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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 the dynamic response of a single degree of freedom system with nonlinear stiffness and nonlinear damping that is subjected to both resonant direct excitation and resonant parametric excitation, with a general phase between the two. This generalizes and expands on previous studies of nonlinear effects on parametric amplification, notably by including the effects of nonlinear damping, which is commonly observed in a large variety of systems, including micro- and nano-scale resonators. Using the method of averaging, a thorough parameter study is carried out that describes the effects of the amplitudes and relative phase of the two forms of excitation. The effects of nonlinear damping on the parametric gain are first derived. The transitions among various topological forms of the frequency response curves, which can include isolae, dual peaks, and loops, are determined, and bifurcation analyses in parameter spaces of interest are carried out. In general, these results provide a complete picture of the system response and allow one to select drive conditions of interest that avoid bistability while providing maximum amplitude gain, maximum phase sensitivity, or a flat resonant peak, in systems with nonlinear damping.