In this Note, we present a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on the Floquet theory and the Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the harmonic-balance method and asymptotic numerical method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. To validate the general methodology, we investigate the dynamical behavior of the forced Duffing oscillator with the developed continuation technique.
Dans cette Note, nous présentons une méthode numérique fréquentielle pour déterminer la stabilité des solutions périodiques d'un système dynamique. La méthode, basée sur la théorie de Floquet et le développement en série de Fourier (méthode de Hill), consiste à extraire les valeurs propres physiques de l'ensemble des valeurs propres numériques du système perturbé étendu dans le domaine fréquentiel. En combinant alors la méthode de l'équilibrage harmonique et la méthode asymptotique numérique avec la précédente méthode de Hill, on obtient un outils de continuation purement fréquentiel où le calcul de la stabilité des solutions suivies est quasiment immédiat. Afin de valider la méthode, nous appliquons la méthode de continuation à un oscillateur de Duffing forcé.
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Mots-clés : Systèmes dynamiques, Stabilité, Méthode de Hill, Méthode de continuation, Méthode de l'équilibrage harmonique
Arnaud Lazarus 1; Olivier Thomas 1
@article{CRMECA_2010__338_9_510_0,
author = {Arnaud Lazarus and Olivier Thomas},
title = {A harmonic-based method for computing the stability of periodic solutions of dynamical systems},
journal = {Comptes Rendus. M\'ecanique},
pages = {510--517},
publisher = {Elsevier},
volume = {338},
number = {9},
year = {2010},
doi = {10.1016/j.crme.2010.07.020},
language = {en},
}
TY - JOUR AU - Arnaud Lazarus AU - Olivier Thomas TI - A harmonic-based method for computing the stability of periodic solutions of dynamical systems JO - Comptes Rendus. Mécanique PY - 2010 SP - 510 EP - 517 VL - 338 IS - 9 PB - Elsevier DO - 10.1016/j.crme.2010.07.020 LA - en ID - CRMECA_2010__338_9_510_0 ER -
Arnaud Lazarus; Olivier Thomas. A harmonic-based method for computing the stability of periodic solutions of dynamical systems. Comptes Rendus. Mécanique, Volume 338 (2010) no. 9, pp. 510-517. doi : 10.1016/j.crme.2010.07.020. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.020/
[1] Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, John Wiley & Sons, Inc., 1995
[2] Nonlinear normal modes. Part II: Toward a practical computation using numerical continuation techniques, Mechanical Systems and Signal Processing, Volume 23 (2009), pp. 195-216
[3] On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Mathematica, Volume 8 (1886) no. 1, pp. 1-36
[4] On the Hill determinant method, Journal of Physics A: Mathematical and General, Volume 19 (1986), pp. 2903-2906
[5] Computing spectra of linear operators using the Floquet–Fourier–Hill method, Journal of Computational Physics, Volume 219 (2006), pp. 296-321
[6] Analysis of stability and bifurcations of limit cycles in Chua's circuit through the harmonic-balance approach, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume 46 (1999), pp. 881-890
[7] Stability and vibration analysis of a complex flexible rotor bearing system, Communications in Nonlinear Science and Numerical Simulation, Volume 13 (2008), pp. 804-821
[8] A. Lazarus, D. Combescure, B. Prabel, A 3D finite element model for the vibration analysis of asymmetric rotating machines, Journal of Sound and Vibration, . | DOI
[9] A high order purely-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, Volume 324 (2009), pp. 243-262
[10] Sur la théorie des équations différentielles, Annales Scientifiques de l'École Normale Supérieure, Volume 8 (1879), pp. 3-132
[11] Ordinary Differential Equations, Society for Industrial and Applied Mathematics, 2002
[12] An Exploration of Chaos, Texts on Computational Mechanics, vol. VII, North-Holland, 1994
[13] Nonlinear Oscillations, Wiley, 1979
[14] Sur les déterminants d'ordre infini, Bulletin de la Société Mathématique de France, Volume 14 (1886), pp. 77-90
[15] Spectral analysis and eigenvalues computation of the harmonic state operators in continuous-time periodic systems, Systems & Control Letters, Volume 53 (2004), pp. 141-155
[16] On the convergence of Hill's method, Mathematics of Computation, Volume 79 (2010), pp. 169-187
[17] The Behavior of Nonlinear Vibrating Systems, vol. 1: Fundamental Concepts and Methods: Applications to Single-Degree-of-Freedom Systems, Kluwer Academic Publishers, 1990
[18] Frequency lock-in is caused by coupled-mode flutter, Journal of Fluids and Structures, Volume 22 (2006), pp. 783-791
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