Nonlinear Dynamic Modelling and Analysis for a Spur Gear Pair Considering Tooth Profile Deviation Based on Tooth Contact Analysis



Transmission error is an important excitation source of gear transmission system. In many previous dynamic models, the transmission error is often treated as a simple harmonic function, ignoring the difference among transmission errors caused by different deviations. In fact, each kinds of gear deviation may lead to transmission error with different characteristics, which has different influence on the dynamic characteristics of the system. To analyze the effect of different deviations on dynamic characteristics, it is necessary to explore the characteristics of each deviation and transmission error.


This paper takes the profile deviation as the research object and, by analyzing its characteristics, builds the mathematical model of spur gear tooth surface including tooth profile deviation. Based on tooth contact analysis (TCA), the characteristics of transmission errors caused by different factors of tooth profile deviation are analyzed and then the influence of different factors of tooth profile deviation on the dynamic characteristics of spur gear system is discussed. To verify the rationality of the dynamic model, the simulation results are compared with the results in the previous literature.


The simulation results show that the effect of these factors of tooth profile deviation on the system is obviously different and it is necessary to consider their comprehensive effect for improving the accuracy of dynamic models. The modelling and analysis method based on TCA in this paper has practical guiding significance for analyzing the influence of different deviations caused by manufacturing and assembling on the dynamic characteristics of gear system.

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  1. 1.

    Gregory R, Harris S, Munro R (1963) Dynamic behaviour of spur gears. Proc Inst Mech Eng 178:207–218

    Article  Google Scholar 

  2. 2.

    Özgüven HN, Houser DR (1988) Mathematical models used in gear dynamics—a review. J Sound Vib 121:383–411

    Article  Google Scholar 

  3. 3.

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

    Article  Google Scholar 

  4. 4.

    Parker R, Vijayakar S, Imajo T (2000) Non-linear dynamic response of a spur gear pair: modelling and experimentalcomparisons. J Sound Vib 237:435–455

    Article  Google Scholar 

  5. 5.

    Velex P (2012) On the modelling of spur and helical gear dynamic behaviour. In: Gokcek M (ed) Mechanical engineering. Rijeka, Croatia, pp 75–106

    Google Scholar 

  6. 6.

    Remmers E (1987) Gear mesh excitation spectra for arbitrary tooth spacing errors, load and design contact ratio. J Mech Des 100:715–722

    Google Scholar 

  7. 7.

    Kahraman A, Singh R (1990) Non-linear dynamics of a spur gear pair. J Sound Vib 142:49–75

    Article  Google Scholar 

  8. 8.

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

    Article  Google Scholar 

  9. 9.

    Chen Q, Ma YB, Huang SW et al (2014) Research on gears’ dynamic performance influenced by gear backlash based on fractal theory. Appl Surf Sci 313:325–332

    Article  Google Scholar 

  10. 10.

    Saghafi A, Farshidianfar A (2016) An analytical study of controlling chaotic dynamics in a spur gear system. Mech Mach Theory 96:179–191

    Article  Google Scholar 

  11. 11.

    Chen Y, Zhu RP, Jin GH, Xiong YP (2019) Influence of crack depth on dynamic characteristics of spur gear system. J Vib Eng Technol 7:227–233

    Article  Google Scholar 

  12. 12.

    Yi Y, Huang K, Xiong YS, Sang M (2019) Nonlinear dynamic modelling and analysis for a spur gear system with time-varying pressure angle and gear backlash. Mech Syst Signal Pr 132:18–34

    Article  Google Scholar 

  13. 13.

    Mark WD (1992) Contributions to the vibratory excitation of gear systems from periodic undulations on tooth running surfaces. J Acoust Soc Am 91:166–186

    Article  Google Scholar 

  14. 14.

    Velex P, Maatar M (1996) A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour. J Sound Vib 191:629–660

    Article  Google Scholar 

  15. 15.

    Bonori G, Pellicano F (2007) Non-smooth dynamics of spur gears with manufacturing errors. J Sound Vib 306:271–283

    Article  Google Scholar 

  16. 16.

    Mucchi E, Dalpiaz G, Rivola A (2010) Elastodynamic analysis of a gear pump. Part II: meshing phenomena and simulation results. Mech Syst Signal Pr 24:2180–2197

    Article  Google Scholar 

  17. 17.

    Fernández A, Iglesias M, de-Juan A, García P, Sancibrián R, Viadero F (2014) Gear transmission dynamic: effects of tooth profile deviations and support flexibility. Appl Acoust 77:138–149

    Article  Google Scholar 

  18. 18.

    International Standard ISO 1328-1 (2013) Cylindrical gears—ISO system of flank tolerance classification; Part 1: Definitions and allowable values of deviations relevant to flanks of gear teeth. Geneva, Switzerland, pp 10–16

  19. 19.

    Munro RG (1990) A review of the theory and measurement of gear transmission error. Gearbox Noise Vib:3–10

  20. 20.

    Litvin FL, Fuentes A (2004) Gear geometry and applied theory, 2nd ed. Cambridge

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This research was funded by Scientific Research Foundation for Talent Introduced (Grant number 2019YQQ005) and Scientific Research Project (Grant number Xjky2020008) of Anhui Polytechnic University, the University Synergy Innovation Program of Anhui Province (Grant number GXXT-2019-048), Anhui university scientific research platform innovation team building projects (2016–2018) and the National Natural Science Foundation of China (Grant number 51775156).

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Correspondence to Rui Xu.

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Xu, R., Zhang, J., Wang, J. et al. Nonlinear Dynamic Modelling and Analysis for a Spur Gear Pair Considering Tooth Profile Deviation Based on Tooth Contact Analysis. J. Vib. Eng. Technol. (2021).

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  • Nonlinear dynamics
  • Tooth profile deviation
  • Tooth contact analysis
  • Transmission error