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Article Dans Une Revue Nonlinear Dynamics Année : 2003

A Wavelet-Balance Method to Investigate the Vibrations of Nonlinear Dynamical Systems

Résumé

The scope of this paper is to introduce a new wavelet-balanceprocedure allowing to give a genuine time-scale representation ofvibrations of nonlinear dynamical systems by adopting a waveletmultiresolution approach. In a former paper, a wavelet-Galerkinoriented procedure was developed to analyze vibrations of lineartime-periodic systems. The topic is here to extend the process tothe nonlinear case using a perturbation technique. The underlyingidea consists in successively balancing the linearized equationsof motion into wavelet spaces with increasing resolution scales.Here we demonstrate the wavelet-balance procedure may accuratelyexhibit both transient and stationary vibrations of any nonlinearproblem in general, whatever smooth nonlinearity shape or externalforcing may be. In addition, wavelets inherit of fairly goodtime-frequency localization properties that are likely to permitthe investigation of strong nonlinear problems. Numericalexperiments achieved on a well known Duffing oscillator involvinga cubic nonlinearity then illustrate the procedure. Simulationsattest the relevance of the method by comparison with eitherpurely numerical results obtained with a Runge-Kutta integrationscheme or with an analytical study based on the multiple scalesmethod. We demonstrate that this semi-analytical semi-numericalperturbation method permits to capture stable limit cycles of theDuffing oscillator and its related amplitude spectrum response orstill responses to pulse-like excitations. Finally, key propertiesof the method are discussed and future prospective works areoutlined.

Dates et versions

hal-00814573 , version 1 (17-04-2013)

Identifiants

Citer

Stéphane Pernot, Claude-Henri Lamarque. A Wavelet-Balance Method to Investigate the Vibrations of Nonlinear Dynamical Systems. Nonlinear Dynamics, 2003, 32 (1), pp.33-70. ⟨10.1023/A:1024263917587⟩. ⟨hal-00814573⟩

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