A spectral characterization of nonlinear normal modes
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Peer-reviewed
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Abstract
This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity.
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Keywords
Nonlinear normal modes, Koopman operator, Spectral, Characterization, Invariant manifolds, Parametrization
Journal Title
Journal of Sound and Vibration
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Journal ISSN
0022-460X
1095-8568
1095-8568
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Publisher
Elsevier BV
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Sponsorship
Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office