  The direct parametrisation method for invariant manifold is a model-order
reduction technique that can be applied to nonlinear systems described by PDEs
and discretised e.g. with a finite element procedure in order to derive
efficient reduced-order models (ROMs). In nonlinear vibrations, it has already
been applied to autonomous and non-autonomous problems to propose ROMs that can
compute backbone and frequency-response curves of structures with geometric
nonlinearity. While previous developments used a first-order expansion to cope
with the non-autonomous term, this assumption is here relaxed by proposing a
different treatment. The key idea is to enlarge the dimension of the
parametrising coordinates with additional entries related to the forcing. A new
algorithm is derived with this starting assumption and, as a key consequence,
the resonance relationships appearing through the homological equations involve
multiple occurrences of the forcing frequency, showing that with this new
development, ROMs for systems exhibiting a superharmonic resonance, can be
derived. The method is implemented and validated on academic test cases
involving beams and arches. It is numerically demonstrated that the method
generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic
resonances, as well as converged results for systems where the first-order
truncation on the non-autonomous term showed a clear limitation.
