Period multiplications and nonlinear coupling in micro-electromechanical resonators

Abstract

Micro-/nanoelectromechanical system (M/NEMS) resonators are of great importance owing to their wide applications in daily life as well as fundamental research. In this dissertation, we will study the novel dynamics induced by the nonlinearities of the resonant systems with these micro-/nano scale resonators.

One major category of the nonlinear phenomena we will focus on is period multiplication. That is, the resonator will oscillate at its eigenfrequency (𝜔₀) by applying periodic excitation with frequency nω₀. The n=3 case was first achieved in a device modeled as a single-degree-of-freedom resonator. In this circumstance, the system can settle to one of three period-period tripled states or a zero amplitude state. The period-tripled states are identical in amplitude but different in phases by 2π/3. By applying a secondary drive near ω₀, the system can controllably switch among the states. The operation shows the potential of using a mechanical resonator as ternary logic or memory element. Besides that, generating of frequency comb or amplifying signal are also demonstrated with it.

The period-tripled oscillations can also be excited with the assistance of efficient energy transfer between modes. To study it, we employ a resonator with two nonlinear coupled modes whose eigenfrequencies are close to a ratio of 1:3. When driving near the resonance of the upper mode (mode 2), mode 1 can oscillate at 1/3 that of the driving frequency. The states of mode 1 are the same as that of the of single mode one. When mode 2 becomes bistable, there will emerge another set of period-tripled states. The results imply the possibility of developing a mechanical resonator as a multi-valued logic gate or memory element (much higher than three). Besides the excitations of the period-tripled states, the energy transfer led by this type of mode coupling can also generate a lot of novel dynamics, e.g., anomalous decay, limit cycles, etc. Some of them are also studied with the same resonator.

When pumped at the coupled mode system at the sum of eigenfrequencies of mode 1 and mode 2, the oscillations in both modes can be aroused. The oscillations are named self-sustained oscillations (SSO). To achieve it, internal resonance is not required. In the resonator with dispersive coupling only, the oscillations are sinusoidal. But they are not coherent with the pump signal. In the attendance of thermal noise, phases of both modes will diffuse. But their sum is highly correlated to the pump. We theoretically analyzed the behavior and proposed a scheme to stabilize the frequency of one of the modes.

When the internal resonance is also introduced to the sideband pumped coupled mode resonator. With the additional energy exchange channel, the SSO here becomes nonsinusoidal. Their spectra become comb shapes. The Teeth of the spectra are equally spaced with highly related to the pump frequency. Besides the SSO, period-quadrupled oscillations emerge in mode 1. The oscillations can be in one of the four different states with the same amplitude but the phases offset by π/2. Meanwhile, the oscillations of mode 2 at 3ω_p/4 also possesses four corresponding states. That not allows the implementation of the mechanical resonator in quadrupled phase shift keying (QPSK), but also the upconversion of the carrier signal of encoded information.

Date of Award	2022
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Ho Bun CHAN (Supervisor)