  The direct computation of the third-order normal form for a geometrically
nonlinear structure discretised with the finite element (FE) method, is
detailed. The procedure allows to define a nonlinear mapping in order to derive
accurate reduced-order models (ROM) relying on invariant manifold theory. The
proposed reduction strategy is direct and simulation free, in the sense that it
allows to pass from physical coordinates (FE nodes) to normal coordinates,
describing the dynamics in an invariant-based span of the phase space. The
number of master modes for the ROM is not a priori limited since a complete
change of coordinate is proposed. The underlying theory ensures the quality of
the predictions thanks to the invariance property of the reduced subspace,
together with their curvatures in phase space that accounts for the nonresonant
nonlinear couplings. The method is applied to a beam discretised with 3D
elements and shows its ability in recovering internal resonance at high energy.
Then a fan blade model is investigated and the correct prediction given by the
ROMs are assessed and discussed. A method is proposed to approximate an
aggregate value for the damping, that takes into account the damping
coefficients of all the slave modes, and also using the Rayleigh damping model
as input. Frequency-response curves for the beam and the blades are then
exhibited, showing the accuracy of the proposed method.
