Stochastic analysis of fractional oscillators by equivalent system definition

Fractional oscillators have been recently proposed as damping devices under the configuration of Fractional Tuned Mass Dampers (FTMD), realized by connecting an oscillating mass to the primary structure through a viscoelastic link with inherent fractional constitutive law. The characteristic tuning frequency for the FTMD has been identified with the Damped Fractional Frequency (DFF), defined as the frequency at which the squared absolute value of the transfer function of the device attains its relative maximum. The definition of the DFF constitutes an interesting step towards the analysis of fractional oscillators in the frequency domain. In this paper, a simplified frequency domain approach is presented for the design of fractional oscillators subjected to stationary white noise. The analysis of the fractional oscillator is performed by using an equivalent single degree of freedom system with linear viscous damping. The aim is to obtain a clear understanding of the physical dynamic effects of the variations in the fractional oscillator parameters, in terms of damping and natural frequencies. Moreover, the use of an equivalent system allows for the straightforward applications of stochastic analysis to determine an approximate closed-form expression of the response variance.