Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 1097–1108 | Cite as

Computerized Design and Optimization of Tooth Modifications on Pinions for Face Gear Drives

  • Xue-zhong Fu
  • Zong-de Fang
  • Xian-long Peng
  • Xiang-ying Hou
  • Jian-hua Li
Research Article - Mechanical Engineering


This paper presents a novel design and optimization method for tooth modifications on pinions for face gear drives that spans a variety of modification types. A tooth profile and an axial modification curve on a spur pinion are designed to consist of a straight line and two sections of parabola, and the topologically modified tooth surface of the pinion is expressed as a superposition of the theoretical tooth surface and the deviation surface determined by fitting cubic B-splines on the tooth surface grid. The design method can accurately control the modification amount, modification lengths and deviation surface. Based on tooth contact analysis and loaded tooth contact analysis of the face gear drives, a multi-objective optimization model is established with a uniform tooth surface load distribution, minimum wave amplitude of loaded transmission error (WALTE) and minimum tooth surface flash temperature (TSFT) as the three objectives. Optimization variables include the eight parameters of the modification curves, and a fast elitist nondominated sorting genetic algorithm (NSGA-II) is applied to solve the model. The simulation results show that the maximum TSFT can be reduced by up to 36.32%, the WALTE can be reduced by up to 59.26%, and the maximum load density can be reduced by up to 9.84%, which proves the proposed method is feasible and satisfactory for face gear drives.


Face gear drive Tooth modification Load distribution Wave amplitude of loaded transmission error Tooth surface flash temperature NSGA-II 

List of symbols


Parameters of the modification curves (\(i=1\ldots 8\))

\(r_{\mathrm{a}}\) and \(r_{\mathrm{h}}\)

Parameters of the diameters of the pinion

\({\varvec{r}}_{\mathrm{r}}\) and \({\varvec{n}}_{\mathrm{r}}\)

Position and normal vectors of the theoretical pinion

\({\varvec{r}}_{1}\) and \({\varvec{n}}_{1}\)

Position and unit normal vectors of the modified pinion

\({\varvec{r}}_{2}\) and \({\varvec{n}}_{2}\)

Position and unit normal vectors of the theoretical face gear


Parabolic times

\(\delta _{\mathrm{p}}\) and \(\delta _{\mathrm{a}}\)

Modification amounts

\(S_{1}\) and \(S_{2}\)

Movable coordinate systems

\(S_{\mathrm{f}}\), \(S_{\mathrm{q}}\), \(S_{\mathrm{d}}\) and \(S_{\mathrm{e}}\)

Fixed coordinate systems

\(\varPhi _{1}\) and \(\varPhi _{2}\)

Parameters of motion

\(\Sigma _{1}\) and \(\Sigma _{2}\)

Tooth surfaces

\(\Delta q\), \(\Delta E\) and \(\Delta \gamma \)

Installation errors


Difference between the radius of the pinion and the virtual shaper

\(\gamma \) and \(\gamma _{\mathrm{m}}\)

Shaft angle and complementary angle

\(L_{0}\), \(L_{1}\) and \(L_{2}\)

Parameters of the diameters of the face gear

\({\varvec{M}}_{\mathrm{f1}}\), \({\varvec{M}}_{\mathrm{f2}}\)

\(4\times 4\) matrices

\({\varvec{L}}_{\mathrm{f1}}\), \({\varvec{L}}_{\mathrm{f}}\)

\(3\times 3\) submatrices

\({{\varvec{w}}}\) and \({\varvec{d}}\)

Initial and final tooth clearance


Integrated flexibility matrix


Tooth approach of the pinion

P and W

Load and load density

T and \(T_{\mathrm{e}}\)

Load transmission error and wave amplitude

\(\theta _{\mathrm{fla}}\)

Tooth surface flash temperature


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The authors disclose receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Nos. 51375384, 51605378). In addition, the authors are grateful to the editors at American Journal Experts for editing this article.

Compliance with Ethical Standards

Conflict of interest

The authors declare no potential conflicts of interest with respect to the research, authorship and/or publication of this article.


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.School of Mechanical EngineeringNorthwestern Polytechnical UniversityXi’anChina
  2. 2.School of Mechanical EngineeringXi’an University of Science and TechnologyXi’anChina

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