Skip to main content
SpringerLink
Log in

Parametric resonances of an electrically actuated piezoelectric nanobeam resonator considering surface effects and intermolecular interactions

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper investigates the nonlinear dynamics of a parametrically excited doubly clamped piezoelectric nanobeam, actuated by a combined AC and DC loadings. Surface effects, intermolecular van der Waals forces, and fringing effects are incorporated in the nonlinear model. The governing equation of motion is obtained using the extended Hamilton principle. The reduced-order model equation (ROM) is obtained based on the Galerkin method. The multiple-scale method is applied directly to the nonlinear equation of motion and associated boundary conditions to obtain the nanobeam response analytically under small AC voltage loads. The influence of van der Waals forces, piezoelectric voltages, and surface effects is investigated on the natural frequencies, static equilibria, pull-in voltages, and principle parametric resonance (subharmonic resonance of order one-half) of the nanoresonator. It is shown the surface effect profoundly affects the nontrivial parametric responses, trivial stability zones, and bifurcation point’s loci, and it is necessary to consider the surface effects for accurate and exact investigation of the system response. The effect of piezoelectric voltage to control the dynamic instability region is also demonstrated. To validate analytical results, ROM equation is integrated numerically. It is seen that the perturbation results are in accordance with numerical results.

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Caruntu, D.I., Martinez, I.: Reduced order model of parametric resonance of electrostatically actuated MEMS cantilever resonators. Int. J. Non-linear Mech. 66, 28–32 (2014)

    Article  Google Scholar 

  2. Rhoads, J.F., Kumar, V., Shaw, S.W., Turner, K.L.: The non-linear dynamics of electromagnetically actuated microbeam resonators with purely parametric excitations. Int. J. Non-linear Mech. 55, 79–89 (2013)

    Article  Google Scholar 

  3. Abdel-Rahman, E.M., Nayfeh, A.H.: Secondary resonances of electrically actuated resonant microsensors. J. Micromech. Microeng. 13, 491–501 (2003). doi:10.1088/0960-1317/13/3/320

    Article  Google Scholar 

  4. Kacem, N., Baguet, S., Hentz, S., Dufour, R.: Pull-in retarding in nonlinear nanoelectromechanical resonators under superharmonic excitation. J. Comput. Nonlinear Dyn. 7(2), 021011 (2012)

    Article  Google Scholar 

  5. Ouakad, H.M., Younis, M.I.: Nonlinear dynamics of electrically actuated carbon nanotube resonators. J. Comput. Nonlinear Dyn. 5(January 2010), 011009 (2010). doi:10.1115/1.4000319

    Article  Google Scholar 

  6. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  7. Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics: Mems Linear and Nonlinear Statics and Dynamics, vol. 20. Springer, Berlin (2010)

    Google Scholar 

  8. Kacem, N., Hentz, S., Pinto, D., Reig, B., Nguyen, V.: Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors. Nanotechnology 20(27), 275501–275501 (2009). doi:10.1088/0957-4484/20/27/275501

    Article  Google Scholar 

  9. Azizi, S., Ghazavi, M.R., Rezazadeh, G., Ahmadian, I., Cetinkaya, C.: Tuning the primary resonances of a micro resonator, using piezoelectric actuation. Nonlinear Dyn. 76, 839–852 (2013). doi:10.1007/s11071-013-1173-4

    Article  MATH  Google Scholar 

  10. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn. 48, 153–163 (2007). doi:10.1007/s11071-006-9079-z

    Article  MATH  Google Scholar 

  11. Najar, F., Nayfeh, A., Abdel-Rahman, E., Choura, S., El-Borgi, S.: Nonlinear analysis of MEMS electrostatic microactuators: primary and secondary resonances of the first mode*. J. Vib. Control 16(9), 1321–1349 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Alsaleem, F.M., Younis, M.I., Ouakad, H.M.: On the nonlinear resonances and dynamic pull-in of electrostatically actuated resonators. J. Micromech. Microeng. 19(4), 45013–45013 (2009)

    Article  Google Scholar 

  13. Pourkiaee, S.M., Khadem, S.E., Shahgholi, M.: Nonlinear vibration and stability analysis of an electrically actuated piezoelectric nanobeam considering surface effects and intermolecular interactions. J. Vib. Control. (2015). doi:10.1177/1077546315603270

  14. Najar, F., Nayfeh, aH, Abdel-Rahman, E.M., Choura, S., El-Borgi, S.: Dynamics and global stability of beam-based electrostatic microactuators. J. Vib. Control 16(5), 721–748 (2010). doi:10.1177/1077546309106521

    Article  MathSciNet  MATH  Google Scholar 

  15. Ouakad, H.M., Younis, M.I.: Natural frequencies and mode shapes of slacked carbon nanotube NEMS resonators. In: ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 645–652. American Society of Mechanical Engineers (2010)

  16. Ouakad, H.M., Younis, M.I.: Forced Vibrations of Slacked Carbon Nanotube Resonators. In: ASME 2010 International Mechanical Engineering Congress and Exposition, pp. 435–443. American Society of Mechanical Engineers (2010)

  17. Ouakad, H.M., Younis, M.I.: Natural frequencies and mode shapes of initially curved carbon nanotube resonators under electric excitation. J. Sound Vib. 330(13), 3182–3195 (2011)

    Article  Google Scholar 

  18. Ouakad, H.M., Younis, M.I.: Dynamic response of slacked single-walled carbon nanotube resonators. Nonlinear Dyn. 67(2), 1419–1436 (2012)

    Article  MathSciNet  Google Scholar 

  19. Rasekh, M., Khadem, S.E.: Pull-in analysis of an electrostatically actuated nano-cantilever beam with nonlinearity in curvature and inertia. Int. J. Mech. Sci. 53(2), 108–115 (2011). doi:10.1016/j.ijmecsci.2010.11.007

    Article  Google Scholar 

  20. Hajnayeb, A., Khadem, S.E.: Nonlinear vibrations of a carbon nanotube resonator under electrical and van der Waals forces. J. Comput. Theor. Nanosci. 8(8), 1527–1534 (2011). doi:10.1166/jctn.2011.1846

    Article  Google Scholar 

  21. Hajnayeb, a, Khadem, S.E.: Nonlinear vibration and stability analysis of a double-walled carbon nanotube under electrostatic actuation. J. Sound Vib. 331(10), 2443–2456 (2012). doi:10.1016/j.jsv.2012.01.008

    Article  Google Scholar 

  22. Ouakad, H.M., Najar, F., Hattab, O.: Nonlinear analysis of electrically actuated carbon nanotube resonator using a novel discretization technique. Math. Probl. Eng. 2013 (2013). doi:10.1155/2013/517695

  23. Alibeigloo, A., Emtehani, A.: Static and free vibration analyses of carbon nanotube-reinforced composite plate using differential quadrature method. Meccanica 50(1), 61–76 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, T., Younis, M.I.: Nonlinear dynamics of carbon nanotubes under large electrostatic force. J. Comput. Nonlinear Dyn. 11(2), 021009 (2016)

    Article  Google Scholar 

  25. Asemi, S., 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 

  26. Alibeigloo, A., Liew, K.: Elasticity solution of free vibration and bending behavior of functionally graded carbon nanotube-reinforced composite beam with thin piezoelectric layers using differential quadrature method. Int. J. Appl. Mech. 7(01), 1550002 (2015)

    Article  Google Scholar 

  27. Ke, L.-L., Wang, Y.-S., Wang, Z.-D.: Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Compos. Struct. 94(6), 2038–2047 (2012)

    Article  Google Scholar 

  28. Arani, A.G., Kolahchi, R., Zarei, M.S.: Visco-surface-nonlocal piezoelasticity effects on nonlinear dynamic stability of graphene sheets integrated with ZnO sensors and actuators using refined zigzag theory. Compos. Struct. 132, 506–526 (2015)

    Article  Google Scholar 

  29. He, J., Lilley, C.M.: Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8, 1798–1802 (2008). doi:10.1021/nl0733233

    Article  Google Scholar 

  30. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000). doi:10.1088/0957-4484/11/3/301

    Article  Google Scholar 

  31. Eom, K., Park, H.S., Yoon, D.S., Kwon, T.: Nanomechanical resonators and their applications in biological/chemical detection: nanomechanics principles. Phys. Rep. 503(4–5), 115–163 (2011). doi:10.1016/j.physrep.2011.03.002

    Article  Google Scholar 

  32. Wang, K., Wang, B.: Timoshenko beam model for the vibration analysis of a cracked nanobeam with surface energy. J. Vib. Control. (2013). doi:10.1177/1077546313513054

  33. Eltaher, Ma., Mahmoud, F.F., Assie, A.E., Meletis, E.I.: Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl. Math. Comput. 224, 760–774 (2013). doi:10.1016/j.amc.2013.09.002

    MathSciNet  MATH  Google Scholar 

  34. Fu, Y., Zhang, J.: Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies. Appl. Math. Model. 35, 941–951 (2011). doi:10.1016/j.apm.2010.07.051

    Article  MathSciNet  Google Scholar 

  35. Ma, J.B., Jiang, L., Asokanthan, S.F.: Influence of surface effects on the pull-in instability of NEMS electrostatic switches. Nanotechnology 21, 505708–505708 (2010). doi:10.1088/0957-4484/21/50/505708

    Article  Google Scholar 

  36. Wang, K.F., Wang, B.L.: Influence of surface energy on the non-linear pull-in instability of nano-switches. Int. J. Non-Linear Mech. 59, 69–75 (2014). doi:10.1016/j.ijnonlinmec.2013.11.004

    Article  Google Scholar 

  37. Wang, G.F., Feng, X.Q.: Surface effects on the buckling of nanowires under uniaxial compression. Appl. Phys. Lett. 94, 141913–141913 (2009). doi:10.1063/1.3117505

    Article  Google Scholar 

  38. Yan, Z., Jiang, L.Y.: Surface effects on the vibration and buckling of piezoelectric nanoplates. Europhys. Lett. 99(July), 27007–27007 (2012). doi:10.1209/0295-5075/99/27007

    Article  Google Scholar 

  39. Zhang, J., Wang, C.: Vibrating piezoelectric nanofilms as sandwich nanoplates. J. Appl. Phys. 111(January), 2–7 (2012). doi:10.1063/1.4709754

    Google Scholar 

  40. Cha, S.N., Seo, J.S., Kim, S.M., Kim, H.J., Park, Y.J., Kim, S.W., Kim, J.M.: Sound-driven piezoelectric nanowire-based nanogenerators. Adv. Mater. 22(42), 4726–4730 (2010)

    Article  Google Scholar 

  41. Kumar, B., Kim, S.-W.: Energy harvesting based on semiconducting piezoelectric ZnO nanostructures. Nano Energy 1(3), 342–355 (2012)

    Article  Google Scholar 

  42. Ramezani, A., Alasty, A., Akbari, J.: Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces. Int. J. Solids Struct. 44(14), 4925–4941 (2007)

    Article  MATH  Google Scholar 

  43. Batra, R.C., Porfiri, M., Spinello, D.: Effects of van der Waals force and thermal stresses on pull-in instability of clamped rectangular microplates. Sensors 8, 1048–1069 (2008). doi:10.3390/s8021048

    Article  Google Scholar 

  44. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)

  45. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)

    Article  MATH  Google Scholar 

  46. Gheshlaghi, B., Hasheminejad, S.M.: Vibration analysis of piezoelectric nanowires with surface and small scale effects. Curr. Appl. Phys. 12(4), 1096–1099 (2012)

    Article  Google Scholar 

  47. Hosseini-Hashemi, S., Nahas, I., Fakher, M., Nazemnezhad, R.: Nonlinear free vibration of piezoelectric nanobeams incorporating surface effects. Smart Mater. Struct. 23(3), 035012 (2014)

    Article  MATH  Google Scholar 

  48. Zhang, J., Wang, C., Adhikari, S.: Surface effect on the buckling of piezoelectric nanofilms. J. Phys. D Appl. Phys. 45(28), 285301 (2012)

    Article  Google Scholar 

  49. Lu, P., He, L., Lee, H., Lu, C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43(16), 4631–4647 (2006)

    Article  MATH  Google Scholar 

  50. Chen, L.-W., Lin, C.-Y., Wang, C.-C.: Dynamic stability analysis and control of a composite beam with piezoelectric layers. Compos. Struct. 56, 97–109 (2002). doi:10.1016/S0263-8223(01)00183-0

    Article  Google Scholar 

  51. Younis, M.I., Nayfeh, A.H.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31, 91–117 (2003). doi:10.1023/A:1022103118330

    Article  MATH  Google Scholar 

  52. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

  53. Zhang, L.L., Liu, J.X., Fang, X.Q., Nie, G.Q.: Effects of surface piezoelectricity and nonlocal scale on wave propagation in piezoelectric nanoplates. Eur. J. Mech. A Solids 46, 22–29 (2014). doi:10.1016/j.euromechsol.2014.01.005

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-177, Tehran, Iran

    S. Mehrdad Pourkiaee & Siamak E. Khadem

  2. Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran

    Majid Shahgholi

Authors
  1. S. Mehrdad Pourkiaee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siamak E. Khadem
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Majid Shahgholi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siamak E. Khadem.

Appendix

Appendix

$$\begin{aligned}&S =3\alpha _3 \varGamma \left( {\phi ,\phi } \right) \phi ^{\prime \prime }+2\alpha _3 \varGamma \left( {\phi ,w_s } \right) {\psi }''_1 \\&\quad +\,2\alpha _3 \varGamma \left( {w_s ,\psi _1 } \right) {\phi }''+2\alpha _3 \varGamma \left( {\phi ,\psi _1 } \right) {w}''_s \\&\quad +\,4\alpha _3 \varGamma \left( {\phi ,w_s } \right) {\psi }''_2 +4\alpha _3 \varGamma \left( {w_s ,\psi _2 } \right) {\phi }''\\&\quad +\,4\alpha _3 \varGamma \left( {\phi ,\psi _2 } \right) {w}''_s \\&\quad +\,6\frac{\alpha _{61} V_D^2 \phi \psi _1 }{\left( {1-w_s } \right) ^{4}}-6\frac{\alpha _{62} V_D^2 \phi \psi _1 }{\left( {1+w_s } \right) ^{4}}+12\frac{\alpha _{61} V_D^2 \phi \psi _2 }{\left( {1-w_s } \right) ^{4}}\\&\quad -\,12\frac{\alpha _{62} V_D^2 \phi \psi _2 }{\left( {1+w_s } \right) ^{4}}+12\frac{\alpha _{61} V_D^2 \phi ^{3}}{\left( {1-w_s } \right) ^{5}}+12\frac{\alpha _{62} V_D^2 \phi ^{3}}{\left( {1+w_s } \right) ^{5}} \\&\quad +\,12\frac{\alpha _{71} V_D^2 \phi \psi _1 }{\left( {1-w_s } \right) ^{5}}-12\frac{\alpha _{72} V_D^2 \phi \psi _1 }{\left( {1+w_s } \right) ^{5}}+24\frac{\alpha _{71} V_D^2 \phi \psi _2 }{\left( {1-w_s } \right) ^{5}}\\&\quad -\,24\frac{\alpha _{72} V_D^2 \phi \psi _2 }{\left( {1+w_s } \right) ^{5}}+30\frac{\alpha _{71} V_D^2 \phi ^{3}}{\left( {1-w_s } \right) ^{6}}+30\frac{\alpha _{72} V_D^2 \phi ^{3}}{\left( {1+w_s } \right) ^{6}} \\&F =2\alpha _3 \varGamma \left( {w_s ,\psi _3 } \right) {\phi }''+2\alpha _3 \varGamma \left( {\phi ,\psi _3 } \right) {w}''_s \\&\quad +\,2\alpha _3 \varGamma \left( {\phi ,w_s } \right) {\psi }''_3 +6\frac{\alpha _{61} V_D^2 \phi \psi _3 }{\left( {1-w_s } \right) ^{4}} \\&\quad -\,6\frac{\alpha _{62} V_D^2 \phi \psi _3 }{\left( {1+w_s } \right) ^{4}}+12\frac{\alpha _{71} V_D^2 \phi \psi _3 }{\left( {1-w_s } \right) ^{5}}-12\frac{\alpha _{72} V_D^2 \phi \psi _3 }{\left( {1+w_s } \right) ^{5}} \\ \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pourkiaee, S.M., Khadem, S.E. & Shahgholi, M. Parametric resonances of an electrically actuated piezoelectric nanobeam resonator considering surface effects and intermolecular interactions. Nonlinear Dyn 84, 1943–1960 (2016). https://doi.org/10.1007/s11071-016-2618-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2618-3

Keywords

Search

Navigation