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Nonlinear vibration analysis of a circular micro-plate in two-sided NEMS/MEMS capacitive system by using harmonic balance method

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Abstract

In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Kármán plate theory including geometrical nonlinearity is used to obtain the deflection of the micro-plate. Galerkin decomposition method is then employed, and nonlinear ordinary differential equations (ODEs) of motion are determined. A harmonic balance method (HBM) is applied to equations and analytical relation for nonlinear frequency response (F–R) curves are derived for two categories (including and neglecting Casimir force) separately. The analytical results for three cases: (1) semi-linear vibration; (2) weakly nonlinear vibration; (3) highly nonlinear vibration, are validated by comparing with the numerical solutions. After validation, the effects of the voltage and Casimir force on the natural frequency of two-sided capacitor system are investigated. It is shown that by assuming Casimir force in small gap distances, reduction of the natural frequency is considerable. The influences of the applied voltage, damping, micro-plate thickness and Casimir force on the frequency response curves have been presented too. The results of this study can be useful for modeling circular parallel-plates in nano/microelectromechanical transducers such as microphones and pressure sensors.

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References

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

    Book  Google Scholar 

  2. Jia, X., Yang, J., Kitipornchai, S., et al.: Resonance frequency response of geometrically nonlinear micro-switches under electrical actuation. J. Sound Vib. 331, 3397–3411 (2012)

    Article  Google Scholar 

  3. Han, J., Zhang, Q., Wang, W.: Static bifurcation and primary resonance analysis of a MEMS resonator actuated by two symmetrical electrodes. Nonlinear Dyn. 80, 1585–1599 (2015)

    Article  Google Scholar 

  4. Zhang, Y., Zhao, Y.P.: Numerical and analytical study on the pull-in instability of micro-structure under electrostatic loading. Sens. Actuators A 127, 366–380 (2006)

    Article  Google Scholar 

  5. Faris, W.F.: Nonlinear Dynamics of Annular and Circular Plates Under Thermal and Electrical Loadings. Virginia Tech, Blacksburg (2003)

    Google Scholar 

  6. Saeedivahdat, A., Abdolkarimzadeh, F., Feyzi, A., et al.: Effect of thermal stresses on stability and frequency response of a capacitive microphone. Microelectron. J. 41, 865–873 (2010)

    Article  Google Scholar 

  7. Talebian, S., Rezazadeh, G., Fathalilou, M., et al.: Effect of temperature on pull-in voltage and natural frequency of an electrostatically actuated microplate. Mechatronics 20, 666–673 (2010)

    Article  Google Scholar 

  8. Vogl, G.W., Nayfeh, A.H.: Primary resonance excitation of electrically actuated clamped circular plates. Nonlinear Dyn. 47, 181–192 (2007)

    Article  MATH  Google Scholar 

  9. Saghir, S., Younis, M.I.: An investigation of the static and dynamic behavior of electrically actuated rectangular microplates. Int. J. Nonlinear Mech. 85, 81–93 (2016)

    Article  Google Scholar 

  10. Batra, R., Porfiri, M., Spinello, D.: Vibrations and pull-in instabilities of microelectromechanical von Kármán elliptic plates incorporating the Casimir force. J. Sound Vib. 315, 939–960 (2008)

    Article  Google Scholar 

  11. Rombach, P., Müllenborn, M., Klein, U., et al.: The first low voltage, low noise differential silicon microphone, technology development and measurement results. Sens. Actuators A 95, 196–201 (2002)

    Article  Google Scholar 

  12. Martin, D.T.: Design, Fabrication, and Characterization of a MEMS Dual-Backplate Capacitive Microphone. University of Florida, Gainesville (2007)

    Google Scholar 

  13. Liu, J.: Nonlinear Dynamics of a Dual-Backplate Capacitive MEMS Microphone. University of Florida, Gainesville (2007)

    Google Scholar 

  14. Mann, B.P., Liu, J., Hazra, S.: Correcting measurement nonlinearity in dynamic nanoindentation. In: ASME 2006 International Mechanical Engineering Congress and Exposition. 2006. American Society of Mechanical Engineers

  15. Batra, R., Porfiri, M., Spinello, D.: Reduced-order models for microelectromechanical rectangular and circular plates incorporating the Casimir force. Int. J. Solids Struct. 45, 3558–3583 (2008)

    Article  MATH  Google Scholar 

  16. Wang, Y.G., Lin, W.H., Li, X.M., et al.: Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force. Appl. Math. Modell. 35, 2348–2357 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kermani, M.M., Dehestani, M.: Anharmonic 1D actuator model including electrostatic and Casimir forces with fractional damping perturbed by an external force. Acta Mech. Sin. 34, 528–541 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, N., Aluru, N.: Effect of intermolecular force on the static/dynamic behaviour of M/NEM devices. Nanotechnology 25, 485204 (2014)

    Article  Google Scholar 

  19. Klimchitskaya, G., Mohideen, U., Mostepanenko, V.: Casimir and van der Waals forces between two plates or a sphere (lens) above a plate made of real metals. Phys. Rev. A: At. Mol. Opt. Phys. 61, 062107 (2000)

    Article  Google Scholar 

  20. Caruntu, D.I., Oyervides, R.: Frequency response reduced order model of primary resonance of electrostatically actuated MEMS circular plate resonators. Commun. Nonlinear Sci. Numer. Simul. 43, 261–270 (2017)

    Article  Google Scholar 

  21. Khodaparast, H.H., Madinei, H., Friswell, M.I., et al.: An extended harmonic balance method based on incremental nonlinear control parameters. Mech. Syst. Sig. Process. 85, 716–729 (2017)

    Article  Google Scholar 

  22. Liang, B., Zhang, L., Wang, B., et al.: A variational size-dependent model for electrostatically actuated NEMS incorporating nonlinearities and Casimir force. Physica E 71, 21–30 (2015)

    Article  Google Scholar 

  23. Peng, Z.K., Meng, G., Lang, Z.Q., et al.: Study of the effects of cubic nonlinear damping on vibration isolations using harmonic balance method. Int. J. Nonlinear Mech. 47, 1073–1080 (2012)

    Article  Google Scholar 

  24. Vogl, G.W.: Nonlinear Dynamics of Circular Plates Under Electrical Loadings for Capacitive Micromachined Ultrasonic Transducers (CMUTs). Virginia Tech, Blacksburg (2006)

    Google Scholar 

  25. Osterberg, P.M., Senturia, S.D.: M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures. J. Microelectromech. Syst. 6, 107–118 (1997)

    Article  Google Scholar 

  26. Zhao, X., Abdel-Rahman, E.M., Nayfeh, A.H.: A reduced-order model for electrically actuated microplates. J. Micromech. Microeng. 14, 900–906 (2004)

    Article  Google Scholar 

  27. Zhang, Y.: Large deflection of clamped circular plate and accuracy of its approximate analytical solutions. Sci. China Phys. Mech. 59, 624602 (2016)

    Article  Google Scholar 

  28. Liu, J., Martin, D.T., Kadirvel, K., et al.: Nonlinear model and system identification of a capacitive dual-backplate MEMS microphone. J. Sound Vib. 309, 276–292 (2008)

    Article  Google Scholar 

  29. Liu, J., Martin, D.T., Nishida, T., et al.: Harmonic balance nonlinear identification of a capacitive dual-backplate MEMS microphone. J. Microelectromech. Syst. 17, 698–708 (2008)

    Article  Google Scholar 

  30. Fang, F., Xia, G., Wang, J.: Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations. Acta Mech. Sin. 34, 561–577 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Firouzi, B., Zamanian, M., Hosseini, S.A.A.: Static and dynamic responses of a microcantilever with a T-shaped tip mass to an electrostatic actuation. Acta. Mech. Sin. 32, 1104–1122 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Caruntu, D.I., Oyervides, R.: Voltage response of primary resonance of electrostatically actuated MEMS clamped circular plate resonators. J. Comput. Nonlinear Dyn. 11, 041021 (2016)

    Article  Google Scholar 

  33. Sheikhlou, M., Shabani, R., Rezazadeh, G.: Nonlinear analysis of electrostatically actuated diaphragm-type micropumps. Nonlinear Dyn. 83, 951–961 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, Z.K., Zhao, L.B., Jiang, Z.D., et al.: Mechanical behavior analysis on electrostatically actuated rectangular microplates. J. Micromech. Microeng. 25, 035007 (2015)

    Article  Google Scholar 

  35. Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley, London (2008)

    MATH  Google Scholar 

  36. Chia, C.Y.: Nonlinear Analysis of Plates. McGraw-Hill, New York (1980)

    Google Scholar 

  37. Guyader, J.L.: Vibration in Continuous Media. Wiley, London (2013)

    Book  Google Scholar 

  38. Nayfeh, A.H., Lacarbonara, W.: On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. Batra, R.C., Porfiri, M., Spinello, D.: Vibrations of narrow microbeams predeformed by an electric field. J. Sound Vib. 309, 600–612 (2008)

    Article  Google Scholar 

  40. Kacem, N., Hentz, S., Pinto, D., et al.: Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors. Nanotechnology 20, 275501 (2009)

    Article  Google Scholar 

  41. Meirovitch, L.: Fundamentals of Vibrations. Waveland Press, New York (2010)

    Google Scholar 

Download references

Acknowledgements

We are deeply grateful to Prof. Jakob Søndergaard Jensen, who is chief of the Center for Acoustic-Mechanical Micro Systems (CAMM) in Technical University of Denmark (DTU), for his kind guidance and revisions that enriched the contents of this manuscript.

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Correspondence to Milad Saadatmand.

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Saadatmand, M., Shooshtari, A. Nonlinear vibration analysis of a circular micro-plate in two-sided NEMS/MEMS capacitive system by using harmonic balance method. Acta Mech. Sin. 35, 129–143 (2019). https://doi.org/10.1007/s10409-018-0794-8

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  • DOI: https://doi.org/10.1007/s10409-018-0794-8

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