The numerical simulation of structural vibration of aircraft engines components is of crucial importance for engine manufacturers both in the design stage and in the assessment of failures. Protracted vibration of rotor blades shortens their lives and might cause blade failure due to high cycle fatigue. To ensure the structural integrity of blades, accurate predictions of their dynamic behaviour are necessary. In this respect, numerical models based on linear approximations are no longer representative of the actual behaviour of the blade that is affected by several nonlinear forces. Finite element models of blades are usually large models with a high number of degrees of freedom and complex geometry. To compute the nonlinear dynamic behaviour of blade in a fast yet accurate manner, reduced order models are necessary. The main focus of this thesis is the assessment and investigation of reduced order modelling methods for geometrically nonlinear structures. It will be shown that reduced order modelling methods based on modal coordinates displays a slow convergence of the basis when applied to structures discretised with three dimensional finite elements. Direct methods based on modal derivatives have instead the advantage of being computed
directly in physical basis and the computation of the reduced dynamics can be done non-intrusively from any generic finite element software. However, the applicability of these methods is restricted to the case of a clear separation between the mode of interest and the modes coupled with it through the nonlinear forces. Invariant manifold based methods are instead able to reproduce the correct dynamics even without a slow/fast separation between modes, thanks to the inclusion of velocity dependent terms in the nonlinear mapping. However, invariant manifold methods are based on modal coordinates thus making their applicability limited to relatively small structures. In this work, a direct method fully in physical coordinates able to compute the invariant manifold through normal form is proposed and its capabilities assessed.