Skip to main content
SpringerLink
Log in

An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

In this study we deal with a microelectromechanical system (MEMS) and develop a dynamical integrity analysis to interpret and predict the experimental response. The device consists of a clamped-clamped polysilicon microbeam, which is electrostatically and electrodynamically actuated. It has non-negligible imperfections, which are a typical consequence of the microfabrication process. A single-mode reduced-order model is derived and extensive numerical simulations are performed in a neighborhood of the first symmetric natural frequency, via frequency response diagrams and behavior chart. The typical softening behavior is observed and the overall scenario is explored, when both the frequency and the electrodynamic voltage are varied. We show that simulations based on direct numerical integration of the equation of motion in time yield satisfactory agreement with the experimental data. Nevertheless, these theoretical predictions are not completely fulfilled in some aspects. In particular, the range of existence of each attractor is smaller in practice than in the simulations. This is because these theoretical curves represent the ideal limit case where disturbances are absent, which never occurs under realistic conditions. A reliable prediction of the actual (and not only theoretical) range of existence of each attractor is essential in applications. To overcome this discrepancy and extend the results to the practical case where disturbances exist, a dynamical integrity analysis is developed. After introducing dynamical integrity concepts, integrity profiles and integrity charts are drawn. They are able to describe if each attractor is robust enough to tolerate the disturbances. Moreover, they detect the parameter range where each branch can be reliably observed in practice and where, instead, becomes vulnerable, i.e. they provide valuable information to operate the device in safe conditions according to the desired outcome and depending on the expected disturbances.

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

Similar content being viewed by others

References

  1. Senturia SD (2001) Microsystem design. Kluwer Academic, Dordrecht

    Google Scholar 

  2. Younis MI (2011) MEMS linear and nonlinear statics and dynamics. Springer, Berlin

    Book  Google Scholar 

  3. Towfighian S, Heppler GR, Abdel Rahman EM (2012) Low-voltage closed loop MEMS actuators. Nonlinear Dyn 69:565–575

    Article  Google Scholar 

  4. Krylov S, Ilic BR, Schreiber D, Seretensky S, Craighead H (2008) The pull-in behavior of electrostatically actuated bistable microbeams. J Micromech Microeng 18:55026

    Article  Google Scholar 

  5. Mestrom RMC, Fey RHB, van Beek JTM, Phan KL, Nijmeijer H (2008) Modelling the dynamics of a MEMS resonator: simulations and experiments. Sens Actuators A 142:306–315

    Article  Google Scholar 

  6. Rhoads JF, Shaw SW, Turner KL, Moehlis J, DeMartini BE (2006) Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators. J Sound Vib 296:797–829

    Article  ADS  Google Scholar 

  7. Gottlieb O, Champneys A (2005) Global bifurcations of nonlinear thermoelastic microbeams subject to electrodynamic actuation. In: Rega G, Vestroni F (eds) IUTAM symp. on chaotic dynamics and control of systems and processes in mechanics solid mechanics and its applications, vol 122. Springer, Berlin, pp 117–126

    Chapter  Google Scholar 

  8. Alsaleem FM, Younis MI, Ouakad HM (2009) On the nonlinear resonances and dynamic pull-in of electrostatically actuated resonators. J Micromech Microeng 19:045013

    Article  Google Scholar 

  9. Hornstein S, Gottlieb O (2012) Nonlinear multimode dynamics and internal resonances of the scan process in noncontacting atomic force microscopy. J Appl Phys 112:074314

    Article  ADS  Google Scholar 

  10. DeMartini BE, Butterfield HE, Moehlis J, Turner KL (2007) Chaos for a microelectromechanical oscillator governed by the nonlinear Mathieu equation. J Microelectromech Syst 16:1314–1323

    Article  Google Scholar 

  11. De SK, Aluru NR (2006) Complex nonlinear oscillations in electrostatically actuated microstructures. J Microelectromech Syst 15:355–369

    Article  Google Scholar 

  12. Chavarette FR, Balthazar JM, Felix JLP, Rafikov M (2009) A reducing of a chaotic movement to a periodic orbit, of a micro-electro-mechanical system, by using an optimal linear control design. Commun Nonlinear Sci Numer Simul 14:1844–1853

    Article  ADS  Google Scholar 

  13. Kniffka TJ, Welte J, Ecker H (2012) Stability analysis of a time-periodic 2-dof MEMS structure. In: AIP conf. proc. ICNPAA international conference on mathematical problems in engineering, aerospace and sciences, vol 1493, pp 559–566

    Google Scholar 

  14. Dick AJ, Balachandran B, Mote CD Jr (2008) Intrinsic localized modes in microresonator arrays and their relationship to nonlinear vibration modes. Nonlinear Dyn 54:13–29

    Article  MATH  Google Scholar 

  15. Cho H, Jeong B, Yu MF, Vakakis AF, McFarland DM, Bergman LA (2012) Nonlinear hardening and softening resonances in micromechanical cantilever-nanotube systems originated from nanoscale geometric nonlinearities. Int J Solids Struct 49:2059–2065

    Article  Google Scholar 

  16. Kozinsky I, Postma HWC, Kogan O, Husain A, Roukes ML (2007) Basins of attraction of a nonlinear nanomechanical resonator. Phys Rev Lett 99:201–207

    Article  Google Scholar 

  17. Virgin LN (2000) Introduction to experimental nonlinear dynamics: a case study in mechanical vibration. Cambridge University Press, Cambridge

    Google Scholar 

  18. Thompson JMT (1989) Chaotic behavior triggering the escape from a potential well. Proc R Soc Lond Ser A 421:195–225

    Article  ADS  MATH  Google Scholar 

  19. Soliman MS, Thompson JMT (1989) Integrity measures quantifying the erosion of smooth and fractal basins of attraction. J Sound Vib 135:453–475

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Lenci S, Rega G (2003) Optimal control of nonregular dynamics in a Duffing oscillator. Nonlinear Dyn 33:71–86

    Article  MathSciNet  MATH  Google Scholar 

  21. Lenci S, Rega G, Ruzziconi L (2013, in press) The dynamical integrity concept for interpreting/predicting experimental behavior: from macro- to nano-mechanics. Philos Trans R Soc

  22. Lansbury AN, Thompson JMT, Stewart HB (1992) Basin erosion in the twin-well Duffing oscillator: two distinct bifurcation scenarios. Int J Bifurc Chaos 2:505–532

    Article  MathSciNet  MATH  Google Scholar 

  23. Lenci S, Rega G (2004) A dynamical systems analysis of the overturning of rigid blocks. In: Proceedings of the XXI international conference of theoretical and applied mechanics, IPPT PAN, ISBN: 83-89687-01-1, Warsaw, Poland, August 15–21

    Google Scholar 

  24. Ruzziconi L, Lenci S, I YM (2013, in press) An imperfect microbeam under an axial load and electric excitation: nonlinear phenomena and dynamical integrity. Int J Bifurc Chaos 23:1350026. doi:10.1142/S0218127413500260

    Article  Google Scholar 

  25. Rega G, Lenci S (2005) Identifying, evaluating, and controlling dynamical integrity measures in nonlinear mechanical oscillators. Nonlinear Anal TMA 63:902–914

    Article  MathSciNet  MATH  Google Scholar 

  26. Soliman MS, Thompson JMT (1991) Transient and steady state analysis of capsize phenomena. Appl Ocean Res 13:82–92

    Article  Google Scholar 

  27. Infeld E, Lenkowska T, Thompson JMT (1993) Erosion of the basin of stability of a floating body as caused by dam breaking. Phys Fluids 5:2315–2316

    Article  ADS  MATH  Google Scholar 

  28. De Souza JR, Rodrigues ML (2002) An investigation into mechanisms of loss of safe basins in a 2 D.O.F. nonlinear oscillator. J Braz Soc Mech Sci Eng 24:93–98

    Google Scholar 

  29. De Freitas MST, Viana RL, Grebogi C (2003) Erosion of the safe basin for the transversal oscillations of a suspension bridge. Chaos Solitons Fractals 18:829–841

    Article  ADS  MATH  Google Scholar 

  30. Gonçalves PB, Silva FMA, Rega G, Lenci S (2011) Global dynamics and integrity of a two-d.o.f. model of a parametrically excited cylindrical shell. Nonlinear Dyn 63:61–82

    Article  MATH  Google Scholar 

  31. Ruzziconi L, Younis MI, Lenci S (submitted). Multistability in an electrically actuated carbon nanotube: a dynamical integrity perspective

  32. Lenci S, Rega G (2011) Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: a dynamical integrity perspective. Physica D 240:814–824

    Article  ADS  MATH  Google Scholar 

  33. Ruzziconi L, Younis MI, Lenci S (2013) Dynamical integrity for interpreting experimental data and ensuring safety in electrostatic MEMS. In: Wiercigroch M, Rega G (eds) IUTAM symp. on nonlinear dynamics for advanced technologies and engineering, vol 32. Springer, Berlin, pp 249–261

    Chapter  Google Scholar 

  34. Alsaleem F, Younis MI, Ruzziconi L (2010) An experimental and theoretical investigation of dynamic pull-in in MEMS resonators actuated electrostatically. J Microelectromech Syst 19:794–806

    Article  Google Scholar 

  35. Settimi V, Rega G (2012) Bifurcations, basin erosion and dynamic integrity in a single-mode model of noncontact atomic force microscopy. In: Proceedings international conference on structural nonlinear dynamics and diagnosis, Marrakesh, Morocco, April 30–May 02, 2012. doi:10.1051/mateconf/20120104006

    Google Scholar 

  36. Lenci S, Rega G (2006) Control of pull-in dynamics in a nonlinear thermoelastic electrically actuated microbeam. J Micromech Microeng 16:390–401

    Article  Google Scholar 

  37. Ruzziconi L, Bataineh AM, Younis MI, Cui W, Lenci S (submitted). An electrically actuated imperfect microbeam resonator. Experimental and theoretical investigation of the nonlinear dynamic response

  38. Ruzziconi L, Younis MI, Lenci S (submitted). Parameter identification of an electrically actuated imperfect microbeam

  39. Nusse HE, Yorke JA (1998) Dynamics. Numerical explorations. Springer, New York

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank Dr. Weili Cui for the fabrication of the MEMS device and Ahmad M. Bataineh for conducting the experimental data. This work has been supported in part by the National Science Foundation through grant # 0846775, and in part by the Italian Ministry of Education, Universities and Research (MIUR) by the PRIN funded program 2010/11 N. 2010MBJK5B.

Author information

Authors and Affiliations

  1. Department of Civil, Building Engineering, and Architecture, Polytechnic University of Marche, via Brecce Bianche, 60131, Ancona, Italy

    Laura Ruzziconi & Stefano Lenci

  2. Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, 13902, NY, USA

    Mohammad I. Younis

  3. Physical Sciences and Engineering Division, King Abdullah University of Science and Technology KAUST, Thuwal, 23955-6900, Saudi Arabia

    Mohammad I. Younis

Authors
  1. Laura Ruzziconi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad I. Younis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Lenci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laura Ruzziconi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ruzziconi, L., Younis, M.I. & Lenci, S. An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response. Meccanica 48, 1761–1775 (2013). https://doi.org/10.1007/s11012-013-9707-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-013-9707-x

Keywords

Search

Navigation