Nonstationary response statistics of fractional oscillators to evolutionary stochastic excitation

Abstract

While studying fractional oscillators to the excitation as an evolutionary stochastic process, this paper develops a novel pole-residue method to compute closed-form solutions for the nonstationary response statistics, including evolutionary power spectrum, correlation function and mean square value. The proposed method is not only suitable to the oscillators with an arbitrary fractional order, but also applies to oscillators with multiple fractional derivative terms. A necessary task of the proposed method is to obtain a pole-residue form of the oscillator’s transfer function (TF), which can be achieved through two steps: (1) obtaining a discrete impulse response function (IRF) from taking the inverse Fourier transform of the oscillator’s frequency response function (FRF), which is readily derived from its exact TF; (2) invoking the Prony-SS method to decompose this IRF signal for getting the poles and residues of the TF. Once the pole-residue TF is available, the remaining derivation can follow a similar pole-residue method that has been developed for ordinary linear oscillators. In the numerical studies, three fractional oscillators with one and two fractional derivative terms, respectively, are chosen to demonstrate the proposed method; considering the input to be a uniformly modulated random process modeled with either white noise or Kanai–Tajimi earthquake spectrum, the correctness of the proposed method is verified by Monte Carlo simulations.