A spectral characterization of nonlinear normal modes. (1st September 2016)

• Record Type: Journal Article
• Authors: Cirillo, G.I.; Mauroy, A.; Renson, L.; Kerschen, G.; Sepulchre, R.
• Is Part Of: Journal of sound and vibration. Volume 377(2016)
• Page Start: 284
• Page End: 301
• Publication Date: 2016-09-01
• Subjects: Nonlinear normal modes -- Koopman operator -- Spectral -- Characterization -- Invariant manifolds -- Parametrization; Sound -- Periodicals; Vibration -- Periodicals; Son -- Périodiques; Vibration -- Périodiques; Sound; Vibration; Periodicals; Electronic journals; 620.205
• DOI: 10.1016/j.jsv.2016.05.016
• Languages: English
• ISSNs: 0022-460X

Abstract

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.