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

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 perioddoubling (PD) bifurcation at subharmonic resonance frequencies.

A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature

“…[14,15], it was found that the nonlinear curvature makes a major contribution to the pull-in voltage and pull-in deflection, and should be considered for effective design. Recently, taking the effect of nonlinear curvature into consideration, Kim and Lee [16], and Souayeh and Kacem [17] investigated the nonlinear vibration of electrostatically actuated cantilever carbon with an attached mass.…”

Multi-function and cascadable MEMS logic device

“…Hence the performance parameters mentioned above can be improved by designing a suitable device with high integration densities, fast switching speed, and low energy consumption per cycle. With the advancements in areas of carbon nano tubes [14] and graphene based resonators [15], integration densities as high as ~10 8 with power ranging in 10 -5 W and a switching speed of ~20Mz are quite viable. Resonators operating at half the resonance frequency respond at resonance frequency when combination resonance is activated.…”