Comparison of nonlinear mappings for reduced-order modelling of vibrating structures: normal form theory and quadratic m
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ORIGINAL PAPER
Comparison of nonlinear mappings for reduced-order modelling of vibrating structures: normal form theory and quadratic manifold method with modal derivatives Alessandra Vizzaccaro Cyril Touzé
· Loïc Salles ·
Received: 17 March 2020 / Accepted: 7 July 2020 © The Author(s) 2020
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. Keywords Reduced-order modelling · Normal form · Quadratic manifold · Modal derivatives
1 Introduction Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11071-020-05813-1) contains supplementary material, which is available to authorized users. A. Vizzaccaro (B)· L. Salles Imperial College London, Exhibition Road, SW7 2AZ London, UK e-mail: [email protected] C. Touzé IMSIA, ENSTA Paris, CNRS, EDF, CEA, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France
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 number of methods have been proposed, with variants driven by either the structure under study and its peculiarity [63], the dynamical behaviour exhibited by the system [64], the model [54] or the discretisation method [33]. Roughly speaking, one can divide the techniques proposed in the literature into two diff
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