Transient Response Analysis of Nonlinear Oscillators With Fractional Derivative Elements Under Gaussian White Noise Using Complex Fractional Moments

Complex fractional moment (CFM), which is defined as the Mellin transform of a probability density function (PDF), has been successfully employed to find the response PDF of a wide variety of integer-order nonlinear oscillators. In this paper, a CFM-based analysis is performed to determine the transient response PDF of nonlinear oscillators with fractional derivative elements under Gaussian white noise. First, an equivalent linear system is introduced for the purpose of deriving the Fokker-Planck (FP) equation for response amplitude. The equivalent natural frequency and equivalent damping coefficient of the system need to be determined, taking into account both the nonlinear and fractional derivative elements of the original oscillator. Moreover, to convert the FP equation into the governing equation of CFMs, these equivalent coefficients must be given in polynomial form of amplitude. This paper proposes formulas for appropriately determining the equivalent coefficients, based on an equivalent linearization technique. Then, applying stochastic averaging, the FP equation is derived from the equivalent linear system. Next, the Mellin transform converts the FP equation into coupled linear ordinary differential equations for amplitude CFMs, which are solved with a constraint corresponding to the normalization condition for a PDF. Finally, the inverse Mellin transform of the CFMs yields the amplitude PDF. The joint PDF of displacement and velocity is also obtained from the amplitude PDF. Three linear and nonlinear fractional oscillators are considered in numerical examples. For all cases, the analytical results are in good agreement with the pertinent Monte Carlo simulation results.