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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Veröffentlicht in:Nonlinear dynamics Jg. 110; H. 1; S. 525 - 571
Hauptverfasser: Vizzaccaro, Alessandra, Opreni, Andrea, Salles, Loïc, Frangi, Attilio, Touzé, Cyril
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Dordrecht Springer Netherlands 01.09.2022
Springer Nature B.V
Springer Verlag
Springer Science and Business Media B.V
Schlagworte:
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
Online-Zugang:Volltext
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Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
scopus-id:2-s2.0-85134336518
ISSN:0924-090X
1573-269X
1573-269X
DOI:10.1007/s11071-022-07651-9