  Many modern engineering structures exhibit nonlinear vibration.
Characterizing such vibrations efficiently is critical to optimizing designs
for reliability and performance. For linear systems, steady-state vibration
occurs only at the forcing frequencies. However, nonlinearities (e.g., contact,
friction, large deformation, etc.) can result in nonlinear vibration behavior
including superharmonics - responses at integer multiples of the forcing
frequency. When the forcing frequency is near an integer fraction of the
natural frequency, superharmonic resonance occurs, and the magnitude of the
superharmonics can exceed that of the fundamental harmonic that is externally
forced. Characterizing such superharmonic resonances is critical to improving
engineering designs. The present work extends the concept of phase resonance
nonlinear modes (PRNM) to be applicable to general nonlinearities, and is
demonstrated for eight different nonlinear forces. The considered forces
include stiffening, softening, contact, damping, and frictional nonlinearities
that have not been previously considered with PRNM. The proposed variable phase
resonance nonlinear modes (VPRNM) method can accurately track superharmonic
resonances for hysteretic nonlinearities that exhibit amplitude dependent phase
resonance conditions that cannot be captured by PRNM. The proposed method
allows for characterization of superharmonic resonances without constructing a
full frequency response curve at every force level with the harmonic balance
method. Thus, the present method allows for analysis of potential failures due
to large amplitudes near the superharmonic resonance with reduced computational
cost. The consideration of single degree of freedom systems in the present
paper provides insights into superharmonic resonances and a basis for
understanding internal resonances for multiple degree of freedom systems.
