  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.
