A Wavelet-Balance Method to Investigate the Vibrations of Nonlinear Dynamical Systems |
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Authors: | Pernot S. Lamarque C.-H. |
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Affiliation: | (1) Ecole Nationale des Travaux Publics de l'Etat, Laboratoire GéoMatériaux, URA-CNRS 1652, 1 rue Maurice Audin, F-69518 Vaulx-en-Velin Cedex, France |
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Abstract: | 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. |
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Keywords: | wavelet-Galerkin procedure nonlinear vibration Duffing oscillator multiresolution analysis |
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