"In this work we consider several nonlinearity-based and/or noise-related phenomena that were observed recently in micro-electromechanical resonators. The main goal here is to closely examine these phenomena, understand their underlying fundamental physics, and find ways to employ them for measurement purposes and/or to improve the performance of specific classes of micro-electromechanical systems (MEMS). The general perspective of this work is based on the acceptance of the fact that nonlinearity and noise represent integral parts of the models for these systems, and the discussion is constructed about the cases where these generally "undesirable" features can be utilized rather than avoided. In this dissertation we consider three different, but related, topics. To start, we analyze the stationary probability distribution of a nonlinear resonator driven by Poisson noise at a non-zero temperature of the environment. We show that Poisson pulses with low pulse rates cause the power-law divergence of the probability density at the resonator equilibrium in the rotating frame both in overdamped and underdamped regimes. We have also found that the shape of the probability distribution away from the equilibrium position is qualitatively different for overdamped and underdamped cases. In particular, in the overdamped regime, the form of the secondary singularity in the probability distribution strongly depends on the reference phase of the resonator response as well as the pulse modulation phase, while in the underdamped regime there are several singular peaks, and their location is determined by the resonator decay rate in the rotating frame. Finally, we show that even weak Gaussian noise affects the probability distribution by smoothing it in the vicinity of singular peaks. Second, we discuss a time-domain technique for characterizing parameters for models of a single vibrational mode of symmetric micromechanical resonators, including coefficients of conservative and dissipative nonlinearities as well as the strengths of noise sources acting on the mode of interest. These nonlinearities result in an amplitude-dependent frequency and a non-exponential decay, while noise sources cause fluctuations in the resonator amplitude and phase. We capture these features in the modal ringdown response. Analysis of the amplitude of the ringdown response allows one to estimate the quality factor and the dissipative nonlinearity, and the zero-crossing points in the ringdown measurement can be used for characterization of the linear natural frequency and the Duffing and quintic nonlinearities of the vibrational mode which arise from a combination of mechanical and electrostatic effects. Additionally, we show that statistical analysis of the zero-crossing points in the resonator response allows us to separate effects of additive, multiplicative, and measurement noises and estimate their corresponding intensities. Finally, we examine the problem of self-induced parametric amplification in ring/disk resonating gyroscopes. We model the dynamics of this type gyroscopes by considering flexural (elliptical) vibrations of a free thin ring and show that the parametric amplification arises naturally from a nonlinear intermodal coupling between the drive and sense modes of the gyroscope. Analysis shows that this modal coupling results in substantial increase in the sensitivity of the gyroscope to the external angular rate. This improvement in the gyroscope performance strongly depends both on the modal coupling strength and the operating point of the gyroscope, and we further explore ways to enhance this effect by changing the shape of the resonator body and attendant electrodes and utilizing electrostatic tuning."--Pages ii-iii.
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