Skip to main content
SpringerLink
Log in

Statistical analysis of the influence of tooth geometry in the performance of a harmonic drive

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

The objective of this research is to determine the influence of gear tooth geometrical variations in the performance of a double wave harmonic drive through statistical analysis. This work incorporates state of the art analytical and numerical models to evaluate kinematic error, load capacity, bending fatigue strength, and pitting. The geometric variables considered in this study include gear modulus, pressure angle, and tooth correction factor. The statistical analysis follows a three-level full-factorial design of experiments. Nonlinear dynamic simulation is accompanied by finite element analysis to estimate contact and bending stresses. Largest bending fatigue strength is also determined. Results demonstrate that gear modulus is the geometric parameter with prevalent influence on the kinematic error, and pitting life is rather high for all geometric variables considered.

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

Similar content being viewed by others

Abbreviations

\( \alpha \) :

Angle in coordinate system attached to the wave generator (rad)

\( \beta \) :

Angle in a reference coordinate system (rad)

\( \delta_{a} \) :

Setup gap of the flexible spline on the track of the wave generator (mm)

\( \xi \) :

Correction factor

\( \xi_{1} \) :

Correction factor for the circular spline

\( \xi_{2} \) :

Correction factor for the flexible spline

\( \theta_{e} \) :

Kinematic transmission error (rad)

\( \theta_{in} \) :

Angular position of the input shaft (rad)

\( \theta_{out} \) :

Angular position of the output shaft (rad)

\( \sigma_{e} \) :

Corrected endurance limit for the flexible spline material (MPa)

\( \sigma_{ut} \) :

Ultimate tension stress (MPa)

\( \sigma_{VMa} \) :

Equivalent amplitude of the von Mises stress (MPa)

\( \sigma_{VMm} \) :

Equivalent mean of the von Mises stress (MPa)

\( \sigma_{z} \) :

Peak normal stress on the teeth of the flexible spline (MPa)

\( \vartheta , \gamma \) :

Pitting strength empirical parameters

\( \varPhi \) :

Pressure angle (°)

\( a \) :

Larger semi-axis of the wave generator (mm)

\( a' \) :

Larger pitch semi-axis (mm)

\( b \) :

Smaller semi-axis of the waver generator (mm)

\( b' \) :

Smaller pitch semi-axis (mm)

\( d_{e} \) :

External diameter of the flexible spline (mm)

\( h_{f} \) :

Thickness of the flexible spline (mm)

\( i \) :

Transmission ratio

\( k \) :

Iteration time

\( m \) :

Modulus (mm)

\( m_{1} \) :

Material elastic properties of the circular spline (MPa−1)

\( m_{2} \) :

Material elastic properties of the flexible spline (MPa−1)

\( r \) :

Pitch of the rigid spline (mm)

\( t \) :

Time (s)

\( D_{i} \) :

Internal diameter of the rigid spline (mm)

\( \varvec{F} \) :

Vector of internal nodal forces (N)

\( \varvec{K} \) :

Jacobian matrix

\( N_{p} \) :

Number of cycles for pitting failure (cycles)

\( P \) :

Reference point on the rigid spline

\( \varvec{R} \) :

Vector of external nodal forces (N)

\( S_{f} \) :

Fatigue safety factor

\( \varvec{U} \) :

Vector of nodal displacements (mm)

\( Z_{1} \) :

Number of teeth of the rigid spline

\( Z_{2} \) :

Number of teeth of the flexible spline

References

  1. Gervini VI, Gomes SCP, Da Rosa VS (2003) A new robotic drive joint friction compensation mechanism using neural networks. J Braz Soc Mech Sci Eng 25(2):129–139

    Article  Google Scholar 

  2. De Lucena SE, Marcelino MA, Grandinetti FJ (2007) Low-cost PWM speed controller for an electric mini-baja type vehicle. J Braz Soc Mech Sci Eng 29(1):21–25

    Article  Google Scholar 

  3. Jeon HS, Oh SH (1999) A study on stress and vibration analysis of a steel and hybrid flexspline for harmonic drive. In: 10th International conference on composite structures, November 15–16 1999. Melbourne, Australia: Elsevier Ltd

  4. Kayabasi O, Erzincanli F (2007) Shape optimization of tooth profile of a flexspline for a harmonic drive by finite element modelling. Mater Des 28(2):441–447

    Article  Google Scholar 

  5. Ostapski W (2010) Analysis of the stress state in the harmonic drive generator-flexspline system in relation to selected structural parameters and manufacturing deviations. Bull Pol Acad Sci: Tech Sci 58(4):683–698

    MATH  Google Scholar 

  6. Gandhi PS, Ghorbel F (2005) High-speed precision tracking with harmonic drive systems using integral manifold control design. Int J Control 78(2):112–121

    Article  MATH  MathSciNet  Google Scholar 

  7. Tuttle TD, Seering WP (1996) Nonlinear model of a harmonic drive gear transmission. IEEE Trans Robot Autom 12(3):368–374

    Article  Google Scholar 

  8. Ostapski W, Mukha I (2007) Stress state analysis of harmonic drive elements by FEM. Bull Pol Acad Sci: Tech Sci 55(1):115–123

    MATH  Google Scholar 

  9. Péter J, Németh G (2012) Results of laboratory tests of harmonic gear drive. Des Mach Struct 2(1):35–51

    Google Scholar 

  10. Dong H, Wang D (2009) Elastic deformation characteristic of the flexspline in harmonic drive. In: 2009 ASME/IFToMM International conference on reconfigurable mechanisms and robots, ReMAR 2009, June 22–24 2009. London, United Kingdom: IEEE computer society

  11. Tjahjowidodo T, Al-Bender F, Van Brussel H (2013) Theoretical modelling and experimental identification of nonlinear torsional behaviour in harmonic drives. Mechatronics 23(5):497–504

    Article  Google Scholar 

  12. Sobieszanski-Sobieski J, Barthelemy JF, Riley KM (1981) Sensitivity of optimum solutions to problem parameters. Collection of technical papers—AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, 1981(Pt 1): 184–205

  13. Mastinu G, Gobbi M, Miano C (2010) Optimal design of complex mechanical systems. Springer, New York

    Google Scholar 

  14. Wilson CE, Sadler JP (2003) Kinematics and dynamics of machinery, 3rd edn. Prentice Hall, Pearson Education, Upper Saddle River, NJ, USA

  15. González Rey G, Frechilla Fernández P, José García Martín R (2007) Coeficiente de corrección en engranajes cilíndricos como factor de conversión entre sistemas AGMA e ISO. Ingeniería Mecánica 10(3):63–69

    Google Scholar 

  16. SolidWorks, Simulation premium: nonlinear handbook. 2010: Massachusetts, USA

  17. Córdoba E et al (2011) Transmisión flexondulatoria armónica. Universidad Nacional de Colombia, Bogotá

    Google Scholar 

  18. Bathe KJ (2003) Finite element procedures

  19. International, A (1998) Metal handbook, 2nd edn. Editorial Advisory Board, USA

    Google Scholar 

  20. Collins JA, Busby H, Staab G (2009) Mechanical design of machine elements and machines, 2nd edn. John Wiley & Sons, Hoboken, NJ, USA

  21. Wilde RA (1976) Failure in gears and related machine components. In: 20th meeting of the mechanical failures prevention group, May 8–10 1974. Washington, DC: National Bureau of Standard

  22. Norton R (2010) Machine design, an integrated approach, 5th edn. Prentice-Hall Inc., USA

    Google Scholar 

  23. Quinones A, et al. (2005) Influence of the friction force, the tooth correction coefficient and the normal force radial component in the form factor and the stress in the feet of spur gear’s teeth. In: Proceedings of the ASME design engineering division 2005, November 5–11 2005. Orlando: American Society of Mechanical Engineers

Download references

Acknowledgments

The National University of Colombia at Bogotá and the Indiana Space Grant Consortium (INSGC) supported this research effort. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the writers and do not necessarily reflect the views of the sponsors.

Author information

Authors and Affiliations

  1. Department of Mechanical and Mechatronic Engineering, National University of Colombia, Carrera 30 45-03, Office 453-403, Bogotá, Colombia

    Dennis León & Nelson Arzola

  2. Department of Mechanical Engineering, Purdue School of Engineering and Technology, Indiana University-Purdue University Indianapolis, 723 West Michigan Street, SL 260N, Indianapolis, IN, 46202-5132, USA

    Andrés Tovar

Authors
  1. Dennis León
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nelson Arzola
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrés Tovar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nelson Arzola.

Additional information

Technical Editor: Marcelo A. Trindade.

Appendix: Fitted models

Appendix: Fitted models

1.1 Load capacity

The equation for the fitted model is:

$$ \begin{gathered} LC = 8.77151 - 0.0160725\upphi - 32.0864m + 0.672273\xi + 0.0000453607\upphi^{2} + 0.0224681\upphi\;m \hfill \\ - 0.00063109\upphi\xi + 28.1708m^{2} - 1.03049m\xi - 0.0327807\xi^{2} \hfill \\ \end{gathered} $$

with R 2 = 99.98 %, mean absolute error = 0.0241476, and Durbin–Watson statistic = 1.21314 (P = 0.0031), confidence interval \( {\text{LC}} = [3.57501; 5.99652] \) (Fig. 12). The response surface is:

Fig. 12
figure 12

Load capacity response for a correction factor \( {\varvec{\upxi}} = 0.25 \) as a function of the module and the pressure angle

Full size image

1.2 Kinematic transmission error

The equation for the fitted model is:

$$ \begin{gathered} \theta_{e} = 0.294959 + 0.00563235\upphi + 0.165081m - 0.3339\xi - 0.000149184\upphi^{2} \hfill \\ + 0.00381561\upphi\,m + 0.0105043\upphi\xi - 0.361116m^{2} + 0.0211167m\xi - 0.00106844\xi^{2} \hfill \\ \end{gathered} $$

with R 2 = 24.33 %, mean absolute error = 0.0352834, and Durbin–Watson statistic = 1.61063 (P = 0.0426), confidence interval \( \theta_{e} = [0.30854; 0.38938] \). The response surfaces are (Figs. 13, 14):

Fig. 13
figure 13

Kinematic transmission error for a correction factor of the module and the pressure angle

Full size image
Fig. 14
figure 14

Kinematic transmission error for a modulus \({m}= 0.40 \) mm as a function of the correction factor and the pressure angle

Full size image

1.3 Fatigue strength

The equation for the fitted model is:

$$ \begin{gathered} S_{f} = - 2.23008 + 0.34134\upphi + 0.960787m + 2.35171\xi - 0.007408\upphi^{2} + 0.0194722\upphi\;m \hfill \\ - 0.05992\upphi\xi - 1.21444m^{2} - 0.117778m\xi - 0.0370667\xi^{2} \hfill \\ \end{gathered} $$

with R 2 = 69.95 %, mean absolute error = 0.135503, Durbin–Watson statistic = 2.60236 (P = 0.7453), confidence interval \( S_{f} = [1.77552; 2.34368] \). The response surfaces are (Figs. 15, 16):

Fig. 15
figure 15

Fatigue safety factor for a correction factor \( \xi = 0.25 \) as a function of the module and the pressure angle

Full size image
Fig. 16
figure 16

Fatigue safety factor for a modulus m = 0.40 mm as a function of the correction factor and the pressure angle

Full size image

1.4 Pitting life

The equation for the fitted model is:

$$ \begin{gathered} N_{p} = - 5.5651E91 + 4.5532E90\upphi - 6.3185E89m - 4.5535E90\xi - 9.1065E88\upphi^{2} + 9.1499E82\upphi\;m \hfill \\ - 1.098E83\upphi\xi + 1.2647E91m^{2} - 2.2766E91m\xi + 1.8213E91\xi^{2} \hfill \\ \end{gathered} $$

with R 2 = 31.73 %, mean absolute error = 1.954E90, Durbin–Watson statistic = 2.94603 (P = 0.9418), confidence interval \( N_{p} = [2.33704{\text{E}}90; 3.85479{\text{E}}90] \). The response surfaces are (Figs. 17, 18):

Fig. 17
figure 17

Pitting life for a correction factor \( \xi = 0.25 \) as a function of the modulus and the pressure angle

Full size image
Fig. 18
figure 18

Pitting life for a modulus \( {m} = 0.40 \) mm as a function of the correction factor and the pressure angle

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

León, D., Arzola, N. & Tovar, A. Statistical analysis of the influence of tooth geometry in the performance of a harmonic drive. J Braz. Soc. Mech. Sci. Eng. 37, 723–735 (2015). https://doi.org/10.1007/s40430-014-0197-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40430-014-0197-0

Keywords

Search

Navigation