含分数阶导数项的随机Duffing振子的稳态响应分析

本文对一个含有分数阶导数项阻尼的、Gaussian白噪声激励下的Duffing振子进行了稳态响应分析。首先，基于能量平衡理论，运用等效线性化方法，计算等效系统的线性阻尼及自然频率，建立统计意义下的等效线性化系统。然后，利用平均法建立随机Ito方程，得到随机响应的Markovian近似；给出描述振子振幅概率密度函数演化的Fokker-Planck方程，并得到它的稳态解。进一步，对于含有响应振幅的等效线性系统，借助由Laplace变换得到的转换函数，得到原系统的条件功率谱密度，结合振幅的稳态概率密度作为权重函数，给出原系统功率谱密度的估计，以及响应的统计量的估计。数值模拟的结果说明所提出的功率谱密度的近似解析表达式是可靠的，它甚至适用于Duffing振子具有强非线性回复力的情形，因为它可以较好的表现出功率谱密度共振频谱加宽及多峰现象的出现。

Stationary Response Analysis for a Stochastic Duffing Oscillator Comprising Fractional Derivative Element

A reliable approximate technique is developed for the determination of the power spectral density (PSD) for the stochastic response of a Duffing oscillator comprising fractional derivative elements. The oscillator excitation is Gaussian white noise. For this purpose, stochastic averaging is applied to the system to obtain a first-order stochastic differential equation governing the evolution of the probability density of the oscillator response amplitude. This procedure leads to a Markovian modeling of the response amplitude, and the Fokker-Plank equation associated with this Markovian model is derived. Furthermore, it is shown that its stationary solution can be determined in a closed form. Moreover, the transfer function of a surrogate linear system endowed with the original fractional derivative term is used to estimate the PSD of the response for the nonlinear oscillator. Pertinent Monte Carlo simulations are included demonstrating the reliability of the proposed technique, even for strongly nonlinear oscillators.