“…In short, whereas normal form expansion, as proposed in [20,21,42,43], first computes the complete nonlinear mapping and then reduces by selecting a few master normal coordinates, the parametrisation method first reduces by selecting the master coordinates, and then computes the expansions, with the added value that different solutions are possible, thus offering the possibility of using either a graph style or a normal form style. With this initial choice, the developments are thus closer to those already reported in [44,45,46], where arbitrary order expansions have already been shown, together with the possibility of using either graph or normal form style. The main differences can be listed as follows: (i) the focus here is on large FE models of mechanical systems for which the damping matrix is diagonalised by the eigenvectors of the conservative system; (ii) thanks to this assumption, displacement and velocity mappings can be treated separately allowing to show the relationship between the two at generic order and to retrieve homological equations in the sole displacement mapping; (iii) a number of implementation details on the treatment of the direct computation are reported in order to decrease the computational burden (e.g.…”