Journal of the Serbian Society for Computational Mechanics / Vol. 6 / No. 1, 2012 / pp. 148-159 (UDC: 621.373.1:517.93) The fractional distributed order oscillator. a numerical solution J. T. Katsikadelis Institute of Structural Analysis and Aseismic Research, School of Civil Engineering, National Technical University of Athens Abstract The response of one-degree-of freedom systems with fractional distributed-order (FDO) damping is studied. The dynamics of such systems constitutes the problem of the fractional distributed-order oscillator. The investigation is achieved by developing an efficient numerical method for solving FDO differential equations. The problem is treated using two approaches. In the first approach, the system of the two coupled equations governing the response of the FDO oscillator is converted into a single FDO differential equation, while in the second approach the equations are treated as a system of FDO differential equations. Numerical examples are presented for free and forced vibrations of the FDO oscillator and useful conclusions are drawn. The resonance phenomenon is also elucidated. Keywords: fractional distributed-order differential equations; fractional distributed-order oscillator; multi-term fractional differential equations. 1. Introduction The one degree-of freedom systems with distributed fractional order dissipation forces have been introduced recently by Atanackovic and his co-workers (Atanackovic 2002, 2003, Atanackovic et al. 2005). They result from the generalization of the multi-term fractional differential viscoelastic model by considering continuous variation of the order of fractional derivative within a closed interval. This model leads to the following initial value problem for the linear fractional distributed-order oscillator (2) 2 () () () () u t t ut gt (1) 1 1 0 0 () () p p c c pD dp p D udp (2) where , u are the displacement and the dissipation force, respectively, a constant parameter; () p and () p specified functions subjected to certain restrictions following from the second law of thermodynamics; the eigenfrequency of the undamped system; () gt the external forcing function and p c D is the Caputo fractional derivative defined as