  We use the recent theory of Spectral Submanifolds (SSM) for model reduction
of nonlinear mechanical systems subject to parametric excitations.
Specifically, we develop expressions for higher-order nonautonomous terms in
the parameterization of SSMs and their reduced dynamics. We provide these
results both for general first-order as well as second-order mechanical systems
under periodic and quasiperiodic excitation using a multi-index based approach,
thereby optimizing memory requirements and the computational procedure. We
further provide theoretical results that simplify the SSM parametrization for
general second-order dynamical systems. More practically, we show how the
reduced dynamics on the SSM can be used to extract the resonance tongues and
the forced response around the principal resonances in parametrically excited
systems. In the case of two-dimensional SSMs, we formulate explicit expressions
for computing the steady-state response as the zero-level set of a
two-dimensional function for systems that are subject to external as well as
parametric excitation. This allows us to parallelize the computation of the
forced response over the range of excitation frequencies. We demonstrate our
results on several examples of varying complexity, including finite-element
type examples of mechanical systems. Furthermore, we provide an open-source
implementation of all these results in the software package SSMTool.
