Voltage effect on amplitude-frequency response of parametric resonance of electrostatically actuated double-walled carbon nanotube resonators

摘要

This work deals with the amplitude-frequency response of coaxial parametric resonance of electrostatically actuated double-walled carbon nanotubes (DWCNTs). Nonlinear forces acting on the DWCNT are intertube van der Waals and electrostatic forces. Soft alternating current (AC) excitation and small viscous damping are assumed. In coaxial vibration, the outer and inner carbon nanotubes move synchronously (in-phase). Euler-Bernoulli beam model is used for DWCNTs of high length-to-diameter ratio. Modal coordinates are used for decoupling the linearized differential equations of motion without damping. The reduced-order model (ROM) method is used in this investigation. All ROMs using one through five modes of vibration (terms) are developed in terms of modal coordinates. ROM using one term is solved and frequency-amplitude response predicted by using the method of multiple scales (MMS). All other ROMs using two through five terms are numerically integrated using MATLAB in order to simulate time responses of the structure and also solved using AUTO-07P, a software package of continuation and bifurcation, in order to predict the frequency-amplitude response. All models and methods are in agreement at lower amplitudes, while in higher amplitudes only ROM with five terms provides reliable results. The effects of voltage and damping on the amplitude-frequency response of electrostatically actuated DWCNTs are reported. It is shown that increasing voltage and/or decreasing damping results in a larger range of frequencies for which pull-in occurs.