Studies in Nonlinear and Stochastic Phenomena and Quality Factor Enhancement in a Nanomechanical Resonator

Abstract

Nonlinear damping has recently been experimentally observed in carbon nanotube and graphene-based nanoelectromechanical (NEMS) resonators and shown to be an effective means to achieve higher quality (Q) factors. Moreover, it has been shown that white noise excitation can be exploited to shrink the resonance width of the frequency response characteristics of the resonator as a pathway to higher Q factors. Motivated thus, this thesis is a study of certain fundamental characteristics of the nonlinear dynamics of a nanoelectromechanical resonator in both the deterministic and stochastic regimes with a focus on the influence of those characteristics on the Q factor. Using a Duffing oscillator based model, this thesis: (1) derives an analytical expression between oscillation amplitude and frequency of a NEMS resonator using the harmonic balance method to study the frequency response characteristics and validates the results using numerical simulation, (2) studies the deterministic dynamics of a NEMS resonator deriving an analytical relationship between the phase angle and maximum oscillation of the resonator response, (3) derives an analytical expression between the resonance frequency and resonance amplitude, (4) studies the hysteresis characteristics both in the stochastic and deterministic regimes elucidating the effects of nonlinear damping and external excitation on the hysteresis region, (5) finds that stochastic excitation with increasing intensity can shrink the hysteresis width, (6) shows that increasing the magnitude of the linear damping coefficient results in the decrease of Q-factors, (7) shows that in the combined presence of both parametric and external excitation, increasing the ratio of pump frequency to external forcing frequency results in lower resonant frequency and lower resonance width, (8) observes that in the parametrically driven nanomechanical resonator, higher parametric oscillation amplitude increases the resonance amplitude with a small impact on the resonance frequency, (9) solves the stochastic model using the Euler-Maruyama method and generates frequency response curves where it is found that higher noise intensity of Levy stable stochastic process can increase the Q factor, (10) finds that the Q factor is increased by decreasing the nonlinear damping and external harmonic driving amplitude. In summary, this thesis presents a set of novel results on the nonlinear, stochastic dynamics of a NEMS resonator and discusses the implications of the results for achieving enhanced Q factors. The results are of interest both from a theoretical viewpoint as well as in sensing applications using a nanoresonator.