Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

  • Zheng Cao
  • Zaigang ChenEmail author
  • Hanjun Jiang
Original paper


Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.


Gear dynamics Mesh stiffness Profile modification Eccentricity error Force-dependent 

List of symbols

\(\delta _{\mathrm{b}}\), \(\delta _{\mathrm{s}}\), \(\delta _{\mathrm{a}}\)

Bending deformation, shear deformation, axial deformation

\(\alpha _{1}\)

Angle between the line of action and the line perpendicular to the center line of the tooth


Defined in Fig. 1


Distance from the point of contact to the center line of the tooth


Young’s modulus


Poisson’s ratio


Tooth width


Moment of inertia and the area of section at the distance x from the base along the tooth center line

\(k_{\mathrm{b}},\, k_{\mathrm{s}},\, k_{\mathrm{a}}\)

Bending stiffness, shearing stiffness, axial stiffness

\(\delta _{\mathrm{f}}\)

Additional deformation of the tooth caused gear body


The stiffness related to the additional deformation of gear body


Coefficients expressing the additional deformation of the tooth


Defined in Fig. 2

\(\delta _{\mathrm{h}}\)

Local contact compliance


Half width of the contact region on the tooth


Radius of curvature at the point of contact of gear and pinion


Poisson’s ratio of pinion and gear


Young’s modulus of pinion and gear


Distance from the point of contact to the center line of the tooth of the gear and pinion along the line of action direction


Local contact stiffness of tooth pair under load F


Mass moments of inertia of pinion and gear


Mesh stiffness of single tooth pair under the load F


The overall mesh stiffness of mesh gears under the load F


Load-independent stiffness including bending, shear, axial stiffness and additional stiffness caused by gear body


Load-independent stiffness of tooth pair i


The local contact stiffness of tooth pair i

\(\delta \)

Total deformation of the tooth at the contact point


Initial separation of the potential contact tooth pair i


Profile error of tooth pair i of pinion and gear

\(\eta _{i\mathrm{p}},\,\eta _{i\mathrm{g}}\)

Profile modification of tooth pair \(i\_\) of pinion and gear

\(\lambda _{\mathrm{p}},\,\lambda _{\mathrm{g}}\)

Eccentricity error of pinion and gear

\(\delta _{i}\)

Deformation of the tooth pair i


Load on the tooth pair i

\(\hbox {Lsf}_i\)

Load sharing of tooth pair i


Input torque on the pinion


Output torque on the gear


Linear damping element


No-load transmission error


Contact coefficient


Radius of the base circle of the pinion and the gear


Rotational frequency of pinion


Mesh frequency



The authors are grateful for the financial support provided by the National Natural Science Foundation of China (Grant Nos. 51775453 and 51605412), the Sichuan Science and Technology Program (Grant No. 2018JY0159).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Liu, G., Parker, R.G.: Dynamic modeling and analysis of tooth profile modification for multimesh gear vibration. J. Mech. Des. 130(12), 121402 (2008)CrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Zhang, T., Chen, Z., Zhai, W., Wang, K.: Establishment and validation of a locomotive-track coupled spatial dynamics model considering dynamic effect of gear transmissions. Mech. Syst. Signal Process. 119, 328–345 (2019)CrossRefGoogle Scholar
  4. 4.
    Özgüven, H.N., Houser, D.R.: Dynamic analysis of high speed gears by using loaded static transmission error. J. Sound Vib. 125(1), 71–83 (1988)CrossRefGoogle Scholar
  5. 5.
    Kahraman, A., Singh, R.: Non-linear dynamics of a spur gear pair. J. Sound Vib. 142(1), 49–75 (1990)CrossRefGoogle Scholar
  6. 6.
    Kahraman, A., Singh, R.: Non-linear dynamics of a geared rotor-bearing system with multiple clearance. J. Sound Vib. 144(3), 469–509 (1991)CrossRefGoogle Scholar
  7. 7.
    Kahraman, A., Singh, R.: Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system. J. Sound Vib. 146(1), 135–156 (1991)CrossRefGoogle Scholar
  8. 8.
    Lin, H.H., Oswald, F.B., Townsend, D.P.: Dynamic loading of spur gears with linear or parabolic tooth profile modifications. Mech. Mach. Theory 29(8), 1115–1129 (1994)CrossRefGoogle Scholar
  9. 9.
    He, S., Gunda, R., Singh, R.: Effect of sliding friction on the dynamics of spur gear pair with realistic time-varying stiffness. J. Sound Vib. 301(3–5), 927–949 (2007)CrossRefGoogle Scholar
  10. 10.
    Velex, P., Maatar, M.: A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour. J. Sound Vib. 191(5), 629–660 (1996)CrossRefGoogle Scholar
  11. 11.
    Yang, D.C.H., Lin, J.Y.: Hertzian damping, tooth friction and bending elasticity in gear impact dynamics. J. Mech. Transm. Autom. Des. 109(2), 189–196 (1987)CrossRefGoogle Scholar
  12. 12.
    Huang, K.J., Wu, M.R., Tseng, J.T.: Dynamic analyses of gear pairs incorporating the effect of time-varying lubrication damping. J. Vib. Control 17(3), 355–363 (2011)CrossRefGoogle Scholar
  13. 13.
    Guilbault, R., Lalonde, S., Thomas, M.: Nonlinear damping calculation in cylindrical gear dynamic modeling. J. Sound Vib. 331(9), 2110–2128 (2012)CrossRefGoogle Scholar
  14. 14.
    Amabili, M., Rivola, A.: Dynamic analysis of spur gear pairs: steady-state response and stability of the SDOF model with time-varying meshing damping. Mech. Syst. Signal Process. 11(3), 375–390 (1997)CrossRefGoogle Scholar
  15. 15.
    Chen, S., Tang, J., Wu, L.: Dynamics analysis of a crowned gear transmission system with impact damping: based on experimental transmission error. Mech. Mach. Theory 74, 354–369 (2014)CrossRefGoogle Scholar
  16. 16.
    Mark, W.D.: Analysis of the vibratory excitation of gear systems. II. Tooth error representations, approximations, and application. J. Acoust. Soc. Am. 66(6), 1758–1787 (1979)CrossRefGoogle Scholar
  17. 17.
    Jia, S., Howard, I.: Comparison of localised spalling and crack damage from dynamic modelling of spur gear vibrations. Mech. Syst. Signal Process. 20(2), 332–349 (2006)CrossRefGoogle Scholar
  18. 18.
    Chen, Z., Shao, Y.: Dynamic features of planetary gear train with tooth errors. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 229(10), 1769–1781 (2015)CrossRefGoogle Scholar
  19. 19.
    Bahk, C.J., Parker, R.G.: Analytical investigation of tooth profile modification effects on planetary gear dynamics. Mech. Mach. Theory 70, 298–319 (2013)CrossRefGoogle Scholar
  20. 20.
    Yu, W., Mechefske, C.K.: Analytical modeling of spur gear corner contact effects. Mech. Mach. Theory 96, 146–164 (2016)CrossRefGoogle Scholar
  21. 21.
    Liu, S., Song, C., Zhu, C., Liang, C., Yang, X.: Investigation on the influence of work holding equipment errors on contact characteristics of face-hobbed hypoid gear. Mech. Mach. Theory 138, 95–111 (2019)CrossRefGoogle Scholar
  22. 22.
    Cornell, R.W.: Compliance and stress sensitivity of spur gear teeth. J. Mech. Des. 103(2), 447–459 (1981)Google Scholar
  23. 23.
    Chen, Z., Shao, Y.: Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth. Eng. Fail. Anal. 18(8), 2149–2164 (2011)CrossRefGoogle Scholar
  24. 24.
    Chen, Z., Zhai, W., Wang, K.: Vibration feature evolution of locomotive with tooth root crack propagation of gear transmission system. Mech. Syst. Signal Process. 115, 29–44 (2019)CrossRefGoogle Scholar
  25. 25.
    Arafa, M.H., Megahed, M.M.: Evaluation of spur gear mesh compliance using the finite element method. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 213(6), 569–579 (1999)CrossRefGoogle Scholar
  26. 26.
    Wang, J., Howard, I.: The torsional stiffness of involute spur gears. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 218(1), 131–142 (2004)CrossRefGoogle Scholar
  27. 27.
    Wang, J., Howard, I.: Finite element analysis of high contact ratio spur gears in mesh. J. Tribol. 127(3), 469–483 (2005)CrossRefGoogle Scholar
  28. 28.
    Vedmar, L., Henriksson, B.: A general approach for determining dynamic forces in spur gears. J. Mech. Des. 120(4), 593–598 (1998)CrossRefGoogle Scholar
  29. 29.
    Del Rincon, A.F., Viadero, F., Iglesias, M., García, P., De-Juan, A., Sancibrian, R.: A model for the study of meshing stiffness in spur gear transmissions. Mech. Mach. Theory 61, 30–58 (2013)CrossRefGoogle Scholar
  30. 30.
    Parker, R.G., Agashe, V., Vijayakar, S.M.: Dynamic response of a planetary gear system using a finite element/contact mechanics model. J. Mech. Des. 122(3), 304–310 (2000)CrossRefGoogle Scholar
  31. 31.
    Weber, C.: The Deformation of Loaded Gears and the Effect on their Load Carrying Capacity. Department of Scientific and Industrial Research, London (1951)Google Scholar
  32. 32.
    Sainsot, P., Velex, P., Duverger, O.: Contribution of gear body to tooth deflections—a new bidimensional analytical formula. J. Mech. Des. 126(4), 748–752 (2004)CrossRefGoogle Scholar
  33. 33.
    Chen, Z., Zhang, J., Zhai, W., Wang, Y., Liu, J.: Improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack. Eng. Fail. Anal. 82, 72–81 (2017)CrossRefGoogle Scholar
  34. 34.
    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)CrossRefGoogle Scholar
  35. 35.
    Tavakoli, M.S., Houser, D.R.: Optimum profile modifications for the minimization of static transmission errors of spur gears. J. Mech. Transm. Autom. Des. 108(1), 86–94 (1986)CrossRefGoogle Scholar
  36. 36.
    Kahraman, A., Blankenship, G.W.: Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters. J. Appl. Mech. 64(1), 217–226 (1997)CrossRefGoogle Scholar
  37. 37.
    Kahraman, A., Blankenship, G.W.: Effect of involute tip relief on dynamic response of spur gear pairs. J. Mech. Des. 121(2), 313–315 (1999)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of Electrical Engineering and AutomationAnhui UniversityHefeiPeople’s Republic of China
  2. 2.State Key Laboratory of Traction PowerSouthwest Jiaotong UniversityChengduPeople’s Republic of China
  3. 3.The State Key Laboratory of Heavy Duty AC Drive Electric Locomotive Systems IntegrationCRRC Zhuzhou Locomotive Co., LTDZhuzhouPeople’s Republic of China
  4. 4.College of Automobile EngineeringYancheng Institute of TechnologyYanchengPeople’s Republic of China

Personalised recommendations