Response spectral density determination for nonlinear systems endowed with fractional derivatives and subject to colored noise

Abstract

A computationally efficient method for determining the response of non-linear stochastic dynamic systems endowed with fractional derivative elements subject to stochastic excitation is presented. The method relies on a spectral representation both for the system excitation and its response. Specifically, first the ordinary non-linear differential equation of motion is transferred into a set of non-linear algebra equations by employing the harmonic balance method. Next, the response Fourier coefficients are determined by solving these non-linear equations. Finally, repeated use of the proposed procedure yields the response power spectral density. Pertinent numerical examples, including a fractional Duffing and a bilinear oscillator, demonstrate the accuracy of the proposed method.