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Experimental Studies on Torsional Stiffness of Cycloid Gear Based on Machining Parameters of Tooth Surfaces

  • Zhifeng Liu
  • Tao Zhang
  • Yida Wang
  • Congbin YangEmail author
  • Yongsheng Zhao
Regular Paper
  • 21 Downloads

Abstract

Estimating the torsional stiffness has always been the primary issue in analyzing the dynamic characteristics of cycloid gears. The traditional method of obtaining torsional rigidity involves calculating the ratio of the input torque and rotation angle, treating the deformation of cycloid gear as a black box. In order to thoroughly understand the rotation angle caused by the local contact deformation of each cycloid pin gear, a Majumdar–Bhushan contact model and the finite element method are combined to express the normal contact stiffness. By multiplying the normal contact stiffness of each pin gear and the arm of normal contact force, the torsional stiffness of the cycloidal pin wheel system can be calculated. Experiments are conducted to establish the relationship between the torsional stiffness and roughness parameters of the machined tooth surface. The effect of input torque on the torsional stiffness has also been analyzed. This study formulates a relationship between the torsional stiffness and surface characteristics of cycloid gears, which can help improve their design and manufacture in the future.

Keywords

Cycloid gear Torsional stiffness Fractal theory M–B contact model 

List of Symbols

\(a^{\prime}\)

Truncated area of a single asperity

\(a^{\prime}_{l}\)

Maximum truncated area of a single asperity

\(a^{\prime}_{c}\)

Critical truncated area between the elastic and elastic–plastic deformation regions

\(A_{0}\)

Contact area of each small grid

\(D\)

Two-dimensional fractal dimension

\(D_{s}\)

Three-dimensional fractal dimension which can be expressed as Ds = D + 1

\(e\)

The eccentricity of the cycloidal gear

\(E\)

Equivalent elastic modulus

\(f_{e}\)

Normal contact force of a single aspect

\(F\)

Normal load of the contact area

\(F_{i}\)

Average contact stress of each contact zone

\(G\)

Fractal roughness parameter

\(H\)

Material hardness

\(k_{n}\)

Normal stiffness of contact area

\(k_{ne}\)

Normal stiffness of a single asperity

\(K_{i}\)

Normal contact stiffness of each contact grid

\(K_{1}\)

Short amplitude coefficient

\(K_{t}\)

Total equivalent torsional stiffness of the cycloidal gear

\(K_{ni}\)

Normal contact stiffness of the ith tooth pair

\(K_{ti}\)

Equivalent torsional stiffness of the ith tooth pair

\(L_{i}\)

Length of the contact force arm at position i

\(n(a^{\prime})\)

Size distribution function of asperities

\(N\)

Number of teeth engaged in the engagement of the needle wheel

\(r_{p}\)

Circular radius of the pin gear

\(z_{c}\)

Tooth number of the cycloidal gear

\(\updelta\)

Normal deformation of a single aspect

\(\psi\)

Domain extension factor for micro-contact size distribution

\(\gamma\)

Dimension parameter of the spectral density

\(\angle O_{i} O_{p} P\)

Angle between the crank and the two center points

Notes

Acknowledgements

The authors would like to thank the National Natural Science Fund No. 51575009, Jing-Hua Talents Project of Beijing University of Technology and Beijing Science and Technology Major Project coded D17110400590000 for supporting the research.

References

  1. 1.
    Lehman, M. (1979). Berechnung der Kräfte im Trochoiden-Getriebe. Antriebstechnik, 18(12), 613–616.Google Scholar
  2. 2.
    Yang, D. C. H., & Blanche, J. G. (1990). Design and application guidelines for cycloid drives with machining tolerances. Mechanism and Machine Theory, 25(5), 487–501.CrossRefGoogle Scholar
  3. 3.
    Ishida, T., Li, S., Yoshida, T., & Hidaka, T. (1996). Tooth load of thin rim cycloidal gear. In Proceedings of the 7th international power transmission and gearing conference, ASME (Vol. 88, pp. 565–571).Google Scholar
  4. 4.
    Ishida, T., Yoshida, T., & Li, S. (1998). Relationships among face width, amount of gear error, gear dimension, applied torque and tooth load in cycloidal gears. Transactions of the Japan Society of Mechanical Engineers C, 64(623), 2711–2717.CrossRefGoogle Scholar
  5. 5.
    Li, S. (2014). Design and strength analysis methods of the trochoidal gear reducers. Mechanism and Machine Theory, 81(11), 140–154.CrossRefGoogle Scholar
  6. 6.
    Zhao, Y., Xu, J., Cai, L., et al. (2016). Contact stiffness determination of high-speed double- locking toolholder-spindle joint based on a macro- micro scale hybrid method. International Journal of Precision Engineering & Manufacturing, 17(6), 741–753.CrossRefGoogle Scholar
  7. 7.
    Majumdar, A., & Bhushan, B. (1991). Fractal model of elastic-plastic contact between rough surfaces. Journal of Tribology, 113(1), 1–11.CrossRefGoogle Scholar
  8. 8.
    Sayles, R. S., & Surface, Thomas T. R. (1978). Topography as a nonstationary random process. Nature, 271, 431–434.CrossRefGoogle Scholar
  9. 9.
    Yan, W., & Komvopoulos, K. (1998). Contact analysis of elastic-platic fractal surfaces. Journal of Applied Physics, 84(7), 3617–3624.CrossRefGoogle Scholar
  10. 10.
    Wang, S., & Komvopoulos, K. (1994). A fractal theory of the interracial temperature distribution in the slow sliding regime: part I—elastic contact and heat transfer analysis. Journal of Tribology, Transactions of ASME, 116(4), 812–823.CrossRefGoogle Scholar
  11. 11.
    Chen, Q., Ma, Y., Huang, S., et al. (2014). Research on gears’ dynamic performance influenced by gear backlash based on fractal theory. Applied Surface Science, 313, 325–332.CrossRefGoogle Scholar
  12. 12.
    Wang, J., Gu, J., & Yan, Y. (2016). Study on the relationship between the stiffness of RV reducer and the profile modification method of cycloid-pin wheel. In International conference on intelligent robotics and applications (pp. 721–735). Cham: Springer.Google Scholar
  13. 13.
    Ren, Z. Y., Mao, S. M., Guo, W. C., et al. (2017). Tooth modification and dynamic performance of the cycloidal drive. Mechanical Systems and Signal Processing, 85, 857–866.CrossRefGoogle Scholar
  14. 14.
    Kim, K. H., Lee, C. S., & Ahn, H. J. (2009). Torsional rigidity of a two-stage cycloid drive. Transactions of the Korean Society of Mechanical Engineers A, 33(11), 1217–1224.CrossRefGoogle Scholar
  15. 15.
    Ciavarella, M., Demelio, G., Barber, J. R., et al. (1994). Linear elastic contact of the weierstrass profile. Proceedings Mathematical Physical & Engineering Sciences, 2000(456), 387–405.zbMATHGoogle Scholar
  16. 16.
    Xu, L. X., & Yang, Y. H. (2016). Dynamic modeling and contact analysis of a cycloid-pin gear mechanism with a turning arm cylindrical roller bearing. Mechanism and Machine Theory, 104, 327–349.CrossRefGoogle Scholar

Copyright information

© Korean Society for Precision Engineering 2019

Authors and Affiliations

  • Zhifeng Liu
    • 1
  • Tao Zhang
    • 1
  • Yida Wang
    • 2
  • Congbin Yang
    • 1
    Email author
  • Yongsheng Zhao
    • 1
  1. 1.Key Laboratory of Advanced Manufacturing TechnologyBeijing University of TechnologyBeijingPeople’s Republic of China
  2. 2.Shandong Institute of Space Electronic TechnologyYantaiPeople’s Republic of China

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