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

Abstract

In the present work reduced order models (ROM) that are independent from the full order finite element models (FOM) considering geometrical non linearities are developed and applied to the dynamic study of a fan. The structure is considered to present nonlinear vibrations around the pre-stressed equilibrium induced by rotation enhancing the classical linearised 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 parametrised 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 linearised ROM solutions and the STEP ROM. The proposed StepC ROM provides the best compromise between accuracy and time consumption of the ROM.

Assessment of Geometric Nonlinearities Influence On NASA Rotor 37 Response to Blade Tip/Casing Rubbing Events

“…The reduced nonlinear internal forces gnl (q) are evaluated with the STiffness Evaluation Procedure (STEP) [21]. This non-intrusive method is compatible with commercial finite element software packages and has proven to be efficient for the modeling of dynamical mechanical systems subjected to large displacements [21,37,9,38]. This method consists in imposing static displacement fields and determining the corresponding internal nonlinear forces.…”

“…Reduction by projection on a reduced space is often used in the literature because of its ease of implementation, but also because it keeps the general expression of the equation of motion so that the same integration algorithms as those used for the full order model can be used. Different reduction bases have been recently investigated in a nonlinear context by different authors: linear bases [9,10], bases obtained from Proper Orthogonal Decomposition (POD) [11,8], linear bases augmented with modal derivatives (MD) [12,13,14] and bases composed of nonlinear complex modes [15,16].…”

Comparative Study of Blades Reduced Order Models With Geometrical Nonlinearities and Contact Interfaces

“…In order to be compatible with commercial finite element software, this paper focuses on non-intrusive methods. In particular, the STiffness Evaluation Procedure (STEP) is chosen here because it has proven to be efficient for the modeling of dynamical mechanical systems subjected to large displacements [20,27,10,28].…”

“…basis composed of linear vibration modes or Craig-Bampton basis) to obtain the generalized forces in the reduced order model may create spurious artifacts in the time response, inducing lower amplitude displacement with higher harmonics. To avoid such behavior, Balmaseda et al [10] suggest to carry out a filtering of the full order model nonlinear forces. In practice, the nonlinear forces are collected from a given number of snapshots that represent a set of characteristic displacements in the response u 1 , ..., u m ,…”

“…The second challenge consists in evaluating the internal nonlinear forces in the reduced space. Different reduction bases have been recently investigated in a nonlinear context by different authors: linear bases [10,11], bases obtained from Proper Orthogonal Decomposition (POD) [12,3], linear bases augmented with modal derivatives (MD) [13,14] and bases composed of nonlinear complex modes [15,16]. The methods for the determination of the reduced nonlinear forces are classified as intrusive and non-intrusive methods.…”