Vol. 4, No. 9, 2009

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ISSN: 1559-3959
Forced vibrations of a nonlinear oscillator with weak fractional damping

Yuriy A. Rossikhin, Marina V. Shitikova and Tatiana Shcheglova

Vol. 4 (2009), No. 9, 1619–1636
Abstract

This article deals with force-driven vibrations of nonlinear mechanical oscillators whose constitutive equations involve fractional derivatives, defined as fractional powers of the conventional time-derivative operator. This definition of fractional derivatives enables one to analyze approximately the vibratory regimes of the oscillator. The assumption of small fractional derivative terms allows one to use the method of multiple time scales, whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and disturbing force terms can be carried out. The relationship between the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when the orders of the small fractional derivative term and of the cubic nonlinearity appearing in the oscillator’s constitutive equation coincide.

Keywords
fractionally damped oscillator, nonlinear fractional oscillator, method of multiple time scales
Milestones
Received: 26 September 2008
Revised: 4 August 2009
Accepted: 18 August 2009
Published: 17 January 2010
Authors
Yuriy A. Rossikhin
Department of Structural Mechanics
Voronezh State University of Architecture and Civil Engineering
84, 20-letija Oktjabrja Street
Voronezh 394006
Russia
Marina V. Shitikova
Department of Structural Mechanics
Voronezh State University of Architecture and Civil Engineering
84, 20-letija Oktjabrja Street
Voronezh 394006
Russia
Tatiana Shcheglova
Department of Structural Mechanics
Voronezh State University of Architecture and Civil Engineering
84, 20-letija Oktjabrja Street
Voronezh 394006
Russia