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

  • Qi ChenEmail author
  • Yadong Wang
  • Wenfeng Tian
  • Yanming Wu
  • Yuanlong Chen
Original Paper


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.


Nonlinear dynamic model Gear pair Tooth surface microscopic features Fractal theory Backlash Surface roughness 

List of symbols


The initial backlash


The characteristic length


Backlash of gear pair

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

Updated backlash of gear pair


The backlash varying with time

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

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


Damping coefficient


Fractal dimension


The fractal dimension of gear 1


The fractal dimension of gear 2


The comprehensive error amplitude


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


The external static load


Dimensionless error amplitude


Dimensionless external load

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

The inertia of the pinion and gear


The first-order harmonic component coefficient


Average meshing stiffness

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

Time-varying stiffness


Equivalent mass of gear pair


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


The arithmetic mean deviation (denoting surface roughness here)


The surface roughness of gear 1


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



\(t^{\prime }\)

Nominal time

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

The new height of surface asperities

\(\zeta _{33}\)

The dimensionless gear mesh damping

\(\xi \)

Scale factor



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

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringHefei University of TechnologyHefeiChina

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