Reduced Order Model of Coaxial Vibrations of Electrostatically Actuated Double Wall Carbon Nanotubes: Frequency Response of Primary Resonance

In this paper, the Reduced Order Method (ROM) and the Method of Multiple Scales (MMS) are used to investigate the influences of dimensionless damping and voltage parameters on the amplitude-frequency response of an electrostatically actuated double-walled carbon nanotube (DWCNT). The forces responsible for the nonlinearities in the vibrational behavior are intertube van der Waals and electrostatic forces. Soft AC excitation and small viscous damping forces are assumed. Herein, the coaxial case is investigated. In this mode of vibration, the outer and inner carbon nanotubes move synchronously (in-phase) with the same maximum tip deflection. The DWCNT structure is modelled as a cantilever beam with Euler-Bernoulli beam assumptions since the DWCNT is characterized with high length-diameter ratio. The results shown assume steady-state solutions in the first-order MMS solution. The analytical approximate solutions provided by MMS are validated numerically by two-term (2T) Time Reponses and AUTO-07P. The two methods in this paper are found to be in excellent agreement at lower amplitudes. Additionally, the two methods are assessed for their advantages and limitations. The importance of the results in this paper are the effect of damping and voltage on the stability of the DWCNT vibration.