  The resonances of forced dynamical systems occur when either the amplitude of
the frequency response undergoes a local maximum (amplitude resonance) or phase
lag quadrature takes places (phase resonance). This study focuses on the phase
resonance of nonlinear mechanical systems subjected to single-point,
single-harmonic excitation. In this context, the main contribution of this
paper is to develop a computational framework which can predict the mode shapes
and oscillation frequencies at phase resonance. The resulting nonlinear modes
are termed phase resonance nonlinear modes (PRNMs). A key property of PRNMs is
that, besides primary resonances, they can accurately characterize
superharmonic, subharmonic and ultra-subharmonic resonances for which, as shall
be shown in this paper, phase lags at resonance may be different from pi/2. The
proposed developments are demonstrated using one- and two-degree-of-freedom
systems featuring a cubic nonlinearity.
