Nonlinear Dynamics

, Volume 94, Issue 4, pp 2665–2683 | Cite as

Nonlinear vibration of an electrostatically actuated micro-beam made of anelastic material considering compressible fluid media

  • Amir Veysi-Gorgabad
  • Ghader RezazadehEmail author
  • Rasoul Shabani
Original Paper


Similar to elasticity and viscoelasticity, anelasticity is one of the rheological models of materials. Like the viscoelastic materials, anelastic deformation is time dependent, while unlike viscoelasticity, it is full recoverable. Anelastic behavior is very sensitive to microstructure and is significant when things become small. In this paper, the nonlinear vibrations of an electrostatically actuated micro-beam made of anelastic material are studied based on the modified couple stress theory and considering compressible fluid media. The constitutive equation of micro-beams has been developed regarding the stress–strain behavior of classic anelastic materials. Considering the dependency of the effective viscosity on variable gap distance, an analytical model of an electrostatically actuated anelastic micro-beam has been developed through the variable Knudsen number. Then, multiple scales method has been applied to obtain an approximate analytical solution for nonlinear resonant curves. Additionally, the effects of different mechanical behaviors of materials including elasticity, viscoelasticity and anelasticity on the nonlinear dynamic response have been studied. Furthermore, influences of some effective parameters such as length scale, anelastic relaxation time, relaxation intensity, mid-plane stretching force, residual axial force, air-gap pressure and compressible fluid media on the nonlinear vibration analysis have been investigated and discussed in detail. The results demonstrate that resonance curves are strongly dependent on the mechanical behavior of materials. It is found that the relaxation intensity, relaxation time and air-gap pressure change the resonant amplitude significantly. Moreover, the results reveal that considering compressible fluid media, both the frequency response and the softening behavior of the system decrease.


Anelasticity Micro-beam Electrostatic actuation Nonlinear analysis Compressible fluid media 


  1. 1.
    Lakes, R.: Viscoelastic Material. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  2. 2.
    Nowick, A.S., Berry, B.S.: Anelastic Relaxation in Crystalline Solids. Academic Press, New York (1972)Google Scholar
  3. 3.
    Cheng, G., Miao, C., Qin, Q., Li, J., Xu, F., Haftbaradaran, H., Dickey, E.C., Gao, H., Zhu, Y.: Large anelasticity and associated energy dissipation in single-crystalline nanowires. Nat. Nanotechnol. 10, 687–691 (2015)CrossRefGoogle Scholar
  4. 4.
    Zener, C.: Elasticity and Anelasticity of Metals. University Chicago Press, Chicago (1948)zbMATHGoogle Scholar
  5. 5.
    Nix, W.D.: Mechanical properties of thin films. Met. Trans. A 20(11), 2217–2245 (1989)CrossRefGoogle Scholar
  6. 6.
    Choi, D., Kim, H., Nix, W.D.: Anelasticity and damping of thin aluminum films on silicon substrates. J. Microelectromech. Syst. 13(2), 230–237 (2004)CrossRefGoogle Scholar
  7. 7.
    Bergers, L.I.J.C., Hoefnagels, J.P.M., Delhey, N.K.R., Geers, M.G.D.: Measuring time-dependent deformations in metallic MEMS. Microelectr. Reliab. 51, 1054–1059 (2011)CrossRefGoogle Scholar
  8. 8.
    Baker, S.P., Vinci, R.P., Arias, T.: Elastic and anelastic behavior of materials in small dimensions. MRS Bull. 27(1), 26–29 (2002)CrossRefGoogle Scholar
  9. 9.
    Gall, K., Kreiner, P., Turner, D., Hulse, M.: Shape-memory polymers for micro electromechanical systems. J. Microelectromech. Syst. 13(3), 472–483 (2004)CrossRefGoogle Scholar
  10. 10.
    Ansari, R., Faraji Oskouie, M., Sadeghi, F., Bazdid-Vahdati, M.: Free vibration of fractional viscoelastic Timoshenko nano-beams using the nonlocal elasticity theory. Phys. E Low-Dimens. Syst. Nanostruct. 74, 318–327 (2015)CrossRefGoogle Scholar
  11. 11.
    Lin, R., Wang, W.: Structural dynamics of microsystems-current state of research and future directions. Mech. Syst. Signal Process. 20(5), 1015–43 (2006)CrossRefGoogle Scholar
  12. 12.
    Batra, R., Porfiri, M., Spinello, D.: Review of modeling electrostatically actuated micro electromechanical systems. Smart Mater. Struct. 16(6), 23–31 (2007)CrossRefGoogle Scholar
  13. 13.
    Mobki, H., Rezazadeh, G., Sadeghi, M., Vakili-Tahami, F., Seyyed-Fakhrabadi, M.: A comprehensive study of stability in an electro-statically actuated micro-beam. Int. J. Non-Linear Mech. 48, 78–85 (2013)CrossRefGoogle Scholar
  14. 14.
    Chaterjee, S., Pohit, G.: Squeeze film damping characteristics of cantilever microresonators for higher modes of flexural vibration. Int. J. Eng. Sci. Technol. 2, 187–199 (2010)CrossRefGoogle Scholar
  15. 15.
    Yagubizade, H., Fathalilou, M., Rezazadeh, G., Talebian, S.: Squeeze-film damping effect on dynamic pull-in voltage of an electrostatically-actuated microbeam. Sens. Trans. 103, 96–101 (2009)Google Scholar
  16. 16.
    McCarthy, B., Adams, G., McGruer, N., Potter, D.: A dynamic model, including contact bounce, of an electrostatically actuated microswitch. J. Microelectromech. Syst. 11, 276–283 (2002)CrossRefGoogle Scholar
  17. 17.
    Zhang, W.M., Meng, G., Wei, K.X.: Dynamics of nonlinear coupled electrostatic micromechanical resonators under two frequency parametric and external excitations. Shock Vib. 17, 759–770 (2010)CrossRefGoogle Scholar
  18. 18.
    Liu, C.-C., Liu, C.-H.: Analysis of nonlinear dynamic behavior of electrically actuated micro-beam with piezoelectric layers and squeeze-film damping effect. Nonlinear Dyn. 77, 1349–1361 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Reduced-order models for MEMS applications. Nonlinear Dyn. 41, 211–236 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Krylov, S., Maimon, R.: Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force. J. Vib. Acoust. 126, 332–342 (2004)CrossRefGoogle Scholar
  21. 21.
    Nayfeh, A.H., Younis, M.I.: A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping. J. Micromech. Microeng. 14, 170–181 (2004)CrossRefGoogle Scholar
  22. 22.
    Pandey, A.K., Pratap, R.: Effect of flexural modes on squeeze film damping in MEMS cantilever resonators. J. Micromech. Microeng. 17, 2475–2484 (2007)CrossRefGoogle Scholar
  23. 23.
    Teufel, L.W.: Prediction of hydraulic fracture azimuth from anelastic strain recovery measurements of oriented core. In: Proceedings 23rd US Symposium on Rock Mechanics. Berkeley, pp. 238–245 (1982)Google Scholar
  24. 24.
    Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)CrossRefGoogle Scholar
  25. 25.
    Sadeghian, H., Rezazadeh, G., Osterberg, P.: Application of the generalized differential quadrature method to the study of pull-in phenomena of MEMS switches. IEEE/ASME J. Microelectromech. Syst. 16(6), 1334–1340 (2007)CrossRefGoogle Scholar
  26. 26.
    Saeedi-Vahdat, A., Rezazadeh, G.: Effects of axial and residual stresses on thermoelastic damping in capacitive micro-beam resonators. J. Frankl. Inst. 348, 622–639 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Xia, W., Wang, L., Yin, L.: Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int. J. Eng. Sci. 48(12), 2044–2053 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hamrock, B.J.: Fundamentals of Fluid Film Lubrication. McGraw Hill, New York (1994)Google Scholar
  29. 29.
    Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, New York (2011)CrossRefGoogle Scholar
  30. 30.
    Veijola, T., Kuisma, H., Lahdenper, J., Ryhänen, T.: Equivalent-circuit model of the squeezed gas film in a silicon accelerometer. Sens. Actuators A Phys. 48, 239–248 (1994)CrossRefGoogle Scholar
  31. 31.
    Blech, J.J.: On isothermal squeeze films. J. Lubr. Technol. 105, 615–620 (1983)CrossRefGoogle Scholar
  32. 32.
    Schaaf, S.A., Chambr Paul, L.: Flow of Rarefied Gases. Princeton University Press, Princeton (1961)zbMATHGoogle Scholar
  33. 33.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)zbMATHGoogle Scholar
  34. 34.
    Caruntu, D.I., Martinez, I., Taylor, K.N.: Voltage–amplitude response of alternating current near half natural frequency electrostatically actuated MEMS resonators. Mech. Res. Commun. 52, 25–31 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUrmia UniversityUrmiaIran

Personalised recommendations