Reduced-order modelling of parameter-dependent systems with invariant manifolds: Application to Hopf bifurcations in follower force problems

de Figueiredo Stabile, André; Vizzaccaro, Alessandra; Salles, Loïc; Colombo, Alessio; Frangi, Attilio; Touzé, Cyril

doi:10.1016/j.ijnonlinmec.2025.105133

Article (Scientific journals)

Reduced-order modelling of parameter-dependent systems with invariant manifolds: Application to Hopf bifurcations in follower force problems

de Figueiredo Stabile, André; Vizzaccaro, Alessandra; Salles, Loïc et al.

2025 • In International Journal of Non-Linear Mechanics, 177, p. 105133

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https://hdl.handle.net/2268/333934

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10.1016/j.ijnonlinmec.2025.105133

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Keywords :

Follower force; Hopf bifurcation; Invariant manifold; Nonlinear normal modes; Nonlinear oscillations; Parameter-dependent system; Parametrisation method; Follower forces; Invariant manifolds; Limit-cycle; Nonlinear oscillation; Parameter-dependent systems; Parametrization method; Parametrizations; Reduced order modelling; Reduced-order model; Mechanics of Materials; Mechanical Engineering; Applied Mathematics

Abstract :

[en] The direct parametrisation method for invariant manifolds is adjusted to consider a varying parameter. More specifically, the case of systems experiencing a Hopf bifurcation in the parameter range of interest is investigated, and the ability to predict the amplitudes of the limit cycle oscillations after the bifurcation is demonstrated. The cases of the Ziegler pendulum and Beck's column, both of which have a follower force, are considered for applications. By comparison with the eigenvalue trajectories in the conservative case, it is advocated that using two master modes to derive the ROM, instead of only considering the unstable one, should give more accurate results. Also, in the specific case where an exceptional bifurcation point is met, a numerical strategy enforcing the presence of Jordan blocks in the Jacobian matrix during the procedure is devised. The ROMs are constructed for the Ziegler pendulum having two and three degrees of freedom, and then Beck's column is investigated, where a finite element procedure is used to spatially discretise the problem. The numerical results show the ability of the ROMs to correctly predict the amplitude of the limit cycles up to a certain range, and it is shown that computing the ROM after the Hopf bifurcation gives the most satisfactory results. This feature is analysed in terms of phase space representations, and the two proposed adjustments are shown to improve the validity range of the ROMs.

Disciplines :

Mechanical engineering

Author, co-author :

de Figueiredo Stabile, André ; Institute of Mechanical Sciences and Industrial Applications (IMSIA), ENSTA - CNRS - EDF, Institut Polytechnique de Paris, Palaiseau, France

Vizzaccaro, Alessandra ; College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, United Kingdom

Salles, Loïc ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M)

Colombo, Alessio ; Department of Civil and Environmental Engineering, Politecnico di Milano, Milan, Italy

Frangi, Attilio ; Department of Civil and Environmental Engineering, Politecnico di Milano, Milan, Italy

Touzé, Cyril ; Institute of Mechanical Sciences and Industrial Applications (IMSIA), ENSTA - CNRS - EDF, Institut Polytechnique de Paris, Palaiseau, France

Language :

English

Title :

Reduced-order modelling of parameter-dependent systems with invariant manifolds: Application to Hopf bifurcations in follower force problems

Publication date :

October 2025

Journal title :

International Journal of Non-Linear Mechanics

ISSN :

0020-7462

eISSN :

1878-5638

Publisher :

Elsevier

Volume :

177

Pages :

105133

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0020746225001210?httpAccept=text/xml

Funders :

EC - European Commission

Funding text :

Attilio Frangi acknowledges the PRIN 2022 Project \u201CDIMIN-DIgital twins of nonlinear MIcrostructures with iNnovative model-order-reduction strategies\u201D (No. 2022XATLT2) funded by the European Union - NextGenerationEU, and Cyril Touz\u00E9 acknowledges the Agence Innovation D\u00E9fense (AID) who contributed to support this work through the funding attributed to the COFLAP project (registered under the number 2023 65 0089).

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