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Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

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Abstract

Gear disengaging, back-side tooth contact or poor dynamic behavior during operating leads to dynamic instability in planetary gear trains (PGTs). A novel nonlinear dynamic model of PGTs with internal and external gear pairs considering multi-state engagement induced by backlash and contact ratio is established. An improved time-varying meshing stiffness model including temperature stiffness is analytically derived. The time-varying meshing stiffness with temperature effect, friction, backlash, time-varying pressure angle, and time-varying friction arm are to be incorporated into the dynamic model of PGTs. Multi-state engaging behavior is efficiently identified by constructing different Poincaré mappings. A method to calculate dynamic instability is proposed in the time-domain trace. The intrinsic relationship between multi-state engaging and dynamic instability is investigated via multi-section bifurcation plots and phase trajectory topology. The global dynamic instability is revealed based on the bifurcation and evolution of coexistence behavior under the parameter-state synergy. The results show that the multi-state engagement is heavily depending on bifurcation and phase trajectory topology, which whereby affects the dynamic instability. Two special phenomena, complete and incomplete bifurcations, are discovered under parameter-state synergy. Complete bifurcation causes global instability and incomplete bifurcation results in local instability and yields coexistence responses. Incomplete bifurcation brings about new bifurcation branches.

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References

  1. Yu, W., Mechefske, C.K., Timusk, M.: Influence of the addendum modification on spur gear back-side mesh stiffness and dynamics. J. Sound Vib. 389, 183–201 (2017)

    Google Scholar 

  2. Fernandez-del-Rincon, A., Diez-Ibarbia, A., Iglesias, M., et al.: Gear rattle dynamics: lubricant force formulation analysis on stationary conditions. Mech. Mach. Theory 142, 103581 (2019)

    Google Scholar 

  3. Shi, J.-F., Gou, X.-F., Zhu, L.-Y.: Generation mechanism and evolution of five-state meshing behavior of a spur gear system considering gear-tooth time-varying contact characteristics. Nonlinear Dyn. 106, 2035–2060 (2021)

    Google Scholar 

  4. Wang, S., Zhu, R.: Research on dynamics and failure mechanism of herringbone planetary gearbox in wind turbine under gear surface pitting. Eng. Fail. Anal. 146, 107130 (2023)

    Google Scholar 

  5. Liu, X.: Vibration modelling and fault evolution symptom analysis of a planetary gear train for sun gear wear status assessment. Mech. Syst. Signal Process. 166, 108403 (2022)

    Google Scholar 

  6. Yang, C., Li, H., Cao, S.: Unknown fault diagnosis of planetary gearbox based on optimal rank nonnegative matrix factorization and improved stochastic resonance of bistable system. Nonlinear Dyn. 111, 217–242 (2023)

    Google Scholar 

  7. Kahraman, A.: Free torsional vibration characteristics of compound planetary gear sets. Mech. Mach. Theory 36(8), 953–971 (2001)

    MATH  Google Scholar 

  8. Bahk, C.J., Parker, R.G.: Analytical solution for the nonlinear dynamics of planetary gears. J. Comput. Nonlinear Dyn. 6(2), 021007 (2011)

    Google Scholar 

  9. Bahk, C.J., Parker, R.G.: Analytical investigation of tooth profile modification effects on planetary gear dynamics. Mech. Mach. Theory 70, 298–319 (2013)

    Google Scholar 

  10. Ericsona, T.M., Parker, R.G.: Experimental measurement and finite element simulation of elastic-body vibration in planetary gears. Mech. Mach. Theory 160, 104264 (2021)

    Google Scholar 

  11. Cooley, C.G., Parker, R.G.: Mechanical stability of high-speed planetary gears. Int. J. Mech. Sci. 69, 59–71 (2013)

    Google Scholar 

  12. Beinstingel, A., Parker, R.G., Marburg, S.: Experimental measurement and numerical computation of parametric instabilities in a planetary gearbox. J. Sound Vib. 536, 117160 (2022)

    Google Scholar 

  13. Wang, C., Parker, R.G.: Nonlinear dynamics of lumped-parameter planetary gears with general mesh phasing. J. Sound Vib. 523, 116682 (2022)

    Google Scholar 

  14. Li, Z., Wen, B., Peng, Z., et al.: Dynamic modeling and analysis of wind turbine drivetrain considering the effects of non-torque loads. Appl. Math. Model. 83, 146–168 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Liu, J., Li, X., Xia, M.: A dynamic model for the planetary bearings in a double planetary gear. Mech. Syst. Signal Process. 194, 110257 (2023)

    Google Scholar 

  16. Zhang, Q., Wang, X., Shijing, Wu., et al.: Nonlinear characteristics of a multi-degree-of-freedom wind turbine’s gear transmission system involving friction. Nonlinear Dyn. 107, 3313–3338 (2022)

    Google Scholar 

  17. Li, S., Qingming, Wu., Zhang, Z.: Bifurcation and chaos analysis of multistage planetary gear train. Nonlinear Dyn. 75, 217–233 (2014)

    MathSciNet  Google Scholar 

  18. Zhu, W., Shijing, Wu., Wang, X., et al.: Harmonic balance method implementation of nonlinear dynamic characteristics for compound planetary gear sets. Nonlinear Dyn. 81, 1511–1522 (2015)

    MATH  Google Scholar 

  19. Zhiliang, Xu., Wennian, Yu., Shao, Y., et al.: Dynamic modeling of the planetary gear set considering the effects of positioning errors on the mesh position and the corner contact. Nonlinear Dyn. 109, 1551–1569 (2015)

    Google Scholar 

  20. Ryali, L., Talbot, D.: A dynamic load distribution model of planetary gear sets. Mech. Mach. Theory 158, 104229 (2021)

    Google Scholar 

  21. Zhang, C., Wei, J., Wang, F., et al.: Dynamic model and load sharing performance of planetary gear system with journal bearing. Mech. Mach. Theory 151, 103898 (2020)

    Google Scholar 

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

    Google Scholar 

  23. Gu, X., Velex, P.: On the dynamic simulation of eccentricity errors in planetary gears. Mech. Mach. Theory 61, 14–29 (2013)

    Google Scholar 

  24. Qiu, X., Han, Q., Chu, F.: Dynamic modeling and analysis of the planetary gear under pitching base motion. Int. J. Mech. Sci. 141, 31–45 (2018)

    Google Scholar 

  25. Zhang, L., Wang, Y., Kai, Wu., et al.: Dynamic modeling and vibration characteristics of a two-stage closed-form planetary gear train. Mech. Mach. Theory 97, 12–28 (2016)

    Google Scholar 

  26. Kim, W., Lee, J.Y., Chung, J.: Dynamic analysis for a planetary gear with time-varying pressure angles and contact ratios. J. Sound Vib. 331, 883–901 (2012)

    Google Scholar 

  27. Xun, C., Long, X., Hua, H.: Effects of random tooth profile errors on the dynamic behaviors of planetary gears. J. Sound Vib. 415, 91–110 (2018)

    Google Scholar 

  28. Jianjun, T., Hao, Li., Hao, T., et al.: Dynamic modeling and analysis of planetary gear train system considering structural flexibility and dynamic multi-teeth mesh process. Mech. Mach. Theory 186, 105348 (2023)

    Google Scholar 

  29. Shuai, Mo., Ting, Z., Guo-guang, J.: Analytical investigation on load sharing characteristics of herringbone planetary gear train with flexible support and floating sun gear. Mech. Mach. Theory 144, 103670 (2020)

    Google Scholar 

  30. Lai, J., Liu, Y., Xiangyang, Xu., et al.: Dynamic modeling and analysis of Ravigneaux planetary gear set with unloaded floating ring gear. Mech. Mach. Theory 170, 104696 (2022)

    Google Scholar 

  31. Zhang, C., Wei, J., Niu, R., et al.: Similarity and experimental prediction on load sharing performance of planetary gear transmission system. Mech. Mach. Theory 180, 105163 (2023)

    Google Scholar 

  32. Tatar, A., Schwingshackl, C.W., Friswell, M.I.: Dynamic behaviour of three-dimensional planetary geared rotor systems. Mech. Mach. Theory 134, 39–56 (2019)

    Google Scholar 

  33. 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, 1695–1709 (2017)

    MathSciNet  Google Scholar 

  34. Sun, Y., Ma, H., Huangfu, Y., et al.: A revised time-varying mesh stiffness model of spur gear pairs with tooth modifications. Mech. Mach. Theory 129, 261–278 (2018)

    Google Scholar 

  35. Huangfu, Y., Chen, K., Ma, H., et al.: Meshing and dynamic characteristics analysis of spalled gear systems: a theoretical and experimental study. Mech. Syst. Sig. Process. 139, 106640 (2020)

    Google Scholar 

  36. 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)

    Google Scholar 

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

    Google Scholar 

  38. Cao, Z., Chen, Z., Jiang, H.: Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness. Nonlinear Dyn. 99, 1227–1241 (2020)

    Google Scholar 

  39. Chen, W., Lei, Y., Yao, Fu.: A study of effects of tooth surface wear on time-varying mesh stiffness of external spur gear considering wear evolution process. Mech. Mach. Theory 155, 104055 (2021)

    Google Scholar 

  40. Shen, Z., Qiao, B., Yang, L., et al.: Evaluating the influence of tooth surface wear on TVMS of planetary gear set. Mech. Mach. Theory 136, 206–223 (2019)

    Google Scholar 

  41. Dai, He., Longa, X., Chen, F.: An improved analytical model for gear mesh stiffness calculation. Mech. Mach. Theory 159, 104262 (2021)

    Google Scholar 

  42. Oztürk, V.Y., Cigeroglu, E., Ozgüven, H.N.: Ideal tooth profile modifications for improving nonlinear dynamic response of planetary gear trains. J. Sound Vib. 500, 116007 (2021)

    Google Scholar 

  43. Pedrero, J.I., Pleguezuelos, M., Sánchez, M.B.: Analytical model for meshing stiffness, load sharing, and transmission error for helical gears with profile modification. Mech. Mach. Theory 185, 105340 (2023)

    Google Scholar 

  44. Wang, J., Yang, J., Lin, Y.: Analytical investigation of profile shifts on the mesh stiffness and dynamic characteristics of spur gears. Mech. Mach. Theory 167, 104529 (2022)

    Google Scholar 

  45. Chakroun, Ala Eddin, Hammami, Chaima, Hammami, Ahmed, et al.: Gear mesh stiffness of polymer-metal spur gear system using generalized Maxwell model. Mech. Mach. Theory 175, 104934 (2022)

    Google Scholar 

  46. Mo, S., Li, Y., Luo, B., et al.: Research on the meshing characteristics of asymmetric gears considering the tooth profile deviation. Mech. Mach. Theory 175, 104926 (2022)

    Google Scholar 

  47. Chen, Z., Shao, Y.: Dynamic simulation of planetary gear with tooth root crack in ring gear. Eng. Fail. Anal. 31, 8–18 (2013)

    Google Scholar 

  48. Yang, Yi., Tang, J., Niaoqing, Hu., et al.: Research on the time-varying mesh stiffness method and dynamic analysis of cracked spur gear system considering the crack position. J. Sound Vib. 548, 117505 (2023)

    Google Scholar 

  49. Zheng, X., Luo, W., Yumei, Hu., et al.: Analytical approach to mesh stiffness modeling of high-speed spur gears. Int. J. Mech. Sci. 224, 107318 (2022)

    Google Scholar 

  50. Abruzzo, M., Beghini, M., Santus, C., Presicce, F.: A dynamic model combining the average and the local meshing stiffnesses and based on the static transmission error for spur gears with profile modification. Mech. Mach. Theory 180, 105139 (2023)

    Google Scholar 

  51. Shi, J.-F., Gou, X.-F., Zhu, L.-Y.: Five-state engaging model and dynamics of gear-rotor-bearing system based on time-varying contact analysis considering gear temperature and lubrication. Appl. Math. Model. 112, 47–77 (2022)

    MathSciNet  MATH  Google Scholar 

  52. Marafona, J.D., Marques, P.M., Martins, R.C., et al.: Mesh stiffness models for cylindrical gears: a detailed review. Mech. Mach. Theory 166, 104472 (2021)

    Google Scholar 

  53. Li, Z., Chen, Z., Zhai, W.: Nonlinear dynamic characteristics of a spur gear pair considering extended tooth contact and coupling effect between gear neighboring teeth. Nonlinear Dyn. 111, 2395–2414 (2023)

    Google Scholar 

  54. Li, Z., Zhu, L., Chen, S., Chen, Z., Gou, X.: Establishment of the integrated safety domain for spur gear pair and its safety characteristics in the domain. Mech. Syst. Signal Process. 178, 109288 (2022)

    Google Scholar 

  55. Shi, J.-F., Gou, X.-F., Jin, W.-Y., et al.: Multi-meshing-state and disengaging-proportion analyses of a gear-bearing system considering deterministic-random excitation based on nonlinear dynamics. J. Sound Vib. 544, 117360 (2023)

    Google Scholar 

  56. Shen, Z., Qiao, B., Yang, L., et al.: Fault mechanism and dynamic modeling of planetary gear with gear wear. Mech. Mach. Theory 155, 104098 (2021)

    Google Scholar 

  57. Tsai, S.-J., Huang, G.-L., Ye, S.-Y.: Gear meshing analysis of planetary gear sets with a floating sun gear. Mech. Mach. Theory 84, 145–163 (2015)

    Google Scholar 

  58. Wang, J., Shan, Z., Chen, S.: Nonlinear dynamics analysis of multifactor low-speed heavy-load gear system with temperature effect considered. Nonlinear Dyn. 110, 257–279 (2022)

    Google Scholar 

  59. Sun, Z., Chen, S., Zehua, Hu., et al.: Vibration response analysis of a gear-rotor-bearing system considering steady-state temperature. Nonlinear Dyn. 107, 477–493 (2022)

    Google Scholar 

  60. Guo, Yi., Parker, R.G.: Dynamic analysis of planetary gears with bearing clearance. J. Comput. Nonlinear Dyn. 7, 041002–041011 (2012)

    Google Scholar 

  61. Liu, C., Qin, D., Lim, T.C., et al.: Dynamic characteristics of the herringbone planetary gear set during the variable speed process. J. Sound Vib. 333, 6498–6515 (2014)

    Google Scholar 

  62. Shi, J.F., Gou, X.F., Zhu, L.Y.: 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)

    Google Scholar 

  63. Mason, J.F., Piiroinen, P.T., Wilson, R.E., et al.: Basins of attraction in non-smooth models of gear rattle. Int. J. Bifur. Chaos 19, 203–224 (2009)

    Google Scholar 

  64. Mason, J.F., Piiroinen, P.T.: Interactions between global and grazing bifurcations in an impacting system. Chaos 21, 013113 (2011)

    MathSciNet  Google Scholar 

  65. Mason, J.F., Piiroinen, P.T.: The effect of codimension-two bifurcations on the global dynamics of a gear model. J. Appl. Dyn. Syst. 8, 1694–1711 (2009)

    MathSciNet  MATH  Google Scholar 

  66. Gou, X.F., Zhu, L.Y., Chen, D.L.: Bifurcation and chaos analysis of spur gear pair in two-parameter plane. Nonlinear Dyn. 79, 2225–2235 (2015)

    Google Scholar 

  67. de Souza, S.L.T., Caldas, I.L.: Basins of attraction and transient chaos in a gear-rattling model. J. Vib. Control 7, 849–862 (2001)

    MATH  Google Scholar 

  68. Mo, S., Zhang, Y., Luo, B., et al.: The global behavior evolution of non-orthogonal face gear-bearing transmission system. Mech. Mach. Theory 175, 104969 (2022)

    Google Scholar 

  69. Zhu, L.-Y., Li, Z.-F., Gou, X.-F., 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)

    Google Scholar 

  70. Shi, J.-F., Gou, X.-F., Zhu, L.-Y.: Bifurcation of multi-stable behaviors in a two-parameter plane for a non-smooth nonlinear system with time-varying parameters. Nonlinear Dyn. 100, 3347–3365 (2020)

    Google Scholar 

  71. Yang, D.C.H., Lin, J.Y.: Hertzian damping, tooth friction and bending elasticity in gear impact dynamics. J. Mech. Des. 109(2), 189–196 (1987)

    Google Scholar 

  72. Tian, X.: Dynamic simulation for system response of gearbox including localized gear faults. Mast. Abstr. Int. 43(3), 0979 (2004)

    Google Scholar 

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Funding

This paper is financially funded by the National Natural Science Foundation of China (No. 52365017), the National Natural Science Foundation of China (Grant No. 12102159), the National Natural Science Foundation of China (Grant No. 51665029), the 2022 Higher Education Innovation Fund Project in Gansu Province of China (Grant No. 2022A-018).

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Authors and Affiliations

  1. School of Mechanical and Electronical Engineering, Lanzhou University of Technology, Lanzhou, 730050, China

    Xue-zong Bai, Jian-fei Shi, De-wang Li & Zong-wen An

  2. Northwest Branch of CNG New Energy Holdings Co., Ltd, Lanzhou, 730070, China

    Hu-zi Qiu

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  1. Xue-zong Bai
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Bai, Xz., Qiu, Hz., Shi, Jf. et al. Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects. Nonlinear Dyn 111, 20843–20868 (2023). https://doi.org/10.1007/s11071-023-08932-7

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