Advertisement

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 FuEmail author
  • Zong-de Fang
  • Xian-long Peng
  • Xiang-ying Hou
  • Jian-hua Li
Research Article - Mechanical Engineering
  • 55 Downloads

Abstract

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.

Keywords

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

List of symbols

\(l_{i}\)

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

k

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

B

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

F

Integrated flexibility matrix

Z

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

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.

References

  1. 1.
    Litvin, F.L.; Fuentes, A.; Zanzi, C.; et al.: Face-gear drive with spur involute pinion: geometry, generation by a worm, stress analysis. Comput. Methods Appl. Mech. 191, 2785–2813 (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Lin, C.; He, C.J.; Gu, S.J.; et al.: Loaded tooth contact analysis of curve-face gear pair. Adv. Mech. Eng. 9, 1–10 (2017)Google Scholar
  3. 3.
    Litvin, F.L.; Gonzalez-Perez, I.; Fuentes, A.; et al.: Design, generation and stress analysis of face-gear drive with helical pinion. Comput. Methods Appl. Mech. 194, 3870–3901 (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    Li, S.T.: Effects of machining errors, assembly errors and tooth modifications on loading capacity, load-sharing ratio and transmission error of a pair of spur gears. Mech. Mach. Theory 42, 698–726 (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Barone, S.; Borgianni, L.; Forte, P.: Evaluation of the effect of misalignment and profile modification in face gear drive by a finite element meshing simulation. J. Mech. Des. 126, 916–924 (2004)CrossRefGoogle Scholar
  6. 6.
    Litvin, F.L.; Fuentes, A.; Zanzi, C.; et al.: Design, generation, and stress analysis of two versions of geometry of face-gear drives. Mech. Mach. Theory 37, 1179–1211 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Zanzi, C.; Pedrero, J.I.: Application of modified geometry of face gear drive. Comput. Methods Appl. Mech. 194, 3047–3066 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Tsay, M.F.; Fong, Z.H.: Novel profile modification methodology for moulded face-gear drives. Proc. Inst. Mech. Eng. C J. Mech. 6, 715–725 (2007)CrossRefGoogle Scholar
  9. 9.
    Tang, J.Y.; Yin, F.; Chen, X.M.: The principle of profile modified face-gear grinding based on disk wheel. Mech. Mach. Theory 70, 1–15 (2013)CrossRefGoogle Scholar
  10. 10.
    Peng, X.L.; Zhang, L.; Fang, Z.D.: Manufacturing process for a face gear drive with local bearing contact and controllable meshing performance based on ease-off surface modification. J. Mech. Des. 138, 043320 (2016)CrossRefGoogle Scholar
  11. 11.
    Wang, Y.Z.; Lan, Z.; Hou, L.W.; Chu, X.M.; et al.: An efficient honing method for face gear with tooth profile modification. Int. J. Adv. Manuf. Technol. 90, 1155–1163 (2016)CrossRefGoogle Scholar
  12. 12.
    Artoni, A.; Kolivand, M.; Kahraman, A.: An ease-off based optimization of the loaded transmission error of hypoid gears. J. Mech. Des. 132, 011010 (2010)CrossRefGoogle Scholar
  13. 13.
    Conry, T.F.; Seireg, A.: A mathematical programming technique for the evaluation of load distribution and optimal modifications for gear systems. J. Eng. Ind. 95, 1115–1122 (1973)CrossRefGoogle Scholar
  14. 14.
    Simon, V.V.: Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears. Mech. Mach. Theory 41, 707–724 (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Imrek, H.; Unuvar, A.: Investigation of influence of load and velocity on scoring of addendum modified gear tooth profiles. Mech. Mach. Theory 44, 938–948 (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Artoni, A.; Gabiccini, M.; Guiggiani, M.; et al.: Multi-objective ease-off optimization of hypoid gears for their efficiency, noise and durability performances. J. Mech. Des. 133, 121007 (2011)CrossRefGoogle Scholar
  17. 17.
    Simon, V.V.: Optimization of face-hobbed hypoid gears. Mech. Mach. Theory 77, 164–181 (2014)CrossRefGoogle Scholar
  18. 18.
    Simon, V.V.: Optimal machine tool settings for the manufacture of face-hobbed spiral bevel gears. J. Mech. Des. 136, 081004-1 (2014)Google Scholar
  19. 19.
    Catmull, E.; Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10, 350–355 (1978)CrossRefGoogle Scholar
  20. 20.
    Zhang, Y.; Fang, Z.D.: Analysis of tooth contact and load distribution of helical gears with crossed axes. Mech. Mach. Theory 34, 41–57 (1999)CrossRefzbMATHGoogle Scholar
  21. 21.
    Conry, T.F.; Seireg, A.: A mathematical programming method for design of elastic bodies in contact. J. Appl. Mech. 38, 387–392 (1971)CrossRefGoogle Scholar
  22. 22.
    Wolfe, P.: The simplex method for quadratic programming. Econometrica 28, 170–170 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Blok, H.: The flash temperature concept. Wear 6, 483–494 (1963)CrossRefGoogle Scholar
  24. 24.
    ISO. Calculation of scuffing load capacity of cylindrical, bevel and hypoid gears-Part 1: flash temperature method, ISO Standard TR 13989-1, International Organization for Standardization, Geneva, Switzerland (2000)Google Scholar
  25. 25.
    Houpert, L.: An engineering approach to hertzian contact elasticity—part I. J. Tribol. Trans. ASME 123, 582–588 (2001)CrossRefGoogle Scholar
  26. 26.
    Deb, K.; Rao, N.U.B.; Karthik, S.: Dynamic Multi-objective Optimization and Decision-Making Using Modified NSGA-II: A Case Study on Hydro-Thermal Power Scheduling. Evolutionary Multi-Criterion Optimization EMO, Berlin (2007)Google Scholar
  27. 27.
    Mejía, J.A.H.; Schütze, O.; Cuate, O.; et al.: RDS-NSGA-II: a memetic algorithm for reference point based multi-objective optimization. Eng. Optim. 49, 828–845 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Deb, K.; Pratap, A.; Agarwal, S.; et al.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE. Trans. Evol. Comput. 6, 182–197 (2002)CrossRefGoogle Scholar

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

Personalised recommendations