Skip to main content
SpringerLink
Log in

An investigation of the mechanical behavior of initially curved microplates under electrostatic actuation

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this article, we investigate the mechanical behavior of initially curved microplates under electrostatic actuation. Microplates are essential components of many Micro-Electro-Mechanical System devices; however, they commonly undergo an initial curvature imperfection, due to the microfabrication process. Initial curvature imperfection significantly affects the mechanical behavior of microplates. In this work, we derive a dynamic analogue of the von Kármán governing equation for such plates. These equations are then used to develop a reduced order model based on the Galerkin procedure to simulate the static and dynamic behavior of the microplate. Two profiles of initial curvature commonly encountered in microfabricated structures are considered, where one assumes a variation in shape along one dimension of the plate only (cylindrical bending shape) while the other assumes a variation in shape along both dimensions of the plate. Their effects on both the static and dynamic responses of the microplates are examined and compared. We validate the reduced order model by comparing the calculated static behavior and the fundamental natural frequency with those computed by a finite element model over a range of the initial plate rise. The static behavior of the microplate is investigated when varying the DC voltage. Then, the dynamic behavior of the microplate is examined under the application of a harmonic AC voltage superimposed to a DC voltage.

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

Similar content being viewed by others

References

  1. Zengerle, R., Ulrich, J., Kluge, S., Richter, M., Richter, A.: A bidirectional silicon micropump. Sens. Actuators A Phys. 50(1), 81–86 (1995)

    Article  Google Scholar 

  2. Zengerle, R., Richter, A., Sandmaier, H.: A micro membrane pump with electrostatic actuation. In: Proceedings of IEEE Micro Electro Mechanical Systems, 1992, MEMS’92, An Investigation of Micro Structures, Sensors, Actuators, Machines, and Robots (1992)

  3. Maillefer, D., van Lintel, H., Rey-Mermet, G., Hirschi, R.: A high-performance silicon micropump for an implantable drug delivery system. In: Twelfth IEEE International Conference on Micro Electro Mechanical Systems MEMS’99 (1999)

  4. Je, C.H., Lee, J., Yang, W.S., Kim, J., Cho, Y.-H.: A surface-micromachined capacitive microphone with improved sensitivity. J. Micromech. Microeng. 23(5), 055018 (2013)

    Article  Google Scholar 

  5. Suresh, K., Uma, G., Umapathy, M.: Design of a resonance-based mass sensor using a self-sensing piezoelectric actuator. Smart Mater. Struct. 21(2), 025015 (2012)

    Article  Google Scholar 

  6. Celep, Z.: Free flexural vibration of initially imperfect thin plates with large elastic amplitudes. ZAMM Z. Angew. Math. Mech. 56(9), 423–428 (1976)

    Article  MATH  Google Scholar 

  7. Celep, Z.: Shear and rotatory inertia effects on the large amplitude vibration of the initially imperfect plates. J. Appl. Mech. 47(3), 662–666 (1980)

    Article  MATH  Google Scholar 

  8. Yamaki, N., Chiba, M.: Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement–Part I: Theory. Thin Walled Struct. 1(1), 3–29 (1983)

    Article  Google Scholar 

  9. Yamaki, N., Otomo, K., Chiba, M.: Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement–Part II: Experiment. Thin Walled Struct. 1(2), 101–119 (1983)

    Article  MATH  Google Scholar 

  10. Marın, J., Perkins, N.C., Vorus, W.: Non-linear response of predeformed plates subject to harmonic in-plane edge loading. J. Sound Vib. 176(4), 515–529 (1994)

    Article  MATH  Google Scholar 

  11. Lin, C., Chen, L.: Large-amplitude vibration of an initially imperfect moderately thick plate. J. Sound Vib. 135(2), 213–224 (1989)

    Article  Google Scholar 

  12. Ostiguy, G.L., Sassi, S.: Effects of initial geometric imperfections on dynamic behavior of rectangular plates. Nonlinear Dyn. 3(3), 165–181 (1992)

    Article  Google Scholar 

  13. Liu, W., Yeh, F.: Non-linear vibrations of initially imperfect, orthotropic, moderately thick plates with edge restraints. J. Sound Vib. 165(1), 101–122 (1993)

    Article  MATH  Google Scholar 

  14. Amabili, M.: Theory and experiments for large-amplitude vibrations of rectangular plates with geometric imperfections. J. Sound Vib. 291(3), 539–565 (2006)

    Article  Google Scholar 

  15. Alijani, F., Amabili, M.: Theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges. J. Sound Vib. 332(14), 3564–3588 (2013)

    Article  Google Scholar 

  16. Chen, C.S., Cheng, W.S., Tan, A.H.: Non-linear vibration of initially stressed plates with initial imperfections. Thin Walled Struct. 43(1), 33–45 (2005)

    Article  Google Scholar 

  17. Chen, C.S., Hsu, C.Y.: Imperfection sensitivity in the nonlinear vibration oscillations of initially stressed plates. Appl. Math. Comput. 190(1), 465–475 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Huang, H.: Large-amplitude vibration of imperfect rectangular, circular and laminated plate with viscous damping (Ph.D. Dissertation).http://scholarworks.uno.edu/td/1924/ (2014)

  19. Faris, W.F.: Nonlinear dynamics of annular and circular plates under thermal and electrical loadings (Ph.D. Dissertation). https://vtechworks.lib.vt.edu/handle/10919/11100 (2003)

  20. Vogl, G.W., Nayfeh, A.H.: A reduced-order model for electrically actuated clamped circular plates. J. Micromech. Microeng. 15(4), 684–690 (2005)

    Article  Google Scholar 

  21. Zhao, X., Abdel-Rahman, E.M., Nayfeh, A.H.: A reduced-order model for electrically actuated microplates. J. Micromech. Microeng. 14(7), 900–906 (2004)

    Article  Google Scholar 

  22. Li, Z., Zhao, L., Jiang, Z., Ye, Z., Dai, L., Zhao, Y.: Mechanical behavior analysis on electrostatically actuated rectangular microplates. J. Micromech. Microeng. 25(3), 035007 (2015)

    Article  Google Scholar 

  23. Jallouli, A., Kacem, N., Bourbon, G., Le Moal, P., Walter, V., Lardies, J.: Pull-in instability tuning in imperfect nonlinear circular microplates under electrostatic actuation. Phys. Lett. A 380(46), 3886–3890 (2016)

    Article  Google Scholar 

  24. Rahman, M.M., Chowdhury, S.: Square diaphragm CMUT capacitance calculation using a new deflection shape function. J. Sens. 2011, Article ID 581910 (2011). https://doi.org/10.1155/2011/581910

  25. Akhbari, S., Sammoura, F., Yang, C., Heidari, A., Horsley, D., Lin, L.: Self-curved diaphragms by stress engineering for highly responsive pMUT. In: 28th IEEE International Conference on Micro Electro Mechanical Systems (MEMS) (2015)

  26. Ventsel, E., Krauthammer, T.: Thin Plates and Shells: Theory: Analysis, and Applications. CRC Press, Boca Raton (2001)

    Book  Google Scholar 

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

    MATH  Google Scholar 

  28. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Hoboken (2008)

    MATH  Google Scholar 

  29. Salmon, R.: Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431–44 (1983)

    Article  MATH  Google Scholar 

  30. Lobitz, D., Nayfeh, A., Mook, D.: Non-linear analysis of vibrations of irregular plates. J. Sound Vib. 50(2), 203–217 (1977)

    Article  MATH  Google Scholar 

  31. Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 12(5), 672–680 (2003)

    Article  Google Scholar 

  32. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Reduced-order models for MEMS applications. Nonlinear Dyn. 41(1–3), 211–236 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Saghir, S., Bellaredj, M., Ramini, A., Younis, M.: Initially curved microplates under electrostatic actuation: theory and experiment. J. Micromech. Microeng. 26(9), 095004 (2016)

    Article  Google Scholar 

  34. COMSOL Multiphysics, (http://www.comsol.com)

Download references

Author information

Authors and Affiliations

  1. 4700 King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

    S. Saghir & M. I. Younis

  2. University of Management and Technology (UMT), Daska Road, Sialkot, Pakistan

    S. Saghir

Authors
  1. S. Saghir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. I. Younis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. I. Younis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saghir, S., Younis, M.I. An investigation of the mechanical behavior of initially curved microplates under electrostatic actuation. Acta Mech 229, 2909–2922 (2018). https://doi.org/10.1007/s00707-018-2141-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2141-3

Search

Navigation