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

Alessandra Vizzaccaro*, Yichang Shen, Loïc Salles, Jiří Blahoš, Cyril Touzé

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)

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 non-resonant 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.

Original languageEnglish
Article number113957
Number of pages35
JournalComputer Methods in Applied Mechanics and Engineering
Volume384
Early online date12 Jun 2021
DOIs
Publication statusPublished - 1 Oct 2021

Bibliographical note

Funding Information:
The author A. Vizzaccaro is thankful to Rolls-Royce plc for the financial support. The author Y. Shen wishes to thank China Scholarship Council (No. 201806230253 ). The author L. Salles is thankful to Rolls-Royce plc and the EPSRC, United Kingdom for the support under the Prosperity Partnership Grant “ Cornerstone: Mechanical Engineering Science to Enable Aero Propulsion Futures, United States ”, Grant Ref: EP/R004951/1 . The author J. Blahoš thank the European Union’s Horizon 2020 Framework Programme research and innovation programme under the Marie Sklodowska-Curie agreement No 721865 .

Funding Information:
The author A. Vizzaccaro is thankful to Rolls-Royce plc for the financial support. The author Y. Shen wishes to thank China Scholarship Council (No. 201806230253). The author L. Salles is thankful to Rolls-Royce plc and the EPSRC, United Kingdom for the support under the Prosperity Partnership Grant ?Cornerstone: Mechanical Engineering Science to Enable Aero Propulsion Futures, United States?, Grant Ref: EP/R004951/1. The author J. Blaho? thank the European Union's Horizon 2020 Framework Programme research and innovation programme under the Marie Sklodowska-Curie agreement No 721865.

Publisher Copyright:
© 2021 Elsevier B.V.

Keywords

  • Geometric nonlinearities
  • Nonlinear mapping
  • Normal form
  • Reduced order modelling

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