Skip to main content
SpringerLink
Log in

Tooth Thickness Error Analysis of Straight Beveloid Gear by Inclined Gear Shaping

  • Regular Paper
  • Published:
International Journal of Precision Engineering and Manufacturing Aims and scope Submit manuscript

Abstract

In this paper, two approaches to calculate the tooth thickness error (TTE) of straight beveloid gear by tilt-type gear shaper were proposed. The first calculation approach of TTE was established by considering the change of pitch circle radius during the gear slotting process. The analytical tooth surface model of straight beveloid gear was derived by inclined gear shaping, and the tooth surface point set was obtained. Then, another calculation methodology of TTE was established based on the analytical straight beveloid gear model. Two approaches were employed to calculate the TTE of internal and external straight beveloid gear, respectively. And they were employed to validate each other, and the results show a good consistency. The influences of design parameters on TTE was analyzed. Results show that the internal/external straight beveloid gears have a convex/concave TTE along the tooth width direction while cutting by tilt-type gear shaper, which makes the tooth thickness of the heel and toe sides of beveloid gear thinned/thickened, respectively. The TTE of internal and external beveloid gear both increase with the increase of design cone angle, and decrease with the increase of modulus and number of teeth. For the same macro gear design parameters, the TTE of external beveloid gear is smaller than that of internal beveloid gear. The results of the two approaches show good consistency at the middle of tooth surface. The maximum difference between the results of two approaches gradually increases away from the middle of tooth surface, and it is positively correlated with the value of TTE.

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Abbreviations

\(\delta \) :

Cone angle of beveloid gear

\({\omega }^{c}\) :

Angular velocity of slotting cutter

\(\omega \) :

Angular velocity of gear blank

\({Z}_{c}\) :

Tooth numbers of slotting cutter

Z :

Tooth numbers of gear blank

\({m}_{n}\) :

Modulus of normal section

\({m}_{t}\) :

Modulus of transverse section

\({\alpha }_{n}\) :

Pressure angle of normal section

\({\alpha }_{t}\) :

Pressure angle of transverse section

\({\alpha }_{t}^{\prime}\) :

Pressure angle at any circle of the gear blank

\({\alpha }_{tp}^{{\prime}}\) :

Pressure angle at pitch circle of the gear blank in the processing section

\({h}_{an}^{\ast}\) :

Addendum coefficient of normal section

\({h}_{at}^{\ast}\) :

Addendum coefficient of transverse section

\({r}_{p}\) :

Working pitch circle radius of gear blank

\({r}_{p}^{c}\) :

Working pitch circle radius of slotting cutter

r :

Reference circle radius of gear blank

\({r}_{t}^{c}\) :

Reference circle radius of slotting cutter

\({r}_{bt}\) :

Base circle radius of gear blank

\({r}_{bt}^{c}\) :

Base circle radius of slotting cutter

\(ym\) :

Radial displacement of rack cutter

\({x}_{M}\) :

Modification coefficient of beveloid gear in the middle section

\({x}_{t}\) :

Modification coefficient of beveloid gear in the section of interest

B :

Distance from the section of interest to the middle section

\({x}_{t}^{c}\) :

Modification coefficient of the slotting cutter cutting section

s :

Tooth thickness at reference circle

\({e}^{c}\) :

Reference circle space width of the slotting cutter

\({e}_{p}^{c}\) :

Pitch circle space width of the slotting cutter

\({r}_{K}\) :

Radius of any circle between tip circle and root circle of the machined gear

\({h}_{k}\) :

Tooth depth coefficient

\({s}_{K}\) :

Tooth thickness at any circle of the gears processed by the rack cutter

\({s}_{K}^{c}\) :

Tooth thickness at any circle of the gears processed by the slotting cutter

\({s}_{K}^{er}\) :

Tooth thickness error at any circle of the gears

\({\varphi }_{c}\) :

Rotated angles of the slotting cutter

\({\varphi }_{g}\) :

Rotated angles of the gear blank

\({n}_{i}^{k}\) :

Common normal vector of the contact point of the unit k in coordinate \({S}_{i}\)

\({v}_{i}^{k}\) :

Velocity vector of the contact point of the unit k in coordinate \({S}_{i}\)

\({x,y,z}_{k}\) :

X/Y/Z-Axis value of the unit k tooth surface

\({n}_{x,y,z}^{c}\) :

X/Y/Z-Axis components of the common normal vector of the cutting edge

References

  1. Chen, Q., Song, C., Zhu, C., Du, X., & Ni, G. (2017). Manufacturing and contact characteristics analysis of internal straight beveloid gear pair. Mechanism and Machine Theory, 114, 60–73.

    Article  Google Scholar 

  2. Mitome, K. (2004). Today and tomorrow of conical gear. JSME Symposium on Motion and Power Transmissions, 11, 270–273.

    Article  Google Scholar 

  3. Sun, R., Song, C., Zhu, C., Wang, Y., & Yang, X. (2021). Computational studies on mesh stiffness of paralleled helical beveloid gear pair. International Journal of Precision Engineering and Manufacturing, 22, 123–137.

    Article  Google Scholar 

  4. Yang, X., Song, C., Zhu, C., & Liu, S. (2018). Tooth surface deviation and mesh analysis of beveloid gears with parallel axis considering machine tool adjustment errors. Journal of Advanced Mechanical Design, Systems, and Manufacturing, 12(4), JAMDSM0082.

    Article  Google Scholar 

  5. Song, C., Zhu, C., Lim, T. C., & Peng, T. (2012). Parametric analysis of gear mesh and dynamic response of loaded helical beveloid transmission with small shaft angle. Journal of Mechanical Design, 134(8), 84501.

    Article  Google Scholar 

  6. Zhu, C., Song, C., Lim, T. C., & Peng, T. (2013). Pitch cone design and influence of misalignments on tooth contact behaviors of crossed beveloid gears. Mechanism and Machine Theory, 59, 48–64.

    Article  Google Scholar 

  7. Song, C., Zhou, Y., Zhu, C., Ni, G., & Liu, S. (2018). Loaded tooth contact analysis of intersected beveloid and cylindrical involute gear pair with small shaft angle. Journal of Advanced Mechanical Design, Systems, and Manufacturing, 12(1), 1881–3054.

    Article  Google Scholar 

  8. Zhu, C., Liu, L., Song, C., Xiang, Y., & Liu, H. (2014). Pitch cone design and tooth contact analysis of intersected beveloid gears for marine transmission. Mechanism and Machine Theory, 82, 141–153.

    Article  Google Scholar 

  9. Liu, C. C., & Tsay, C. B. (2002). Mathematical models and contact simulations of concave beveloid gears. Journal of Mechanical Design, 124(4), 753–760.

    Article  Google Scholar 

  10. Brecher, C., Brumm, M., Hübner, F., & Henser, J. (2013). Influence of the manufacturing method on the running behavior of beveloid gears. Production Engineering (Berlin, Germany), 7(2), 265–274.

    Google Scholar 

  11. Zhao, J., Huang, J., Wen, J., Bian, Y., & Sun, J. (2014). Gear hobbing of space beveloid gears. Coal Mine Machinery, 35(10), 135–137.

    Google Scholar 

  12. Ni, G., Zhu, C., Song, C., Du, X., & Zhou, Y. (2017). Tooth contact analysis of crossed beveloid gear transmission with parabolic modification. Mechanism and Machine Theory, 113, 40–52.

    Article  Google Scholar 

  13. Liu, S., Song, C., Zhu, C., & Ni, G. (2018). Effects of tooth modifications on mesh characteristics of crossed beveloid gear pair with small shaft angle. Mechanism and Machine Theory, 119, 142–160.

    Article  Google Scholar 

  14. Liu, S., Song, C., Zhu, C., Ni, G., & Ullah, N. (2019). Concave and convex modifications analysis for skewed beveloid gears considering misalignments. Mechanism and Machine Theory, 133, 127–149.

    Article  Google Scholar 

  15. Wu, J., Li, G., & Li, H. (2000). A new slotting technology for a pair of inside engaged gears. Journal of Xi’an Petroleum Institute (Natural Science Edition), 15(03), 45–48.

    Google Scholar 

  16. Wu, J. (1999). Development of backlash control beveloid gear RV reducer for robots. PhD Thesis, Harbin Institute of Technology.

  17. Li, G., Wen, J., Liu, F., & Li, X. (2004). Profile errors and axial errors noninnvolute beveloid gears with crossed axes. Journal of Nanjing University of Science and Technology, 028(6), 585–589.

    Google Scholar 

  18. Li, G., Li, X., Wen, J., Xin, Z., & Yu, L. (2004). Meshing theory and simulation of noninvolute beveloid gears. Mechanism and Machine Theory, 39(8), 883–892.

    Article  Google Scholar 

  19. Hu, D. (2017). Study on theory and processing technology of beveloid internal gear enveloping external-rotor crown worm drive. Master Thesis, Xihua University.

  20. Liu, C. C. (2005). The mathematical model and tooth surface deviations of internal conical gears. In Proceedings of the ASME international design engineering technical conferences and computers and information in engineering conference (DETC2005).

  21. Liu, C. C., & Wang, S. F. (2007). Tooth contact analysis and contact ellipse simulation of internal conical gear pairs. In 12th IFToMM world congress, Besancon (France), 18–21 June 2007.

  22. Hu, R., Du, X., & Zhu, C. (2020). Design of shaper cutter for precision matching internal beveloid gears. China Mechanical Engineering, 31(549), 19–25.

    Google Scholar 

  23. Litvin, F. L. (2004). Gear geometry and applied theory. Cambridge University Press.

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the National Key R&D Program of China (Grant No. 2019YFB2004700)

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400030, China

    Feihong Zhu, Chaosheng Song, Caichao Zhu & Xuesong Du

Authors
  1. Feihong Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chaosheng Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Caichao Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xuesong Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chaosheng Song.

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

Zhu, F., Song, C., Zhu, C. et al. Tooth Thickness Error Analysis of Straight Beveloid Gear by Inclined Gear Shaping. Int. J. Precis. Eng. Manuf. 23, 429–443 (2022). https://doi.org/10.1007/s12541-022-00625-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12541-022-00625-2

Keywords

Search

Navigation