elastohy- drodynamic lubrication with provision for surface roughness. J. Tribol. 134, 011503 (2012)

    Article  Google Scholar 

  32. Blok, H.: The flash temperature concept. Wear 6, 483–494 (1963)

    Article  Google Scholar 

Download references

Acknowledgements

This investigation is financially supported by the Open Project of State Key Laboratory of Traction Power, Southwest Jiaotong University, China (Grant No. TPL2101), by the National Natural Science Foundation of China (Grant No. 52022083, 51365025).

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Authors and Affiliations

  1. State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031, China

    Zheng-Fa Li, Shi-Qian Chen & Zai-Gang Chen

  2. School of Mechanical Engineering, Tiangong University, Tianjin, 300387, China

    Ling-Yun Zhu & Xiang-Feng Gou

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  1. Zheng-Fa Li
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  2. Ling-Yun Zhu
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  3. Shi-Qian Chen
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  4. Zai-Gang Chen
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  5. Xiang-Feng Gou
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Correspondence to Zai-Gang Chen or Xiang-Feng Gou.

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Li, ZF., Zhu, LY., Chen, SQ. et al. Study on safety characteristics of the spur gear pair considering time-varying backlash in the established multi-level safety domains. Nonlinear Dyn 109, 1297–1324 (2022). https://doi.org/10.1007/s11071-022-07467-7

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  • DOI: https://doi.org/10.1007/s11071-022-07467-7

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