Coaxial vibrations of electrostatically actuated DWCNT resonators: Amplitude–voltage​ response of parametric resonance

This paper investigates the amplitude–voltage response of parametric resonance of coaxial vibrations of double-walled carbon nanotubes (DWCNTs) under electrostatic actuation. The system under investigation consists of a DWCNT parallel to a ground plate and under AC voltage. This voltage produces a nonlinear electrostatic force leading the DWCNT into vibrations. There is a nonlinear intertube van der Waals force between the two coaxial carbon-nanotubes. In coaxial vibration, the two concentric nanotubes move together synchronously. The AC frequency is near the fundamental coaxial natural frequency of the DWCNT. This leads to parametric resonance. The case of small damping, soft electrostatic actuation, and DWCNTs with a high length to diameter ratio (Euler–Bernoulli beam model), is considered. Modal analysis is performed to decouple the equations of free vibrations in their linear part. The solution of the nonlinear problem is then found in terms of modal coordinates. Reduced Order Models (ROMs) using from one to five modes of vibration are used for investigation. Three methods are used to solve these models, (1) the Method of Multiple Scales used to solve the ROM using one mode of vibration, (2) continuation and bifurcation analysis of the five modes of vibration (5T) Reduced Order Model (ROM) using AUTO-07P, and (3) numerical integration of 5T ROM using Matlab. All models and methods are in excellent agreement for amplitudes lower than 0.4 of the gap, while at amplitudes larger than 0.4 only 5T-ROM provide reliable results. The effects of detuning frequency and damping on the amplitude–voltage response of DWCNTs under electrostatic actuation are reported. The importance of the results in this paper are the effect of damping and detuning frequency on the subcritical and supercritical bifurcations, as they define the voltage intervals DWCNT reaches nonzero steady-state amplitudes.