Multifidelity modeling and comparative analysis of electrically coupled microbeams under squeeze-film damping effect


We investigate the nonlinear dynamic response of a device made of two electrically coupled cantilever microbeams. The vibrations of the microbeams triggered by the electric actuation lead to the redistribution of the air flow in the gap separating them and induce a damping effect, known as the squeeze-film damping. This nonlinear dissipation mechanism is prominent when encapsulating and operating the microstructure under high gas pressure. We present different modeling approaches to analyze the impact of the squeeze-film damping on the dynamic behavior of the microsystem. We first develop a nonlinear multi-physics model of the device by coupling Euler–Bernoulli beam equations with the nonlinear Reynolds equation and use the Galerkin decomposition and differential quadrature method to discretize the structural and fluidic domains, respectively. We consider also another modeling approach based on approximating the squeeze-film damping force by a nonlinear analytical expression. This approach is widely used in the literature and referred to as partially coupled model in this paper. We conduct a comparative study of the nonlinear dynamic responses obtained from the two models under different operating conditions in terms of electric actuation and applied pressure. The simulated frequency and force-response curves show the limitations of the partially coupled model to capture properly the microsystem dynamics, especially when approaching the onset of the pull-in instability and exciting the microsystem with an AC voltage near resonance. As such, we propose a correction factor to the partially coupled model which is much less computationally demanding to obtain good match with the fully coupled model. The selection of the correction factor depends on the thickness ratio, the ambient pressure, and the excitation frequency. The influence of the ambient pressure and the thickness ratio between the two microbeams were also examined. We observe that operating the microsystem at a reduced ambient pressure or when reducing one of the microbeams’ thickness can lead to a premature instability of the dynamic solution which reduces the maximum amplitude of the vibrating microbeams. This feature can be exploited for switching applications but it constitutes an undesirable effect for resonators.

This is a preview of subscription content, access via your institution.

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


  1. 1.

    Nayfeh, A.H., Ouakad, H.M., Najar, F., Choura, S., Abdel-Rahman, E.M.: Nonlinear dynamics of a resonant gas sensor. Nonlinear Dyn. 59, 607–618 (2010)

    Article  Google Scholar 

  2. 2.

    Ghommem, M., Abdelkefi, A.: Novel design of microgyroscopes employing electrostatic actuation and resistance-change based sensing. J. Sound Vib. 411, 278–288 (2017)

    Article  Google Scholar 

  3. 3.

    Ghommem, M., Abdelkefi, A.: Nonlinear analysis of rotating nanocrystalline silicon microbeams for microgyroscope applications. Microsyst. Technol. 23, 5931–5946 (2017)

    Article  Google Scholar 

  4. 4.

    Ghommem, M., Abdelkefi, A.: Nonlinear reduced-order modeling and effectiveness of electrically-actuated microbeams for bio-mass sensing applications. Int. J. Mech. Mater. Design 15, 125–143 (2019)

    Article  Google Scholar 

  5. 5.

    Ben Sassi, S., Khater, M.E., Najar, F., Abdel-Rahman, E.M.: A square wave is the most efficient and reliable waveform for resonant actuation of micro switches. J. Micromech. Microeng. 28, 1–14 (2018)

    Article  Google Scholar 

  6. 6.

    Samaali, H., Najar, F.: Design of a capacitive MEMS double beam switch using dynamic pull-in actuation at very low voltage. Microsyst. Technol. 23, 5317–5327 (2017)

    Article  Google Scholar 

  7. 7.

    Bouchaala, A., Jaber, N., Yassine, O., Shekhah, O., Chernikova, V., Eddaoudi, M., Younis, M.I.: Nonlinear-based MEMS sensors and active switches for gas detection. Sensors 16, 758 (2016)

    Article  Google Scholar 

  8. 8.

    Jrad, M., Younis, M.I., Najar, F.: Modeling and design of an electrically actuated resonant microswitch. J. Vib. Control 22, 559–569 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ouakad, H., Younis, M.: On using the dynamic snap-through motion of MEMS initially curved microbeams for filtering applications. J. Sound Vib. 2, 555–568 (2014)

    Article  Google Scholar 

  10. 10.

    Samaali, H., Najar, F., Choura, S., Nayfeh, A.H., Masmoudi, M.: A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage. Nonlinear Dyn. 63, 719–734 (2011)

    Article  Google Scholar 

  11. 11.

    Hammad, B.K., Abdel-Rahman, E.M., Nayfeh, A.H.: Modeling and analysis of electrostatic MEMS filters. Nonlinear Dyn. 60, 385–401 (2010)

    Article  Google Scholar 

  12. 12.

    Ilyas, S., Jaber, N., Younis, M.I.: MEMS logic using mixed-frequency excitation. J. Microelectromech. Syst. 26, 1140–1146 (2017)

    Article  Google Scholar 

  13. 13.

    Chappanda, K.N., Ilyas, S., Younis, M.I.: Micro-mechanical resonators for dynamically reconfigurable reduced voltage logic gates. J. Micromech. Microeng. 28, 055009 (2018)

    Article  Google Scholar 

  14. 14.

    Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, Berlin (2011)

    Book  Google Scholar 

  15. 15.

    Ahmed, M.S., Ghommem, M., Abdelkefi, A.: Shock response of electrostatically coupled microbeams under the squeeze-film damping effect. Acta Mech. 229(12), 5051–5065 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Yagubizade, H., Younis, M.I.: The effect of squeeze-film damping on the shock response of clamped-clamped microbeams. J. Dyn. Syst. Meas. Control 134(1), 011017 (2012)

    Article  Google Scholar 

  17. 17.

    Mo, Y., Du, L., Qu, B., Peng, B., Yang, J.: Squeeze film air damping ratio analysis of a silicon capacitive micromechanical accelerometer. Microsyst. Technol. 24, 1089–1095 (2014)

    Article  Google Scholar 

  18. 18.

    Ouakad, H., Al-Qahtani, H., Hawwa, M.A.: Influence of squeeze-film damping on the dynamic behavior of a curved micro-beam. Adv. Mech. Eng. 8, 1–8 (2016)

    Article  Google Scholar 

  19. 19.

    Liu, C.-C., Wang, C.-C.: Numerical investigation into nonlinear dynamic behavior of electrically-actuated clamped-clamped micro-beam with squeeze-film damping effect. Appl. Math. Model. 38, 3269–3280 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ilyas, S., Al-Hafiz, M.A., Chappanda, K.N., Ramini, A., Younis, M.: An experimental and theoretical investigation of electrostatically-coupled cantilever microbeams. Sens. Actuators A Phys. 247, 368–378 (2016)

    Article  Google Scholar 

  21. 21.

    Ahmed, M.S., Ghommem, M., Abdelkefi, A.: Nonlinear analysis and characteristics of electrically-coupled microbeams under mechanical shock. Microsyst. Technol. 25(3), 829–843 (2019)

    Article  Google Scholar 

  22. 22.

    Meirovitch, L., Parker, R.G.: Fundamentals of vibrations. Appl. Mech. Rev. 54, B100 (2001)

    Article  Google Scholar 

  23. 23.

    Sassi, S.B., Najar, F.: Strong nonlinear dynamics of MEMS and NEMS structures based on semi-analytical approaches. Commun. Nonlinear Sci. Numer. Simul. 61, 1–21 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Samaali, H., Najar, F., Choura, S.: Dynamic study of a capacitive MEMS switch with double clamped-clamped microbeams. Shock Vib. 2014, 1–7 (2014).

    Article  Google Scholar 

  25. 25.

    Khater, M.E., Vummidi, K., Abdel-Rahman, E.M., Nayfeh, A.H., Raman, S.: Dynamic actuation methods for capacitive MEMS shunt switches. J. Micromech. Microeng. 21(3), 035009 (2011)

    Article  Google Scholar 

  26. 26.

    Pandey, A.K., Pratap, R.: Effect of flexural modes on squeeze film damping in MEMS cantilever resonators. J. Micromech. Microeng. 17(12), 2475 (2007)

    Article  Google Scholar 

  27. 27.

    Najar, F., Nayfeh, A.H., Abdel-Rahman, E.M., Choura, S., El-Borgi, S.: Nonlinear analysis of MEMS electrostatic microactuators: primary and secondary resonances of the first mode. J. Vib. Control 16(9), 1321–1349 (2010)

    MathSciNet  Article  Google Scholar 

Download references


The author M. Ghommem gratefully acknowledges the financial support via the Biosciences and Bioengineering Research Institute and American University of Sharjah Grant Number EN0277-BBRI18.

Author information



Corresponding author

Correspondence to Mehdi Ghommem.

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

Verify currency and authenticity via CrossMark

Cite this article

Najar, F., Ghommem, M. & Abdelkefi, A. Multifidelity modeling and comparative analysis of electrically coupled microbeams under squeeze-film damping effect. Nonlinear Dyn 99, 445–460 (2020).

Download citation


  • Electric coupling
  • Squeeze-film damping
  • Nonlinear modeling
  • Dynamic pull-in