New approach for the analysis of damped vibrations of fractional oscillators

Article type: Research Article

Authors: Rossikhin, Yuriy A. | Shitikova, Marina V.

Affiliations: Voronezh State University of Architecture and Civil Engineering, Voronezh 394006, Russia

Note: [] Corresponding author. E-mail: [email protected]

Abstract: The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. 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 nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

Keywords: Fractional oscillator, linear and nonlinear vibrations, method of multiple time scales

DOI: 10.3233/SAV-2009-0475

Journal: Shock and Vibration, vol. 16, no. 4, pp. 365-387, 2009