Abstract

The objective of this contribution is to compare two methods proposed recently in order to build efficient reduced-order models for geometrically nonlinear structures. The first method relies on the normal form theory that allows one to obtain a nonlinear change of coordinates for expressing the reduced-order dynamics in an invariant-based span of the phase space. The second method is the modal derivative approach, and more specifically, the quadratic manifold defined in order to derive a second-order nonlinear change of coordinates. Both methods share a common point of view, willing to introduce a nonlinear mapping to better define a reduced-order model that could take more properly into account the nonlinear restoring forces. However, the calculation methods are different and the quadratic manifold approach has not the invariance property embedded in its definition. Modal derivatives and static modal derivatives are investigated, and their distinctive features in the treatment of the quadratic nonlinearity are underlined. Assuming a slow/fast decomposition allows understanding how the three methods tend to share equivalent properties. While they give proper estimations for flat symmetric structures having a specific shape of nonlinearities and a clear slow/fast decomposition between flexural and in-plane modes, the treatment of the quadratic nonlinearity makes the predictions different in the case of curved structures such as arches and shells. In the more general case, normal form approach appears preferable since it allows correct predictions of a number of important nonlinear features, including the hardening/softening behaviour, whatever the relationships between slave and master coordinates are.

Highlights

  • Reduced-order modelling of thin structures experiencing large-amplitude vibration is a topic that has attracted a large amount of researches in the last years

  • A particular attention is paid on writing the differences one can await when using the quadratic manifold with modal derivatives (MDs) and static modal derivatives (SMDs), in comparison with the results provided by normal form theory in mind, giving rise to new developments

  • The dynamics onto the reduced subspaces is given by Eq (36) when using the MD approach, Eq (37) if one considers SMD instead, and Eq (38) with the normal form method, with R1 the master coordinates

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Summary

Introduction

Reduced-order modelling of thin structures experiencing large-amplitude vibration is a topic that has attracted a large amount of researches in the last years. The invariance property of reduction spaces is encapsulated in their definition, ensuring that the dynamical solutions computed from a reduced-order model (ROM) exist for the full system [17,45,50,52] This key ingredient allows deriving accurate ROMs, which, for example, are able to predict the correct hardening/softening behaviour of nonlinear structure, which is not the case for their linear counterparts [57]. The invariance property is directly inherited from the definition of an NNM as an invariant manifold in phase space, while the invariance of the quadratic manifold computed from MDs is not at hand The aim of this contribution is to investigate more properly the common points and differences in the Comparison of nonlinear mappings for reduced-order two methods and explain their advantages and drawbacks in the context of building reduced-order models for geometrically nonlinear structures.

Framework
NNMs and normal form
Modal derivatives
Definition of modal derivatives and static modal derivatives
Quadratic manifold
Reduced-order model obtained with quadratic manifold
Drift and mode shapes
Comparison on two-degree-of-freedom systems
Presentation of the model
Results
Equations of motion
Presentation of the test cases
Amplitude–frequency relationships
Nonlinear mode shapes
Conclusion
Compliance with ethical standards
Full Text
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