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Research Papers

Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis

[+] Author and Article Information
Mikhail Guskov1

Processes and Engineering inMechanics and Materials Laboratory, (PIMM, CNRS UMR 8006), Arts & Métiers ParisTech, 151 boulevard de l’Hôpital, 75013 Paris, Francemikhail.guskov@paris.ensam.fr

Fabrice Thouverez

Laboratory of Tribology and Dynamics of Systems, (LTDS, CNRS UMR 5513), Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69130 Ecully cedex, Francefabrice.thouverez@ec-lyon.fr

1

Corresponding author.

J. Vib. Acoust 134(3), 031003 (Apr 24, 2012) (11 pages) doi:10.1115/1.4005823 History: Received July 23, 2010; Revised October 01, 2011; Published April 23, 2012; Online April 24, 2012

Quasi-periodic motions and their stability are addressed from the point of view of different harmonic balance-based approaches. Two numerical methods are used: a generalized multidimensional version of harmonic balance and a modification of a classical solution by harmonic balance. The application to the case of a nonlinear response of a Duffing oscillator under a bi-periodic excitation has allowed a comparison of computational costs and stability evaluation results. The solutions issued from both methods are close to one another and time marching tests showing a good agreement with the harmonic balance results confirm these nonlinear responses. Besides the overall adequacy verification, the observation comparisons would underline the fact that while the 2D approach features better performance in resolution cost, the stability computation turns out to be of more interest to be conducted by the modified 1D approach.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Multidimensional time domain concept

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Figure 2

Harmonic indices sets (M  =  2, N  =  5)

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Figure 3

Second order Poincaré section computation schematic

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Figure 4

Harmonic terms frequency distribution for MHBM and AHBM for the example frequency ratio in Eqs. 31,33

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Figure 5

Response of the Duffing oscillator (Eq. 30) to each harmonic component of the excitation taken separately (Euclidean norm of the harmonic solution x̃)

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Figure 6

Response curves for N = 13 by the AHBM and MHBM

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Figure 7

Response of the Duffing oscillator (Eq. 30) to a bi-periodic excitation on the resonance peaks : 2D time and frequency representation

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Figure 8

Response of the Duffing oscillator (Eq. 30) to a bi-periodic excitation: MHBM convergence study

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Figure 9

Stability results for the Duffing oscillator (Eq. 30 under bi-harmonic excitation (— response curve, ∘ stable)

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Figure 10

Time marching (TM) test for several points of the response curve (Fig. 9) and respective FFT

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