Reduced Order Models for Nonlinear Dynamic Analysis With Application to a Fan Blade

Abstract

In this work, reduced order models (ROM) that are independent from the full order finite element models (FOM) considering geometrical nonlinearities are developed and applied to the dynamic study of a fan. The structure is considered to present nonlinear vibrations around the prestressed equilibrium induced by rotation enhancing the classical linearized approach. The reduced nonlinear forces are represented by a polynomial expansion obtained by the stiffness evaluation procedure (STEP) and then corrected by means of a proper orthogonal decomposition (POD) that filters the full order nonlinear forces (StepC ROM). The linear normal modes (LNM) and Craig-Bampton (C-B) type reduced basis are considered here. The latter are parameterized with respect to the rotating velocity. The periodic solutions obtained with the StepC ROM are in good agreement with the solutions of the FOM and are more accurate than the linearized ROM solutions and the STEP ROM. The proposed StepC ROM provides the best compromise between accuracy and time consumption of the ROM.

Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures

“…7, it recovers directly the correct softening behaviour, as it could have been expected from previous theoretical derivations [15,30]. This good prediction is an excellent result since many reduction method encounters difficulties in retrieving such a behaviour [47,69]. However, one can see that, when vibration amplitude at the forcing point reach values at about 0.1 m, the single NNM solution starts to depart from the reference solution, underlining that a new stiffening effect comes into play and needs to be taken into account on order to obtain a ROM prediction closer to the reference.…”

“…As a matter of fact, no clear numerical demonstration that a reduced-order model is able to accurately compute the non-linear vibration of a Fan Blade has been proposed so far. The existing methods published in literature for vibration of large 3D models cannot achieve a better accuracy that 10% of error, in the best case, as compared to the full model [69].…”

“…A second ROM is built by adding nonlinear damping terms, following Eq. (37). However a simplification is introduced.…”

“…Improvements of the method have been proposed, see e.g. the dual modes proposed in [36], the combination of STEP with a POD (Proper Orthogonal Decomposition) [37], or a modified STEP where displacements are imposed on selected degrees of freedom only [38,39]. Implicit condensation and expansion (ICE) have been proposed in order to eliminate the high-frequency components [40,41,42] thanks to a two-step procedure where a stress manifold is fitted after a series of static loading computations.…”

Reduced Order Model of Nonlinear Structures for Turbomachinery Aeroelasticity

“…Nonlinear internal forces are extracted from these computations and used to evaluate the nonlinear coefficients of the polynomial. Although this method is efficient for 2D structures, specific corrections are needed for 3D structures [19,20] since perturbations are introduced by possible conflicts between the natural volumetric dilatation/compression of the structure and the one imposed by the prescribed displacement. The second method, which is not sensitive to these artifacts, is the Implicit Condensation (IC) [12] and its Expansion (ICE) [13].…”