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Abstract In contrast to the well-known phenomenon of frequency stabilization in a synchronized noisy nonlinear oscillator, little is known about its amplitude stability. In this paper, we investigate experimentally and theoretically the amplitude evolution and stability of a nonlinear nanomechanical self-sustained oscillator that is synchronized with an external harmonic drive. We show that the phase difference between the tones plays a critical role on the amplitude level, and we demonstrate that in the strongly nonlinear regime, its amplitude fluctuations are reduced considerably. These findings bring to light a new facet of the synchronization phenomenon, extending its range of applications beyond the field of clock-references and suggesting a new means to enhance oscillator amplitude stability. 

Abstract In this work we demonstrate how one can improve the angular rate sensitivity of ring/disk resonating gyroscopes by tailoring their nonlinear behavior by systematic shaping of the gyroscope body and electrodes, and by the tuning of bias voltages on segmented electrodes. Of specific interest are the drive and sense mode Duffing nonlinearities, which limit their dynamic ranges, and the intermodal dispersive coupling between these modes that provides parametric amplification of the sense mode output signal. These two effects have the same physical origins and are in competition in terms of system performance, which naturally calls for optimization considerations. The present analysis is based on a systematic modeling of the nonlinear response of these devices by which we explore ways in which one can optimize the angular rate sensitivity by manipulating the mechanical and electrostatic contributions to the nonlinearities. In particular, non-uniform modifications of the gyroscope body thickness are employed to affect the mechanical contributions to these parameters, while the electrostatic components are manipulated via shaping of the resonator-electrode gap and by applying non-uniform bias voltages among segmented electrodes around the gyroscope body. These models predict that such relatively simple alterations can achieve improvements in gain by about an order of magnitude when compared to devices with uniform layouts. 

This paper describes a hybrid approach for modeling nonlinear vibrations and determining essential (normal form) coefficients that govern a reduced-order model of a structure. Incorporating both computational and analytical tools, this blended method is demonstrated by considering a micro-electro-mechanical vibrating gyroscopic rate sensor that is actuated by segmented DC electrodes. Two characterization methods are expatiated, where one is more favorable in computational tools and the other can be used in experiments. Using the reduced model, it is shown that tuning the nonuniform DC bias results in favorable changes in Duffing and mode-coupling nonlinearities which can improve the gyroscope angular rate sensitivity by two orders of magnitude. 

Abstract This paper considers nonlinear interactions between vibration modes with a focus on recent studies relevant to micro- and nanoscale mechanical resonators. Due to their inherently small damping and high susceptibility to nonlinearity, these devices have brought to light new phenomena and offer the potential for novel applications. Nonlinear interactions between vibration modes are well known to have the potential for generating a “zoo” of complicated bifurcation patterns and a wide variety of dynamic behaviors, including chaos. Here, we focus on more regular, robust, and predictable aspects of their dynamics, since these are most relevant to applications. The investigation is based on relatively simple two-mode models that are able to capture and predict a wide range of transient and sustained dynamical behaviors. The paper emphasizes modeling and analysis that has been done in support of recent experimental investigations and describes in full detail the analysis and attendant insights obtained from the models that are briefly described in the experimental papers. Standard analytical tools are employed, but the questions posed and the conclusions drawn are novel, as motivated by observations from experiments. The paper considers transient dynamics, response to harmonic forcing, and self-excited systems and describes phenomena such as extended coherence time during transient decay, zero dispersion response, and nonlinear frequency veering. The paper closes with some suggested directions for future studies in this area.