High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to m...

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Vydáno v:	Nonlinear dynamics Ročník 110; číslo 1; s. 525 - 571
Hlavní autoři:	Vizzaccaro, Alessandra, Opreni, Andrea, Salles, Loïc, Frangi, Attilio, Touzé, Cyril
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Dordrecht Springer Netherlands 01.09.2022 Springer Nature B.V Springer Verlag Springer Science and Business Media B.V
Témata:	Aerospace Engineering Amplitudes Applied Mathematics Automation Automotive Engineering Canonical forms Cantilever beams Clamping Classical Mechanics Control Control and Systems Engineering Damping Dynamical Systems Eigenvectors Electrical and Electronic Engineering Engineering Engineering Sciences Engineering, computing & technology Finite element method Folding Foldings Geometric non-linearity Geometric nonlinearities High-order Higher-order Ingénierie mécanique Ingénierie, informatique & technologie Invariant manifolds Invariants Large amplitude Manifold folding Manifolds Mathematical analysis Mathematics Mechanical Engineering Mechanics Methods Microelectromechanical systems Model order reduction Model reduction Nonlinear Sciences Normal form Ocean Engineering Original Paper Parameterization Parametrizations Partial differential equations Structural mechanics Vibration Vibrations Finite element method Manifold folding Geometric nonlinearities Model order reduction Normal form finite element method geometric nonlinearities normal form manifold folding model order reduction
ISSN:	0924-090X, 1573-269X, 1573-269X
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Shrnutí:	This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to mechanical problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion, under the assumption that the damping matrix is diagonalised by the conservative linear eigenvectors. The same development is performed on the reduced-order dynamics which is computed at generic order following different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clamped-clamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that invariant manifold of the first mode shows a folding point at large amplitudes. This exemplifies the failure of the graph style due to the folding point on a real structure, whereas the normal form style is able to pass over the folding. Finally, a MEMS (Micro Electro Mechanical System) micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 scopus-id:2-s2.0-85134336518
ISSN:	0924-090X 1573-269X 1573-269X
DOI:	10.1007/s11071-022-07651-9