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 - 2022
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
Finite element method; Geometric nonlinearities; Manifold folding; Model order reduction; Normal form; Foldings; Geometric non-linearity; High-order; Higher-order; Invariant manifolds; Large amplitude; Parametrizations; Control and Systems Engineering; Aerospace Engineering; Ocean Engineering; Mechanical Engineering; Electrical and Electronic Engineering; Applied Mathematics
Abstract :
[en] 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.
Disciplines :
Mechanical engineering
Author, co-author :
Vizzaccaro, Alessandra ; Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom ; Department of Mechanical Engineering, Imperial College London, London, United Kingdom
Opreni, Andrea; Department of Civil and Environmental Engineering, Politecnico di Milano, Milano, Italy
Salles, Loïc ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M) ; Department of Mechanical Engineering, Imperial College London, London, United Kingdom
Frangi, Attilio; Department of Civil and Environmental Engineering, Politecnico di Milano, Milano, Italy
Touzé, Cyril; Institute of Mechanical Sciences and Industrial Applications (IMSIA), ENSTA Paris - CNRS - EDF - CEA, Institut Polytechnique de Paris, Palaiseau cedex, France
Language :
English
Title :
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
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