The effects of nonlinear damping on degenerate parametric amplification
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ORIGINAL PAPER
The effects of nonlinear damping on degenerate parametric amplification Donghao Li
· Steven W. Shaw
Received: 20 August 2020 / Accepted: 11 November 2020 © The Author(s) 2020
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. D. Li (B) · S. W. Shaw Department of Mechanical and Civil Engineering, Florida Institute of Technology, Florida 32901, USA e-mail: [email protected] S. W. Shaw e-mail: [email protected]
Keywords Parametric amplification · Nonlinear damping · Bifurcation analysis · MEMS Mathematics Subject Classification 70K28 · 70K40 · 70K50
1 Introduction Parametric amplification (PA) is the use of a resonant parametric excitation to enhance the response of a resonantly driven oscillator. This approach allows one to alter the effective damping of the system, even to the limit of zero damping at the point of parametric instability, thus bringing benefits of spectral narrowing and higher-frequency selectivity to resonant systems [1,2]. Specifically, the amplification, deamplification, and thermal noise squeezing have been analyzed for a Josephson parametric amplifier (JPA) [3,4]. In a classic study of a mechanical device, PA and thermomechanical noise squeezing were observed in a vibrating microcantilever and were analyzed using a linear model [5]. These studies were based on a linear model, which demonstrated the main effects. Studies on the impacts of stiffness nonlinearity on PA and similar systems have shown that the response can be quite rich [6–8]. The Duffing nonlinearity is especially of interest, as it is oftentimes exhibited in systems with large vibration amplitude, resulting in both opportunities for improved performance and challenges due to the complexity in dynamic responses [9,10].
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D. Li, S. W. Shaw
PA is used in a wide variety of applications, especially in the realm of nano- and m
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