Skip to main content
SpringerLink
Log in

An improved nonlinear dynamic model of gear pair with tooth surface microscopic features

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

Abstract

In view of the issue that current gear dynamics model contains no parameters about tooth surface topography, this paper puts forward an improved nonlinear dynamic model of gear pair with tooth surface microscopic features through revision of the backlash equation by W–M function from fractal theory and combination with the tradition gear torsional model. The model sets up a mathematical relationship between gear dynamic characteristics and surface microscopic parameters such as surface roughness and fractal dimension. Results of the numerical simulations indicate that as surface roughness decreases, meshing stiffness increases and viscous damping rises, the gear dynamic performance tends to be better, which is consistent with the existing research reports. Furthermore, it is found that dropping of fractal dimension is good to improve gear dynamic performance, so gear dynamics can be enhanced by decreasing the fractal dimension if surface roughness is set or cannot be decreased anymore. Moreover, it is also shown that initial backlash has little impact on the rule of gear dynamics response but influences the size of start-up or stop shock. Finally, the model is validated by a series of simulations and comparison with experimental data and existing model. The theory here opens up a mathematical methodology to analyze gear dynamics with respect to tooth surface microscopic features, which lays a theoretical basis for design of tooth surface topography to obtain better performance of gear transmission in the future.

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

Similar content being viewed by others

Abbreviations

\(b_0\) :

The initial backlash

\(b_\mathrm{c}\) :

The characteristic length

\(b_\mathrm{h}\) :

Backlash of gear pair

\(b_\mathrm{h}^{\prime }\) :

Updated backlash of gear pair

b(t):

The backlash varying with time

\(b^{\prime }(t)\) :

The backlash with a scale factor (\(\xi \))

\(c_\mathrm{h}\) :

Damping coefficient

D :

Fractal dimension

\(D_1\) :

The fractal dimension of gear 1

\(D_2\) :

The fractal dimension of gear 2

\(e_\mathrm{r}\) :

The comprehensive error amplitude

e(t):

Static transmission error

\(f_\mathrm{h} (p)\) :

Backlash function

\(f^{\prime }_h (p)\) :

Modified backlash function

\(f^{\prime }_h ({p^{\prime }})\) :

The dimensionless modified backlash function

\(f_\mathrm{m}\) :

The external static load

\(F_{\mathrm{ah}}\) :

Dimensionless error amplitude

\(F_\mathrm{m}\) :

Dimensionless external load

\(I_i ({i=p,g})\) :

The inertia of the pinion and gear

\(k_\mathrm{har}\) :

The first-order harmonic component coefficient

\(k_{\mathrm{hm}}\) :

Average meshing stiffness

\(k_\mathrm{h} (t)\) :

Time-varying stiffness

\(m_{\mathrm{e}}\) :

Equivalent mass of gear pair

p(t):

The difference between the dynamic transmission error and static transmission error changing with time (t)

\(p^{\prime } ({t^{\prime }})\) :

The dimensionless difference between the dynamic transmission error and static transmission error changing with nominal time (\(t^{\prime }\))

\(R_\mathrm{a}\) :

The arithmetic mean deviation (denoting surface roughness here)

\(R_{\mathrm{a}1}\) :

The surface roughness of gear 1

\(R_{\mathrm{a}2}\) :

The surface roughness of gear 2

\(R_\mathrm{a} D (D)\) :

Function to get the corresponding \(R_{\mathrm{a}}\) with a definite D

\(R_i ({i=p, g})\) :

Basis radius of the pinion and gear

\(T_i ({i=p,g})\) :

Torsion of the pinion and gear

\(\varepsilon \) :

Time-varying stiffness coefficient

\(\theta _i ({i=p,g})\) :

Torsional vibration displacement of the pinion and gear

\(\omega _\mathrm{h}\) :

Gear meshing frequency

\({\varOmega }_\mathrm{h}\) :

Dimensionless excitation frequency

\(\omega _\mathrm{n}\) :

Intermediate variable

\({\phi }_\mathrm{h}\) :

The initial phase of time-varying stiffness

\({\phi }_\mathrm{e}\) :

The initial phase of static transmission error

\(\lambda \) :

The characteristic scale coefficient

t :

Time

\(t^{\prime }\) :

Nominal time

\(z^{\prime }(t)\) :

The new height of surface asperities

\(\zeta _{33}\) :

The dimensionless gear mesh damping

\(\xi \) :

Scale factor

References

  1. Gregory, R., Harris, S., Munro, R.: Dynamic behaviour of spur gears. Proc. Inst. Mech. Eng. 178(1), 207–218 (1963)

    Article  Google Scholar 

  2. Özgüven, H.N., Houser, D.R.: Mathematical models used in gear dynamics—a review. J. Sound Vib. 121(3), 383–411 (1988)

    Article  Google Scholar 

  3. 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 

  4. Parker, R., Vijayakar, S., Imajo, T.: Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons. J. Sound Vib. 237(3), 435–455 (2000)

    Article  Google Scholar 

  5. 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)

    Article  Google Scholar 

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

    Google Scholar 

  7. Kahraman, A., Singh, R.: Non-linear dynamics of a spur gear pair. J. Sound Vib. 142(1), 49–75 (1990)

    Article  Google Scholar 

  8. 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)

    Article  Google Scholar 

  9. 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)

    Article  Google Scholar 

  10. Özgüven, H.: A non-linear mathematical model for dynamic analysis of spur gears including shaft and bearing dynamics. J. Sound Vib. 145(2), 239–260 (1991)

    Article  Google Scholar 

  11. Wang, J., Lim, T.C.: Effect of tooth mesh stiffness asymmetric nonlinearity for drive and coast sides on hypoid gear dynamics. J. Sound Vib. 319(3–5), 885–903 (2009)

    Article  Google Scholar 

  12. Kim, W., Yoo, H.H., Chung, J.: Dynamic analysis for a pair of spur gears with translational motion due to bearing deformation. J. Sound Vib. 329(21), 4409–4421 (2010)

    Article  Google Scholar 

  13. 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)

    Article  Google Scholar 

  14. 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 

  15. Walha, L., Fakhfakh, T., Haddar, M.: Nonlinear dynamics of a two-stage gear system with mesh stiffness fluctuation, bearing flexibility and backlash. Mech. Mach. Theory 44(5), 1058–1069 (2009)

    Article  MATH  Google Scholar 

  16. Moradi, H., Salarieh, H.: Analysis of nonlinear oscillations in spur gear pairs with approximated modelling of backlash nonlinearity. Mech. Mach. Theory 51, 14–31 (2012)

    Article  Google Scholar 

  17. Chen, S., Tang, J., Luo, C., Wang, Q.: Nonlinear dynamic characteristics of geared rotor bearing systems with dynamic backlash and friction. Mech. Mach. Theory 46(4), 466–478 (2011)

    Article  MATH  Google Scholar 

  18. Fang, Y., Liang, X., Zuo, M.J.: Effects of friction and stochastic load on transient characteristics of a spur gear pair. Nonlinear Dyn. 93, 599–609 (2018)

    Article  Google Scholar 

  19. Chen, Q., Ma, Y., Huang, S., Zhai, H.: Research on gears’ dynamic performance influenced by gear backlash based on fractal theory. Appl. Surf. Sci. 313, 325–332 (2014)

    Article  Google Scholar 

  20. 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 (2016)

    Article  MathSciNet  Google Scholar 

  21. Huang, K., Xiong, Y., Wang, T., Chen, Q.: Research on the dynamic response of high-contact-ratio spur gears influenced by surface roughness under EHL condition. Appl. Surf. Sci. 392, 8–18 (2017)

    Article  Google Scholar 

  22. Chen, S., Tang, J.: Effect of backlash on dynamics of spur gear pair system with friction and time-varying stiffness. J. Mech. Eng. 45(8), 119–124 (2009)

    Article  Google Scholar 

  23. Bhushan, B.: Introduction to Tribology. Wiley, London (2013)

    Book  Google Scholar 

  24. Zhu, H., Ge, S., Huang, X., Zhang, D., Liu, J.: Experimental study on the characterization of worn surface topography with characteristic roughness parameter. Wear 255(1–6), 309–314 (2003)

    Article  Google Scholar 

  25. Hasegawa, M., Liu, J., Okuda, K., Nunobiki, M.: Calculation of the fractal dimensions of machined surface profiles. Wear 192(1–2), 40–45 (1996)

    Article  Google Scholar 

  26. Majumdar, A., Bhushan, B.: Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J. Tribol. 112(2), 205–216 (1990)

    Article  Google Scholar 

  27. Majumdar, A., Tien, C.: Fractal characterization and simulation of rough surfaces. Wear 136(2), 313–327 (1990)

    Article  Google Scholar 

  28. Majumdar, A., Bhushan, B.: Fractal model of elastic–plastic contact between rough surfaces. J. Tribol. 113(1), 1–11 (1991)

    Article  Google Scholar 

  29. Gentili, E., Tabaglio, L., Aggogeri, F.: Review on micromachining techniques. In: Kuljanic, E. (ed.) AMST’05 Advanced Manufacturing Systems and Technology, pp. 387–396. Springer, Berlin (2005)

    Chapter  Google Scholar 

  30. Kubo, A., Yamada, K., Aida, T., Sato, S.: Research on ultra high speed gear devices. Trans. Jpn. Soc. Mech. Eng. 38, 2692–2715 (1972)

    Article  Google Scholar 

Download references

Funding

This study was funded by the Natural Science Foundation of China (Nos. 51775158, 51775161).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Hefei University of Technology, Hefei, 230009, China

    Qi Chen, Yadong Wang, Wenfeng Tian, Yanming Wu & Yuanlong Chen

Authors
  1. Qi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yadong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenfeng Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yanming Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yuanlong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Chen, Q., Wang, Y., Tian, W. et al. An improved nonlinear dynamic model of gear pair with tooth surface microscopic features. Nonlinear Dyn 96, 1615–1634 (2019). https://doi.org/10.1007/s11071-019-04874-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-04874-1

Keywords

Search

Navigation