Exact analytic solutions for the damped Duffing nonlinear oscillator

Dimitrios E. Panayotounakos; Efstathios E. Theotokoglou; Michalis P. Markakis

doi:10.1016/j.crme.2006.03.009

Exact analytic solutions for the damped Duffing nonlinear oscillator
[Les solutions analytiques exactes pour le amortissable non linéaire oscillateur de Duffing]

Présenté par : Maurice Roseau

Dimitrios E. Panayotounakos ¹ ; Efstathios E. Theotokoglou ¹ ; Michalis P. Markakis ¹

¹ School of Applied Mathematical and Physical Sciences (SEMFE), National Technical University of Athens, NTUA, 5, Heroes of Polythechniou Avenue, Zographou, 157 73, Athens, Greece

Comptes Rendus. Mécanique, Volume 334 (2006) no. 5, pp. 311-316.

Résumé
Abstract

Nous montrons que l'oscillateur de Duffing amorti peut être réduit à une équation équivalente à la forme normale d'équation d'Abel de seconde espèce. Sur la base d'une méthode développée récemment pour construction des solutions analytiques exactes de ce type d'équations d'Abel, des solutions analytiques exactes sont obtenues pour l'oscillateur de Duffing amorti, satisfaisant aux conditions initiales conformes au problème physique sousjacent. Pour illustrer la généralité de la méthode, une application à l'oscillateur de van der Pol est brièvement discutée.

We prove that the second-order damped nonlinear Duffing oscillator is reduced to an equivalent equation of the normal Abel form of the second kind. Based on a recently developed mathematical methodology for the construction of exact analytic solutions of Abel's equation, exact analytic solutions are obtained for the nonlinear damped Duffing oscillator obeying the initial conditions adapted to the physical problem. To improve the general developed methodology an application concerning the nonlinear Van der Pol free oscillator is briefly discussed.

Reçu le : 2005-09-16
Accepté le : 2006-02-23
Publié le : 2006-05-01

DOI : 10.1016/j.crme.2006.03.009

Keywords: Analytical mechanics, Damped Duffing nonlinear oscillator, Abel's equation
Mots-clés : Mécanique analytique, Non linéaire oscillateur de Duffing, Équation d'Abel

Affiliations des auteurs :

Dimitrios E. Panayotounakos ¹ ; Efstathios E. Theotokoglou ¹ ; Michalis P. Markakis ¹

¹ School of Applied Mathematical and Physical Sciences (SEMFE), National Technical University of Athens, NTUA, 5, Heroes of Polythechniou Avenue, Zographou, 157 73, Athens, Greece

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     author = {Dimitrios E. Panayotounakos and Efstathios E. Theotokoglou and Michalis P. Markakis},
     title = {Exact analytic solutions for the damped {Duffing} nonlinear oscillator},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {311--316},
     publisher = {Elsevier},
     volume = {334},
     number = {5},
     year = {2006},
     doi = {10.1016/j.crme.2006.03.009},
     language = {en},
}

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Dimitrios E. Panayotounakos; Efstathios E. Theotokoglou; Michalis P. Markakis. Exact analytic solutions for the damped Duffing nonlinear oscillator. Comptes Rendus. Mécanique, Volume 334 (2006) no. 5, pp. 311-316. doi : 10.1016/j.crme.2006.03.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.03.009/

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