The effects of nonlinear damping on degenerate parametric amplification

TL;DR

This study investigates how nonlinear damping influences the dynamic response and parametric amplification in systems with nonlinear stiffness, providing insights into optimizing drive conditions for desired amplification characteristics.

Contribution

It extends previous research by incorporating nonlinear damping effects into the analysis of parametric amplification, offering a comprehensive parameter study and bifurcation analysis.

Findings

Nonlinear damping significantly affects parametric gain.

Transitions among response curve topologies are characterized.

Guidelines for selecting drive conditions to optimize amplification.

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.