Nonlinear dynamics of electrically actuated carbon nanotube resonators

Hassen M. Ouakad, Mohammad I. Younis*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

131 Scopus citations


This work presents an investigation of the nonlinear dynamics of carbon nanotubes (CNTs) when actuated by a dc load superimposed to an ac harmonic load. Cantilevered and clamped-clamped CNTs are studied. The carbon nanotube is described by an Euler-Bernoulli beam model that accounts for the geometric nonlinearity and the nonlinear electrostatic force. A reduced-order model based on the Galerkin method is developed and utilized to simulate the static and dynamic responses of the carbon nanotube. The free-vibration problem is solved using both the reduced-order model and by solving directly the coupled in-plane and out-of-plane boundary-value problems governing the motion of the nanotube. Comparison of the results generated by these two methods to published data of a more complicated molecular dynamics model shows good agreement. Dynamic analysis is conducted to explore the nonlinear oscillation of the carbon nanotube near its fundamental natural frequency (primary-resonance) and near one-half, twice, and three times its natural frequency (secondary-resonances). The nonlinear analysis is carried out using a shooting technique to capture periodic orbits combined with the Floquet theory to analyze their stability. The nonlinear resonance frequency of the CNTs is calculated as a function of the ac load. Subharmonic-resonances are found to be activated over a wide range of frequencies, which is a unique property of CNTs. The results show that these resonances can lead to complex nonlinear dynamics phenomena, such as hysteresis, dynamic pull-in, hardening and softening behaviors, and frequency bands with an inevitable escape from a potential well.

Original languageEnglish (US)
Pages (from-to)1-13
Number of pages13
JournalJournal of Computational and Nonlinear Dynamics
Issue number1
StatePublished - Jan 2010
Externally publishedYes


  • Carbon nanotube
  • Dynamic pull-in
  • Electrostatic force
  • Escape phenomenon
  • Fundamental natural frequency
  • Galerkin
  • Reduced-order model
  • Shooting

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Mechanical Engineering
  • Applied Mathematics


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