Analytical Compliance Model for Right Circle Flexure Hinge Considering the Stress Concentration Effect

  • Weixiao Tuo
  • Xingfei LiEmail author
  • Yue Ji
  • Tengfei Wu
  • Ziming Xie
Regular Paper


In this paper, an analytical compliance model for right circle flexure hinge (RCFH) is presented with the stress concentration in consideration. The stress concentration caused by changes in RCFH’s cross-section usually happens at the weakest point. It has been shown to seriously affect RCFH’s compliance calculation. Based on the virtual work theory, superposition relationship of the deformation, as well as Castigliano’s second theorem, RCFH’s analytical compliance model considering the stress concentration effect is established. The model is calculated as a series of closed-form equations which are related with geometric dimensions and employed material. Complicated definite integrals existing in these compliance equations are proved to be correctly calculated through comparisons with other literatures. Finally, in order to examine the validity of the established model, finite element analysis (FEA) is conducted. The relative errors between the theoretical values obtained by the established model and FEA results are found within 20% for a wide range of geometric dimensions.


Right circle flexure hinge Compliance model Stress concentration Finite element analysis 



This research was funded by the Foundation of the National Natural Science Foundation of China (No. 61733012), the Foundation of the National Natural Science Foundation of China (No. 61703303), the Natural Science Foundation of Tianjin City (No. 17JCQNJC04100), the Project supported by State Key Laboratory of Precision Measuring Technology and Instruments (No. PILAB1705) and the Research Project of Tianjin Municipal Education Committee (No. 2017KJ086).


  1. 1.
    Lobontiu, N. (2002). Compliant mechanisms: Design of flexure hinges. Boca Raton: CRC Press.CrossRefGoogle Scholar
  2. 2.
    Lobontiu, N., & Garcia, E. (2003). Two-axis flexure hinges with axially-collocated and symmetric notches. Computers & Structures,81(13), 1329–1341.CrossRefGoogle Scholar
  3. 3.
    Liu, P. B., Yan, P., Zhang, Z., & Leng, T. T. (2015). Flexure-hinges guided nano-stage for precision manipulations: Design, modeling and control. International Journal of Precision Engineering and Manufacturing,16(11), 2245–2254.CrossRefGoogle Scholar
  4. 4.
    Bhagat, U., Shirinzadeh, B., Clark, L., Chea, P., Qin, Y., Tian, Y., et al. (2014). Design and analysis of a novel flexure-based 3-DOF mechanism. Mechanism and Machine Theory,74, 173–187.CrossRefGoogle Scholar
  5. 5.
    Bhattacharya, S., Chattaraj, R., Das, M., Patra, A., Bepari, B., & Bhaumik, S. (2015). Simultaneous parametric optimization of IPMC actuator for compliant gripper. International Journal of Precision Engineering and Manufacturing,16(11), 2289–2297.CrossRefGoogle Scholar
  6. 6.
    Sattler, R., Plötz, F., Fattinger, G., & Wachutka, G. (2002). Modeling of an electrostatic torsional actuator: Demonstrated with an RF MEMS switch. Sensors and Actuators, A: Physical,97, 337–346.CrossRefGoogle Scholar
  7. 7.
    Tian, Y., Shirinzadeh, B., & Zhang, D. (2010). Closed-form compliance equations of filleted V-shaped flexure hinges for compliant mechanism design. Precision Engineering,34(3), 408–418.CrossRefGoogle Scholar
  8. 8.
    Tian, Y., Shirinzadeh, B., Zhang, D., & Zhong, Y. (2010). Three flexure hinges for compliant mechanism designs based on dimensionless graph analysis. Precision Engineering,34(1), 92–100.CrossRefGoogle Scholar
  9. 9.
    Qin, Y. D., Zhao, X., Shirinzadeh, B., Tian, Y. L., & Zhang, D. W. (2018). Closed-form modeling and analysis of an XY flexure-based nano-manipulator. Chinese Journal of Mechanical Engineering,31(1), 7.CrossRefGoogle Scholar
  10. 10.
    Wang, D. H., Yang, Q., & Dong, H. M. (2011). A monolithic compliant piezoelectric-driven microgripper: Design, modeling, and testing. IEEE/ASME Transactions on Mechatronics,18(1), 138–147.CrossRefGoogle Scholar
  11. 11.
    Wang, J., Yang, Y., Yang, R., Feng, P., & Guo, P. (2019). On the validity of compliance-based matrix method in output compliance modeling of flexure-hinge mechanism. Precision Engineering. Scholar
  12. 12.
    Zettl, B., Szyszkowski, W., & Zhang, W. J. (2005). On systematic errors of two-dimensional finite element modeling of right circular planar flexure hinges. Journal of Mechanical Design,127(4), 782–787.CrossRefGoogle Scholar
  13. 13.
    Shen, Y., Chen, X., Jiang, W., & Luo, X. (2014). Spatial force-based non-prismatic beam element for static and dynamic analyses of circular flexure hinges in compliant mechanisms. Precision Engineering,38(2), 311–320.CrossRefGoogle Scholar
  14. 14.
    Zettl, B., Szyszkowski, W., & Zhang, W. J. (2005). Accurate low DOF modeling of a planar compliant mechanism with flexure hinges: The equivalent beam methodology. Precision Engineering,29(2), 237–245.CrossRefGoogle Scholar
  15. 15.
    Paros, J. M., & Weisbord, L. (1965). How to design flexure hinges. Machine Design,37(27), 151–156.Google Scholar
  16. 16.
    Valentini, P. P., & Pennestrì, E. (2017). Elasto-kinematic comparison of flexure hinges undergoing large displacement. Mechanism and Machine Theory,110, 50–60.CrossRefGoogle Scholar
  17. 17.
    Yang, M., Du, Z., & Dong, W. (2016). Modeling and analysis of planar symmetric superelastic flexure hinges. Precision Engineering,46, 177–183.CrossRefGoogle Scholar
  18. 18.
    Li, Y., Xiao, S., Xi, L., & Wu, Z. (2014). Design, modeling, control and experiment for a 2-DOF compliant micro-motion stage. International Journal of Precision Engineering and Manufacturing,15(4), 735–744.CrossRefGoogle Scholar
  19. 19.
    Wu, Y., & Zhou, Z. (2002). Design calculations for flexure hinges. Review of Scientific Instruments,73(8), 3101–3106.CrossRefGoogle Scholar
  20. 20.
    Lobontiu, N., Garcia, E., & Canfield, S. (2003). Torsional stiffness of several variable rectangular cross-section flexure hinges for macro-scale and MEMS applications. Smart Materials and Structures,13(1), 12–19.CrossRefGoogle Scholar
  21. 21.
    Tseytlin, Y. M. (2002). Notch flexure hinges: An effective theory. Review of Scientific Instruments,73(9), 3363–3368.CrossRefGoogle Scholar
  22. 22.
    Yong, Y. K., Lu, T. F., & Handley, D. C. (2008). Review of circular flexure hinge design equations and derivation of empirical formulations. Precision Engineering,32(2), 63–70.CrossRefGoogle Scholar
  23. 23.
    Chen, G., & Howell, L. L. (2009). Two general solutions of torsional compliance for variable rectangular cross-section hinges in compliant mechanisms. Precision Engineering,33(3), 268–274.CrossRefGoogle Scholar
  24. 24.
    Hearn, E. J. (1997). Mechanics of materials, 3th. Oxford: Butterworth-Heinemann.Google Scholar
  25. 25.
    Xu, N., Dai, M., & Zhou, X. (2017). Analysis and design of symmetric notch flexure hinges. Advances in Mechanical Engineering. Scholar
  26. 26.
    Li, T. M., Zhang, J. L., & Jiang, Y. (2015). Derivation of empirical compliance equations for circular flexure hinge considering the effect of stress concentration. International Journal of Precision Engineering and Manufacturing,16(8), 1735–1743.CrossRefGoogle Scholar
  27. 27.
    Awtar, S., Slocum, A. H., & Sevincer, E. (2007). Characteristics of beam-based flexure modules. Journal of Mechanical Design,129(6), 625–639.CrossRefGoogle Scholar
  28. 28.
    Linß, S., Schorr, P., & Zentner, L. (2017). General design equations for the rotational stiffness, maximal angular deflection and rotational precision of various notch flexure hinges. Mechanical Sciences,8(1), 29.CrossRefGoogle Scholar
  29. 29.
    Meng, Q. (2012). A design method for flexure-based compliant mechanisms on the basis of stiffness and stress characteristics. Doctoral dissertation, University of Bologna.Google Scholar
  30. 30.
    Young, W. C., Budynas, R. G., & Sadegh, A. M. (2002). Roark’s formulas for stress and strain. New York: McGraw-Hill.Google Scholar

Copyright information

© Korean Society for Precision Engineering 2020

Authors and Affiliations

  1. 1.State Key Laboratory of Precision Measuring Technology and InstrumentsTianjin UniversityTianjinChina
  2. 2.Key Laboratory of Advanced Electrical Engineering and Energy TechnologyTianjin Polytechnic UniversityTianjinChina

Personalised recommendations