  The recently developed generalized Fourier-Galerkin method is complemented by
a numerical continuation with respect to the kinetic energy, which extends the
framework to the investigation of modal interactions resulting in folds of the
nonlinear modes. In order to enhance the practicability regarding the
investigation of complex large-scale systems, it is proposed to provide
analytical gradients and exploit sparsity of the nonlinear part of the
governing algebraic equations. A novel reduced order model (ROM) is developed
for those regimes where internal resonances are absent. The approach allows for
an accurate approximation of the multi-harmonic content of the resonant mode
and accounts for the contributions of the off-resonant modes in their
linearized forms. The ROM facilitates the efficient analysis of self-excited
limit cycle oscillations, frequency response functions and the direct tracing
of forced resonances. The ROM is equipped with a large parameter space
including parameters associated with linear damping and near-resonant harmonic
forcing terms. An important objective of this paper is to demonstrate the broad
applicability of the proposed overall methodology. This is achieved by selected
numerical examples including finite element models of structures with strongly
nonlinear, non-conservative contact constraints.
