  The objective of this contribution is to compare two methods proposed
recently in order to build efficient reduced-order models for geometrically
nonlinear structures. The first method relies on the normal form theory that
allows one to obtain a nonlinear change of coordinates for expressing the
reduced-order dynamics in an invariant-based span of the phase space. The
second method is the modal derivative (MD) approach, and more specifically the
quadratic manifold defined in order to derive a second-order nonlinear change
of coordinates. Both methods share a common point of view, willing to introduce
a nonlinear mapping to better define a reduced order model that could take more
properly into account the nonlinear restoring forces. However the calculation
methods are different and the quadratic manifold approach has not the
invariance property embedded in its definition. Modal derivatives and static
modal derivatives are investigated, and their distinctive features in the
treatment of the quadratic nonlinearity is underlined. Assuming a slow/fast
decomposition allows understanding how the three methods tend to share
equivalent properties. While they give proper estimations for flat symmetric
structures having a specific shape of nonlinearities and a clear slow/fast
decomposition between flexural and in-plane modes, the treatment of the
quadratic nonlinearity makes the predictions different in the case of curved
structures such as arches and shells. In the more general case, normal form
approach appears preferable since it allows correct predictions of a number of
important nonlinear features, including for example the hardening/softening
behaviour, whatever the relationships between slave and master coordinates are.
