Skip to main content
SpringerLink
Log in

Modeling and approximate analytical solution of nonlinear behaviors for a self-excited electrostatic actuator

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Analytical modeling and solution of nonlinear vibration behaviors are vitally important for the optimization of vibratory actuators. In this paper, the nonlinear vibration characteristics of a self-excited electrostatic actuator are investigated. The actuator consists of two plate electrodes and a metal cantilever, which is fixed on an isolated insulating base and placed in the middle of the electrodes. When a DC voltage is applied to the electrodes, the cantilever is excited into reciprocating vibration autonomously and collides with the electrodes with strongly nonlinear features. During modeling, the collision is considered as a transient process and only the velocities of the cantilever before and after the collisions are concerned. By using the phase plane method, the limit cycle and boundary conditions of the self-excited oscillation of the cantilever are obtained and the strongly nonlinear problem of the entire system is converted into a weakly nonlinear problem of the cantilever moving between the electrodes. The nonlinear equations are then solved by using the averaging method, and the approximate analytical results demonstrate good consistency with the numerical and experimental results. Besides, the approximate analytical results are also adopted in the optimization of the actuator for enhanced power density and efficiency. The modeling and solution methods provide a framework for further optimization of the electrostatic actuator in the application field of centimeter-scale robotics.

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

Data availability

All data needed to evaluate the conclusions are included in the paper.

Code availability

The code that supports the findings of this study is available from the author upon reasonable request.

References

  1. Wood, R.J., Steltz, E., Fearing, R.: Optimal energy density piezoelectric bending actuators. Sens. Actuators A 119(2), 476–488 (2005). https://doi.org/10.1016/j.sna.2004.10.024

    Article  Google Scholar 

  2. Zou, Y., Zhang, W., Zhang, Z.: Liftoff of an electromagnetically driven insect-inspired flapping-wing robot. IEEE Trans. Robot. 32(5), 1285–1289 (2016). https://doi.org/10.1109/Tro.2016.2593449

    Article  Google Scholar 

  3. Zhao, H., Hussain, A.M., Duduta, M., Vogt, D.M., Wood, R.J., Clarke, D.R.: Compact dielectric elastomer linear actuators. Adv. Funct. Mater. 28(42), 1804328 (2018). https://doi.org/10.1002/adfm.201804328

    Article  Google Scholar 

  4. Chen, Y., Zhao, H., Mao, J., Chirarattananon, P., Helbling, E.F., Hyun, NsP, Clarke, D.R., Wood, R.J.: Controlled flight of a microrobot powered by soft artificial muscles. Nature 575(7782), 324–329 (2019). https://doi.org/10.1038/s41586-019-1737-7

    Article  Google Scholar 

  5. Jafferis, N.T., Helbling, E.F., Karpelson, M., Wood, R.J.: Untethered flight of an insect-sized flapping-wing microscale aerial vehicle. Nature 570(7762), 491–495 (2019). https://doi.org/10.1038/s41586-019-1322-0

    Article  Google Scholar 

  6. Goldberg, B., Zufferey, R., Doshi, N., Helbling, E.F., Whittredge, G., Kovac, M., Wood, R.J.: Power and control autonomy for high-speed locomotion with an insect-scale legged robot. IEEE Robot. Autom. Lett. 3(2), 987–993 (2018). https://doi.org/10.1109/lra.2018.2793355

    Article  Google Scholar 

  7. Rahmanian, S., Hosseini-Hashemi, S., SoltanRezaee, M.: Efficient large amplitude primary resonance in in-extensional nanocapacitors: nonlinear mean curvature component. Sci. Rep. 9(1), 1–18 (2019). https://doi.org/10.1038/s41598-019-56726-y

    Article  Google Scholar 

  8. SoltanRezaee, M., Bodaghi, M.: Nonlinear dynamic stability of piezoelectric thermoelastic electromechanical resonators. Sci. Rep. 10(1), 1–14 (2020). https://doi.org/10.1038/s41598-020-59836-0

    Article  Google Scholar 

  9. Belhaq, M., Fiedler, B., Lakrad, F.: Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt–Poincaré method. Nonlinear Dyn. 23(1), 67–86 (2000). https://doi.org/10.1023/A:1008316010341

    Article  MATH  Google Scholar 

  10. D’Urso, B., Van Handel, R., Odom, B., Hanneke, D., Gabrielse, G.: Single-particle self-excited oscillator. Phys. Rev. Lett. 94(11), 113002 (2005). https://doi.org/10.1103/PhysRevLett.94.113002

    Article  Google Scholar 

  11. Warminski, J.: Nonlinear dynamics of self-, parametric, and externally excited oscillator with time delay: van der pol versus Rayleigh models. Nonlinear Dyn. 99(1), 35–56 (2020). https://doi.org/10.1007/s11071-019-05076-5

    Article  MATH  Google Scholar 

  12. Di Nino, S., Luongo, A.: Nonlinear aeroelastic behavior of a base-isolated beam under steady wind flow. Int. J. Nonlinear Mech. 119, 103340 (2020). https://doi.org/10.1016/j.ijnonlinmec.2019.103340

    Article  Google Scholar 

  13. Liao, S.J.: An analytic approximate approach for free oscillations of self-excited systems. Int. J. Nonlinear Mech. 39(2), 271–280 (2004). https://doi.org/10.1016/S0020-7462(02)00174-9

    Article  MathSciNet  MATH  Google Scholar 

  14. Sun, K., Qiu, J., Karimi, H.R., Gao, H.: A novel finite-time control for nonstrict feedback saturated nonlinear systems with tracking error constraint. IEEE Trans. Syst. Man Cybern. Syst. (2019). https://doi.org/10.1109/TSMC.2019.2958072

    Article  Google Scholar 

  15. Sun, K., Jianbin, Q., Karimi, H.R., Fu, Y.: Event-triggered robust fuzzy adaptive finite-time control of nonlinear systems with prescribed performance. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2979129

    Article  Google Scholar 

  16. Sun, K., Liu, L., Qiu, J., Feng, G.: Fuzzy adaptive finite-time fault-tolerant control for strict-feedback nonlinear systems. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2965890

    Article  Google Scholar 

  17. Yan, X., Qi, M., Lin, L.: An autonomous impact resonator with metal beam between a pair of parallel-plate electrodes. Sens. Actuators A 199, 366–371 (2013). https://doi.org/10.1016/j.sna.2013.06.012

    Article  Google Scholar 

  18. Zhu, Y., Yan, X., Qi, M., Liu, Z., Zhang, X., Lin, L.: A dc drive electrostatic comb actuator based on self-excited vibration. In: 2018 IEEE Micro Electro Mechanical Systems (MEMS), pp. 592–595. IEEE (2018). https://doi.org/10.1109/MEMSYS.2018.8346623

  19. Qi, M., Zhu, Y., Liu, Z., Zhang, X., Yan, X., Lin, L.: A fast-moving electrostatic crawling insect. In: 2017 IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), pp. 761–764. IEEE (2017). https://doi.org/10.1109/memsys.2017.7863519

  20. Yan, X., Qi, M., Lin, L.: Self-lifting artificial insect wings via electrostatic flapping actuators. In: 2015 28th IEEE International Conference on Micro Electro Mechanical Systems (MEMS), pp. 22–25. IEEE (2015). https://doi.org/10.1109/memsys.2015.7050876

  21. Ding, W.: Self-Excited Vibration: Theory, Paradigms, and Research Methods. Springer, Berlin (2010)

    Book  Google Scholar 

  22. Liu, Z., Qi, M., Zhu, Y., Huang, D., Zhang, X., Lin, L., Yan, X.: Mechanical response of the isolated cantilever with a floating potential in steady electrostatic field. Int. J. Mech. Sci. 161, 105066 (2019). https://doi.org/10.1016/j.ijmecsci.2019.105066

    Article  Google Scholar 

  23. Liu, Z., Yan, X., Qi, M., Zhu, Y., Zhang, X., Lin, L.: Asymmetric charge transfer phenomenon and its mechanism in self-excited electrostatic actuator. In: 2018 IEEE Micro Electro Mechanical Systems (MEMS), pp. 588–591. IEEE (2018). https://doi.org/10.1109/MEMSYS.2018.8346622

  24. Shmulevich, S., Hotzen, I., Elata, D.: The electromechanical response of a self-excited mems franklin oscillator. In: 2015 28th IEEE International Conference on Micro Electro Mechanical Systems (MEMS), pp. 41–44. IEEE (2015). https://doi.org/10.1109/MEMSYS.2015.7050881

  25. McMillan, A.: A non-linear friction model for self-excited vibrations. J. Sound Vib. 205(3), 323–335 (1997). https://doi.org/10.1006/jsvi.1997.1053

    Article  MATH  Google Scholar 

  26. White, F.M., Corfield, I.: Viscous Fluid Flow, vol. 3. McGraw-Hill, New York (2006)

    Google Scholar 

  27. Deng, X., Schenato, L., Wu, W.C., Sastry, S.S.: Flapping flight for biomimetic robotic insects: Part I-system modeling. IEEE Trans. Robot. 22(4), 776–788 (2006). https://doi.org/10.1109/TRO.2006.875480

    Article  Google Scholar 

  28. Palmer, H.B.: The capacitance of a parallel-plate capacitor by the Schwartz–Christoffel transformation. Electr. Eng. 56(3), 363–368 (1937). https://doi.org/10.1109/EE.1937.6540485

    Article  Google Scholar 

  29. Hosseini, M., Zhu, G., Peter, Y.A.: A new formulation of fringing capacitance and its application to the control of parallel-plate electrostatic micro actuators. Analog Integr. Circ. Signal Process. 53(2–3), 119–128 (2007). https://doi.org/10.1007/s10470-007-9067-3

    Article  Google Scholar 

  30. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1995)

    Book  Google Scholar 

  31. Tang, B., Brennan, M.: A comparison of the effects of nonlinear damping on the free vibration of a single-degree-of-freedom system. J. Vib. Acoust. (2012). https://doi.org/10.1115/1.4005010

    Article  Google Scholar 

  32. Nabae, H., Ikeda, K.: Effect of elastic element on self-excited electrostatic actuator. Sens. Actuators A 279, 725–732 (2018). https://doi.org/10.1016/j.sna.2018.06.045

    Article  Google Scholar 

Download references

Funding

This work is supported by National Natural Science Foundation of China (Grant No. 12002017), China Postdoctoral Science Foundation (Grant No. 2019M650441) and the 111 Project (Grant No. B08009).

Author information

Authors and Affiliations

  1. School of Energy and Power Engineering, Beihang University, Beijing, China

    Yangsheng Zhu, Zhiwei Liu, Mingjing Qi & Xiaojun Yan

  2. Collaborative Innovation Center of Advanced Aero-Engine, Beijing, China

    Mingjing Qi & Xiaojun Yan

  3. National Key Laboratory of Science and Technology on Aero-Engine Aero-thermodynamics, Beijing, China

    Mingjing Qi & Xiaojun Yan

  4. Beijing Key Laboratory of Aero-Engine Structure and Strength, Beijing, China

    Mingjing Qi & Xiaojun Yan

Authors
  1. Yangsheng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mingjing Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaojun Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaojun Yan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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, Y., Liu, Z., Qi, M. et al. Modeling and approximate analytical solution of nonlinear behaviors for a self-excited electrostatic actuator. Nonlinear Dyn 103, 279–292 (2021). https://doi.org/10.1007/s11071-020-06157-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-06157-6

Keywords

Search

Navigation