  We show how spectral submanifold theory can be used to provide analytic
predictions for the response of periodically forced multi-degree-of-freedom
mechanical systems. These predictions include an explicit criterion for the
existence of isolated forced responses that will generally be missed by
numerical continuation techniques. Our analytic predictions can be refined to
arbitrary precision via an algorithm that does not require the numerical
solutions of the mechanical system. We illustrate all these results on low- and
high-dimensional nonlinear vibration problems. We find that our SSM-based
forced-response predictions remain accurate in high-dimensional systems, in
which numerical continuation of the periodic response is no longer feasible.
