Size-dependent nonlinear vibration of an electrostatic nanobeam actuator considering surface effects and inter-molecular interactions

  • Saman Esfahani
  • Siamak Esmaeilzade KhademEmail author
  • Ali Ebrahimi Mamaghani


In this paper, the size-dependent nonlinear vibration of an electrostatic nanobeam actuator is investigated based on the nonlocal strain gradient theory, incorporating surface effects. A comprehensive model regarding the von Karman geometrical nonlinearity, inter-molecular forces and both components of the electrostatic excitation (AC and DC) is proposed to explore the system behavior near the primary resonance. Utilizing Hamilton’s principle, the nonlinear equation of motion of the system is derived. The natural frequency and dynamic response of the system, comprising frequency and force response diagrams, are obtained analytically via multiple scales technique in conjunction with the differential quadrature method and validated through a numerical approach. The roles of the nonlocal and strain gradient parameters, surface elasticity, inter-molecular forces and quality factor on the system oscillations are examined. The acquired results unveiled that the size-dependent parameters can significantly displace the multi-valued portions and instability thresholds of the dynamical response. Furthermore, it is deduced that the surface effects induce the stiffness hardening of the nanobeam, whereas the inter-molecular forces impose the stiffness softening effect.


Nanobeam actuator Nonlocal strain gradient theory Surface effects Inter-molecular forces Nonlinear vibration Differential quadrature method (DQM) 


  1. Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30(10), 1279–1299 (1992)zbMATHGoogle Scholar
  2. 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(1), 839–852 (2014)zbMATHGoogle Scholar
  3. Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49(1), 1–28 (1996)Google Scholar
  4. Chen, X., Meguid, S.: Snap-through buckling of initially curved microbeam subject to an electrostatic force. Proc. R. Soc. A 471(2177), 20150072 (2015)MathSciNetzbMATHGoogle Scholar
  5. Chen, X., Meguid, S.: Asymmetric bifurcation of thermally and electrically actuated functionally graded material microbeam. Proc. R. Soc. A 472(2186), 20150597 (2016)Google Scholar
  6. Chen, X., Meguid, S.: Dynamic behavior of micro-resonator under alternating current voltage. Int. J. Mech. Mater. Des. 13(4), 481–497 (2017a)Google Scholar
  7. Chen, X., Meguid, S.: Nonlinear vibration analysis of a microbeam subject to electrostatic force. Acta Mech. 228(4), 1343–1361 (2017b)MathSciNetzbMATHGoogle Scholar
  8. Dequesnes, M., Rotkin, S., Aluru, N.: Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches. Nanotechnology 13(1), 120 (2002)Google Scholar
  9. Ebrahimi, F., Barati, M.R.: Damping vibration behavior of visco-elastically coupled double-layered graphene sheets based on nonlocal strain gradient theory. Microsyst. Technol. 24(3), 1643–1658 (2018)Google Scholar
  10. Eltaher, M., Agwa, M., Mahmoud, F.: Nanobeam sensor for measuring a zeptogram mass. Int. J. Mech. Mater. Des. 12(2), 211–221 (2016)Google Scholar
  11. Eringen, A.C., Edelen, D.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972)MathSciNetzbMATHGoogle Scholar
  12. Fleck, N., Hutchinson, J.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41(12), 1825–1857 (1993)MathSciNetzbMATHGoogle Scholar
  13. Fleck, N., Muller, G., Ashby, M., Hutchinson, J.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1994)Google Scholar
  14. Gupta, R.K.: Electrostaticpull-in test structure design for in-situ mechanical property measurements of microelectromechanical systems (MEMS). Ph.D. thesis, Citeseer (1998)Google Scholar
  15. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)MathSciNetzbMATHGoogle Scholar
  16. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)zbMATHGoogle Scholar
  17. Hoseinzadeh, M., Khadem, S.: A nonlocal shell theory model for evaluation of thermoelastic damping in the vibration of a double-walled carbon nanotube. Phys. E 57, 6–11 (2014)Google Scholar
  18. Hosseini-Hashemi, S., Nazemnezhad, R.: An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects. Compos. Part B Eng. 52, 199–206 (2013)Google Scholar
  19. Israelachvili, J.N.: Intermolecular and Surface Forces. Academic Press, Cambridge (2011)Google Scholar
  20. 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 (2009)Google Scholar
  21. Kambali, P.N., Nikhil, V., Pandey, A.K.: Surface and nonlocal effects on response of linear and nonlinear NEMS devices. Appl. Math. Modell. 43, 252–267 (2017)MathSciNetGoogle Scholar
  22. Lam, D.C., Yang, F., Chong, A., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)zbMATHGoogle Scholar
  23. Lamoreaux, S.K.: The Casimir force: background, experiments, and applications. Rep. Prog. Phys. 68(1), 201 (2004)Google Scholar
  24. Lim, C., Zhang, G., Reddy, J.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)MathSciNetzbMATHGoogle Scholar
  25. Liu, C.C.: Dynamic behavior analysis of cantilever-type nano-mechanical electrostatic actuator. Int. J. Non-Linear Mech. 82, 124–130 (2016)Google Scholar
  26. Ma, J.B., Jiang, L., Asokanthan, S.F.: Influence of surface effects on the pull-in instability of nems electrostatic switches. Nanotechnology 21(50), 505708 (2010)Google Scholar
  27. Mehrdad Pourkiaee, S., 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 23(12), 1873–1889 (2017)MathSciNetzbMATHGoogle Scholar
  28. Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N., Fathi, M.: Nano-resonator frequency response based on strain gradient theory. J. Phys. D Appl. Phys. 47(36), 365303 (2014)Google Scholar
  29. Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N.: Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams. Microsyst. Technol. 21(2), 457–464 (2015)zbMATHGoogle Scholar
  30. Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)Google Scholar
  31. Moghimi Zand, M., Ahmadian, M.: Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 224(9), 2037–2047 (2010)Google Scholar
  32. Mohammadi, M., Eghtesad, M., Mohammadi, H.: Stochastic analysis of dynamic characteristics and pull-in instability of FGM micro-switches with uncertain parameters in thermal environments. Int. J. Mech. Mater. Des. 14(3), 417–442 (2018)Google Scholar
  33. Najar, F., El-Borgi, S., Reddy, J., Mrabet, K.: Nonlinear nonlocal analysis of electrostatic nanoactuators. Compos. Struct. 120, 117–128 (2015)Google Scholar
  34. Nayfeh, A.: Introduction to perturbation techniques. Wiley classics library, Wiley. (1981). Accessed 10 Sept 2018
  35. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn. 48(1–2), 153–163 (2007)zbMATHGoogle Scholar
  36. Nikpourian, A., Ghazavi, M.R., Azizi, S.: On the nonlinear dynamics of a piezoelectrically tuned micro-resonator based on non-classical elasticity theories. Int. J. Mech. Mater. Des. 14, 1–19 (2016)Google Scholar
  37. Ouakad, H.M., El-Borgi, S., Mousavi, S.M., Friswell, M.I.: Static and dynamic response of CNT nanobeam using nonlocal strain and velocity gradient theory. Appl. Math. Modell. 62, 207–222 (2018)MathSciNetGoogle Scholar
  38. Pradiptya, I., Ouakad, H.M.: Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory. Int. J. Mech. Mater. Des. 14(3), 393–415 (2018)Google Scholar
  39. Sharabiani, P.A., Yazdi, M.R.H.: Nonlinear free vibrations of functionally graded nanobeams with surface effects. Compos. Part B Eng. 45(1), 581–586 (2013)Google Scholar
  40. Şimşek, M.: Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016)MathSciNetzbMATHGoogle Scholar
  41. Vatankhah, R., Kahrobaiyan, M., Alasty, A., Ahmadian, M.: Nonlinear forced vibration of strain gradient microbeams. Appl. Math. Modell. 37(18–19), 8363–8382 (2013)MathSciNetzbMATHGoogle Scholar
  42. Wang, K., Wang, B.: Influence of surface energy on the non-linear pull-in instability of nano-switches. Int. J. Non-Linear Mech. 59, 69–75 (2014)Google Scholar
  43. Wang, X.: Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications. Elsevier Science. (2015). Accessed 10 Sept 2018
  44. Yang, J., Jia, X., Kitipornchai, S.: Pull-in instability of nano-switches using nonlocal elasticity theory. J. Phys. D Appl. Phys. 41(3), 035103 (2008)Google Scholar
  45. Yang, W., Wang, X.: Nonlinear pull-in instability of carbon nanotubes reinforced nano-actuator with thermally corrected Casimir force and surface effect. Int. J. Mech. Sci. 107, 34–42 (2016)Google Scholar
  46. Yang, W., Li, Y., Wang, X.: Scale-dependent dynamic-pull-in of functionally graded carbon nanotubes reinforced nanodevice with piezoelectric layer. J. Aerosp. Eng. 30(3), 04016096 (2016a)Google Scholar
  47. Yang, W., Yang, F., Wang, X.: Coupling influences of nonlocal stress and strain gradients on dynamic pull-in of functionally graded nanotubes reinforced nano-actuator with damping effects. Sensors Actuators A Phys. 248, 10–21 (2016b)Google Scholar
  48. Yang, Y.T., Callegari, C., Feng, X., Ekinci, K.L., Roukes, M.L.: Zeptogram-scale nanomechanical mass sensing. Nanoletters 6(4), 583–586 (2006)Google Scholar
  49. Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics, vol. 20. Springer, Berlin (2011)Google Scholar
  50. Younis, M.I., Nayfeh, A.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31(1), 91–117 (2003)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTarbiat Modares UniversityTehranIran

Personalised recommendations