Theoretical investigation of nonlinear resonances in a carbon nanotube cantilever with a tip-mass under electrostatic excitation

I K Kim,S I Lee

doi:10.1063/1.4820577

Abstract

The nonlinear dynamics of a resonating carbon nanotube (CNT) cantilever having an attached mass at the tip (“tip mass”) were investigated by incorporating electrostatic forces and intermolecular interactions between the CNT and a conducting plane surface. This work enables applications of CNT resonating sensors for tiny mass detection and provides a better understanding of the dynamics of CNT cantilevers. The effect of tip mass on a resonating CNT cantilever is normally characterized by the fundamental frequency shift in the linear resonance regime. However, there are more complex dynamics in the nonlinear resonance regime, such as secondary resonances with parametric excitation. The latter have been limited to nano-cantilevers without tip mass or to axially excited micro-beams. To analyze the nonlinear dynamics, we developed a differential equation model that includes both geometric and inertial nonlinear terms for the large vibration amplitudes at increasing drive forces. In our approach, we used Galerkin discretization techniques and numerical integration methods. The CNT cantilever exhibited complex nonlinear responses due to the applied AC and DC voltages and various tip masses. The nonlinear model had a softer response for increasing tip mass than those of the linear model with the same driving conditions. At low applied voltages, the cantilever had linear amplitude and phase responses at primary and secondary superharmonic resonance frequencies. The response branches were softened at the primary resonance through saddle-node (SN) bifurcation from harmonic electrostatic excitation at higher applied voltages. After SN bifurcation, the lower branch of the solution near resonance became unstable. In addition, theoretical analyses were performed on more complex nonlinear responses and stability changes with tip mass variations, such as period-doubling (PD) bifurcation at subharmonic resonance frequencies.

Highlights

With excellent mechanical and electrical properties, carbon nanotubes (CNTs) are ideal materials for nanoelectromechanical systems
The effect of tip mass on a resonating CNT cantilever is normally characterized by the fundamental frequency shift in the linear resonance regime
We analyzed the nonlinear responses of a nanoresonator in the frequency and time domains with tip mass effects and various excitation conditions

Summary

INTRODUCTION

With excellent mechanical and electrical properties, carbon nanotubes (CNTs) are ideal materials for nanoelectromechanical systems. The dynamics of nano-resonators has been examined to predict the linear or nonlinear responses of a cantilevered CNT. Other studies include those on mass- or bio-sensors for detecting metal atoms, proteins, or bacteria.. A mass sensor for detecting a metal atom can be used in vacuum or in air, but a biosensor needs to be operated in an environment that preserves the shape, functions, and mechanical or chemical properties of biological entities. In this respect, Sawano et al. examined the frequency response of the CNT resonator in water or air. The dynamic stability of the nanoresonator was determined by frequency and phase responses; we verified the stability using transient and steady-state time responses with varying tip masses

NANO-RESONATOR MODELING

NONLINEAR DYNAMIC RESPONSE OF CNT CANTILEVER MODEL

CONCLUSIONS