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Sten Ponsioen

Bio: Sten Ponsioen is an academic researcher from ETH Zurich. The author has contributed to research in topics: Nonlinear system & Model order reduction. The author has an hindex of 7, co-authored 10 publications receiving 238 citations.

Papers
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Journal ArticleDOI
George Haller1, Sten Ponsioen1
TL;DR: In this paper, a unified approach to nonlinear modal analysis in dissipative oscillatory systems is proposed, which covers both autonomous and time-dependent systems and provides exact mathematical existence, uniqueness and robustness results.
Abstract: We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation; thus, the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.

139 citations

Journal ArticleDOI
George Haller1, Sten Ponsioen1
TL;DR: In this article, the authors derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the softer degrees of freedom, called slow-fast decomposition (SFD).
Abstract: We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the softer degrees of freedom. This slow–fast decomposition (SFD) enslaves exponentially fast the stiffer degrees of freedom to the softer ones as all oscillations converge to the reduced model defined on a slow manifold. We obtain an expression for the domain boundary beyond which the reduced model ceases to be relevant due to a generic loss of stability of the slow manifold. We also find that near equilibria, the SFD gives a mathematical justification for two modal reduction methods used in structural dynamics: static condensation and modal derivatives. These formal reduction procedures, however, are also found to return incorrect results when the SFD conditions do not hold. We illustrate all these results on mechanical examples.

46 citations

Journal ArticleDOI
TL;DR: In this paper, spectral submanifold (SSM) theory is used to extract forced response curves without any numerical simulation in high-degree-of-freedom, periodically forced mechanical systems.

41 citations

Journal ArticleDOI
TL;DR: In this paper, the smoothest nonlinear continuations of modal subspaces of the linearized system are constructed up to arbitrary orders of accuracy, using the parameterization method.

41 citations

Posted Content
George Haller1, Sten Ponsioen1
TL;DR: In this article, a unified approach to nonlinear modal analysis in dissipative oscillatory systems is proposed, which covers both autonomous and time-dependent systems, and provides exact mathematical existence, uniqueness and robustness results.
Abstract: We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems, and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation, thus the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.

40 citations


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01 Jan 2016
TL;DR: The applications of centre manifold theory is universally compatible with any devices to read, and is available in the book collection an online access to it is set as public so you can get it instantly.
Abstract: Thank you very much for downloading applications of centre manifold theory. As you may know, people have look numerous times for their chosen novels like this applications of centre manifold theory, but end up in malicious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they are facing with some harmful virus inside their laptop. applications of centre manifold theory is available in our book collection an online access to it is set as public so you can get it instantly. Our book servers hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the applications of centre manifold theory is universally compatible with any devices to read.

153 citations

Journal ArticleDOI
TL;DR: In this paper, the authors describe the use of a quadratic manifold for the model order reduction of structural dynamics problems featuring geometric nonlinearities, where the manifold is tangent to a subspace spanned by the most relevant vibration modes, and its curvature is provided by modal derivatives obtained by sensitivity analysis.

76 citations

Journal ArticleDOI
TL;DR: This article proposes an efficient experimental strategy to measure the backbone curve of a particular nonlinear mode and uses it to identify the free parameters of the reduced order model and validate the procedure by comparison to available theoretical models as well as to other experimental identification methods.

56 citations

Journal ArticleDOI
TL;DR: In this article, a review of nonlinear methods for model order reduction in structures with geometric nonlinearity is presented, with a special emphasis on the techniques based on invariant manifold theory.
Abstract: This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures discretised with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.

54 citations

Journal ArticleDOI
TL;DR: This work applies two recently formulated mathematical techniques, Slow-Fast Decomposition (SFD) and Spectral Submanifold (SSM) reduction, to a von Karman beam with geometric nonlinearities and viscoelastic damping to results in a drastic reduction of the finite-element beam model to a one-degree-of freedom nonlinear oscillator.

53 citations