Skip to main content
SpringerLink
Log in

Size-dependent nonlinear dynamic modeling and vibration analysis of piezo-laminated nanomechanical resonators using perturbation method

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This article investigates size-dependent nonlinear dynamic modeling and vibration analysis of nanomechanical resonators equipped with piezoelectric layers employing perturbation methods. For nonlinear dynamic modeling, the nanomechanical beam resonator is modeled based on the nonlocal elasticity theory and von-kármán type of nonlinear formulation. Then, the Hamilton’s principle is employed to derive the nonlinear vibration equation of the piezoelectric laminated beam resonator. The Galerkin separation approach is employed to develop the governing equation as a time-dependent nonlinear differential one which is solved by the multiple scales perturbation method. Obtaining an analytical formulation, a detailed study is conducted to investigate the size effects on nonlinear free vibration of the beam at several modes of vibration. The obtained results indicate that the nonlocal parameter causes a reduction effect on the nonlinear frequencies particularly for its higher values and higher modes of vibration. Also, it is observed that the nonlocal effect on the natural frequencies diminishes with an increase in the vibration amplitude.

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

Similar content being viewed by others

References

  1. Bakhtiari-Nejad, F., Nazemizadeh, M.: Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory. Acta Mech. 227(5), 1363–1379 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Liebold, C., Müller, W.H.: Strain maps on statically bend (001) silicon microbeams using AFM-integrated Raman spectroscopy. Arch. Appl. Mech. 85(9–10), 1353–1362 (2015)

    Article  Google Scholar 

  3. Taheri, M.: Investigation and sensitivity analysis of dimensional parameters and velocity in the 3D nanomanipulation dynamics of carbon nanotubes using statistical Sobol method. Mod. Mech. Eng. 19(1), 125–135 (2019)

    Google Scholar 

  4. Wang, F.C., Zhao, Y.P.: Structural evolution of the silicon nanowire via molecular dynamics simulations: the double-strand atomic chain and the monatomic chain. Arch. Appl. Mech. 85(3), 323–329 (2015)

    Article  Google Scholar 

  5. Erbts, P., Hartmann, S., Düster, A.: A partitioned solution approach for electro–thermo–mechanical problems. Arch. Appl. Mech. 85(8), 1075–1101 (2015)

    Article  MATH  Google Scholar 

  6. Wang, K.F., Wang, B.L., Gao, Y., Zhou, J.Y.: Nonlinear analysis of piezoelectric wind energy harvesters with different geometrical shapes. Arch. Appl. Mech. (2019). https://doi.org/10.1007/s00419-019-01636-8

    Article  Google Scholar 

  7. Briscoe, J., Dunn, St: Piezoelectric nanogenerators—a review of nanostructured piezoelectric energy harvesters. Nano Energy 14, 15–29 (2015)

    Article  Google Scholar 

  8. Ma, C., Cao, L., Li, L., Shao, M., Jing, D., Guo, Z.: Nonlinear behavior of electrostatically actuated microbeams with coupled longitudinal-transversal vibration. Micromachines 10(5), 315 (2019)

    Article  Google Scholar 

  9. Demir, C., Civalek, Ö.: On the analysis of microbeams. Int. J. Eng. Sci. 121, 14–33 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Taheri, M.: Using of spherical contact models in 3D manipulation modeling of Au nanoparticles using atomic force microscopy to calculate the critical force and time. Modares Mech. Eng. 48(2), 175–184 (2018)

    Google Scholar 

  11. Feng, C., Jiang, L., Lau, W.M.: Dynamic characteristics of a dielectric elastomer-based microbeam resonator with small vibration amplitude. J. Micromech. Microeng. 21(9), 095002 (2011)

    Article  Google Scholar 

  12. Damircheli, M.: Investigating the effects of mechanical parameters of fluid, cantilever orientation, and tip position on the sensitivity of higher modes of atomic force microscope to sample stiffness in liquid medium. Proc. Int. Mech. Eng. K: J. Multi-body Dyn. 229(2), 166–176 (2015)

    Google Scholar 

  13. Zhang, C., Xu, G., Jiang, Q.: Characterization of the squeeze film damping effect on the quality factor of a microbeam resonator. J. Micromech. Microeng. 14(10), 1302 (2004)

    Article  Google Scholar 

  14. Hosseini, R., Hamedi, M., Golparvar, H., Zargar, O.: Analytical and experimental investigation into increasing operating bandwidth of piezoelectric energy harvesters. AUT J. Mech. Eng. 3(1), 113–122 (2019)

    Google Scholar 

  15. Yan, Z., Jiang, L.Y.: Flexoelectric effect on the electroelastic responses of bending piezoelectric nanobeams. J. Appl. Phys. 113(18), 194102 (2013)

    Article  Google Scholar 

  16. Firdaus, S.M., Azid, I.A., Sidek, O., Ibrahim, K., Hussien, M.: Enhancing the sensitivity of a mass-based piezoresistive micro-electro-mechanical systems cantilever sensor. Micro Nano Lett. 5(2), 85–90 (2010)

    Article  Google Scholar 

  17. Sansa, M., Fernández-Regúlez, M., Llobet, J., San Paulo, Á., Pérez-Murano, F.: High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators. Nat. commun. 5(1), 1–9 (2014)

    Article  Google Scholar 

  18. Lifshitz, R., Cross, M.C.: Nonlinear dynamics of nanomechanical and micromechanical resonators. Rev. Nonlinear Dyn. Comp. 1, 1–52 (2008)

    MATH  Google Scholar 

  19. Villanueva, L.G., Karabalin, R.B., Matheny, M.H., Chi, D., Sader, J.E., Roukes, M.L.: Nonlinearity in nanomechanical cantilevers. Phys. Rev. B 87(2), 024304 (2013)

    Article  Google Scholar 

  20. Sari, G., Pakdemirli, M.: Non-Linear vibrations of a microbeam resting on an elastic foundation. Arab. J. Sci. Eng. 38(5), 1191–1199 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mahdavi, M.H., Jiang, L.Y., Sun, X.: Nonlinear vibration of a single-walled carbon nanotube embedded in a polymer matrix aroused by interfacial van der Waals forces. J. Appl. Phys. 106(11), 114309 (2009)

    Article  Google Scholar 

  22. Wen, Y.H., Zhu, Z.Z., Zhu, R., Shao, G.F.: Size effects on the melting of nickel nanowires: a molecular dynamics study. Phys. E Low Dimens. Syst. 25(1), 47–54 (2004)

    Article  Google Scholar 

  23. Feng, C., Jiang, L.Y.: Molecular dynamics simulation of squeeze-film damping effect on nano resonators in the free molecular regime. Phys. E Low Dimens. Syst. 43(9), 1605–1609 (2011)

    Article  MathSciNet  Google Scholar 

  24. Zhu, Y., Qin, Q., Xu, F., Fan, F., Ding, Y., Zhang, T., Wang, Z.L.: Size effects on elasticity, yielding, and fracture of silver nanowires: In situ experiments. Phys. Rev. B 85(4), 045443 (2012)

    Article  Google Scholar 

  25. Zhao, B., Liu, T., Chen, J., Peng, X., Song, Z.: A new Bernoulli–Euler beam model based on modified gradient elasticity. Arch. Appl. Mech. 89(2), 277–289 (2019)

    Article  Google Scholar 

  26. Barretta, R., Čanadija, M., de Sciarra, F.M.: A higher-order Eringen model for Bernoulli–Euler nanobeams. Arch. Appl. Mech. 86(3), 483–495 (2016)

    Article  MATH  Google Scholar 

  27. Kandaz, M., Dal, H.: A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams. Arch. Appl. Mech. 88(11), 2051–2070 (2018)

    Article  Google Scholar 

  28. Mercan, K., Numanoglu, H.M., Akgöz, B., Demir, C., Civalek, Ö.: Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Arch. Appl. Mech. 87(11), 1797–1814 (2017)

    Article  Google Scholar 

  29. Jalali, M.H., Zargar, O., Baghani, M.: Size-dependent vibration analysis of FG microbeams in thermal environment based on modified couple stress theory. Iran. J. Sci. Technol. IJST-Trans. Mech. Eng. 43(1), 761–771 (2019)

    Article  Google Scholar 

  30. Jiang, L.Y., Yan, Z.: Timoshenko beam model for static bending of nanowires with surface effects. Phys. E Low Dimens. Syst. 42(9), 2274–2279 (2010)

    Article  Google Scholar 

  31. Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10(5), 425–435 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  32. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41(3), 305–312 (2003)

    Article  Google Scholar 

  33. Janghorban, M.: Two different types of differential quadrature methods for static analysis of microbeams based on nonlocal thermal elasticity theory in thermal environment. Arch. Appl. Mech. 82(5), 669–675 (2012)

    Article  MATH  Google Scholar 

  34. Zhang, Y., Pang, M., Chen, W.: Non-local modelling on the buckling of a weakened nanobeam. Micro Nano Lett. 8(2), 102–106 (2013)

    Article  Google Scholar 

  35. Asemi, S.R., Farajpour, A., Mohammadi, M.: Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory. Compos. Struct. 116, 703–712 (2014)

    Article  Google Scholar 

  36. Nazemizadeh, M., Bakhtiari-Nejad, F.: Size-dependent free vibration of nano/microbeams with piezo-layered actuators. Micro Nano Lett. 10(2), 93–98 (2015)

    Article  Google Scholar 

  37. Nazemizadeh, M., Bakhtiari-Nejad, F.: A general formulation of quality factor for composite micro/nano beams in the air environment based on the nonlocal elasticity theory. Compos. Struct. 132(15), 772–783 (2015)

    Article  Google Scholar 

  38. Demir, C., Mercan, K., Numanoglu, H.M., Civalek, O.: Bending response of nanobeams resting on elastic foundation. J. Appl. Comput. Mech. 4(2), 105–114 (2018)

    Google Scholar 

  39. Zhou, Z.G., Wu, L.Z., Du, S.Y.: Non-local theory solution for a Mode I crack in piezoelectric materials. Eur J. Mech. A-Solids 25(5), 793–807 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, H., Preidikman, S., Balachandran, B., Mote Jr., C.D.: Nonlinear free and forced oscillations of piezoelectric microresonators. J. Micromech. Microeng. 16(2), 356 (2006)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  42. Lestari, W., Hanagud, S.: Nonlinear vibration of buckled beams: some exact solutions. Int. J. Solids Struct. 38(25), 4741–4757 (2001)

    Article  MATH  Google Scholar 

  43. Rao, G.V., Raju, K.K., Raju, I.S.: Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates. Comput. Struct. 6(3), 169–172 (1976)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics, Malek Ashtar University of Technology, Shahin Shahr, Iran

    Mostafa Nazemizadeh, Abbas Assadi & Behrooz Shahriari

  2. Department of Mechanical Engineering, University of Maryland, Maryland, USA

    Firooz Bakhtiari-Nejad

  3. Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

    Firooz Bakhtiari-Nejad

Authors
  1. Mostafa Nazemizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Firooz Bakhtiari-Nejad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abbas Assadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Behrooz Shahriari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mostafa Nazemizadeh.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nazemizadeh, M., Bakhtiari-Nejad, F., Assadi, A. et al. Size-dependent nonlinear dynamic modeling and vibration analysis of piezo-laminated nanomechanical resonators using perturbation method. Arch Appl Mech 90, 1659–1672 (2020). https://doi.org/10.1007/s00419-020-01678-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01678-3

Keywords

Search

Navigation