Nonstationary stochastic response determination of nonlinear oscillators endowed with fractional derivatives

Abstract

Fractional calculus has been broadly used in diverse engineering applications. In this regard, vibrations of fractional oscillators subject to stochastic loads have attracted considerable attention. This paper proposes a semi-analytical approach to determine the nonstationary response statistics of nonlinear oscillators endowed with fractional derivatives. Specifically, the fractional derivative term is represented by introducing a discretization scheme and associated first-order differential equations. This leads to an augmented dimension dynamic system. Further, in conjunction with the statistical linearization technique, the evolution of the system response is captured by a set of coupled ordinary differential equations with time-dependent coefficients. Furthermore, solving the associated Lyapunov equation for the randomly excited dynamic system yields the nonstationary statistics of the oscillator response. The reliability of the proposed method is demonstrated by Monte Carlo simulations pertaining to classical nonlinear oscillators.