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

TL;DR

This paper presents a direct, simulation-free method to compute nonlinear mappings for reduced-order models of finite element nonlinear structures using normal form theory, improving accuracy and capturing internal resonances.

Contribution

It introduces a novel direct computation approach for third-order normal forms that enables accurate, invariant-based reduced-order modeling without limiting the number of master modes.

Findings

Successfully applied to a 3D beam model, capturing internal resonance.

Accurately predicted dynamics of a fan blade model.

Demonstrated high accuracy in frequency-response curves.

Abstract

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.