周期与色噪声联合作用下分数阶Duffing振子非平稳响应的无记忆方法

A memory-free method for fractional-order Duffing systems subjected to combined periodic and colored excitation

基于统计线性化提出了一种求解周期与色噪声激励联合作用下分数阶Duffing系统非平稳响应的无记忆方法。将系统响应分解为确定性周期和零均值随机分量之和,则原非线性运动方程可等效地化为一组耦合的、分别以确定性和随机动力响应为未知量的分数阶微分方程。利用无记忆化方法将确定性和随机分数阶微分方程转化为相应的常微分方程。利用统计线性化方法处理随机常微分方程,得到关于随机响应二阶矩的李雅普诺夫方程。利用数值算法联立求解李雅普诺夫微分方程和确定性常微分方程。通过Monte Carlo模拟,验证此方法的适用性和精度。

A statistical linearization-based memory-free method is proposed for determining the non-stationary response of singledegree-of-freedom Duffing systems endowed with fractional elements and subjected to excitation combined with periodic and col⁃ored noise.Specifically,by decomposing the system response as a combination of a periodic and of a zero-mean stochastic compo⁃nent,the original nonlinear motion equation can be equivalently transformed into two coupled fractional-order differential sub-equa⁃tions,governing the deterministic and the stochastic component,respectively.Relying on a memory-free method,these fractionalorder stochastic/deterministic differential equations are transformed into a set of ordinary differential equations without fractional de⁃rivatives.The Lyapunov differential equation governing the second moment of the stochastic response component is obtained by re⁃sorting to the statistical linearization method for the derived stochastic ordinary differential equations.The Lyapunov differential equation and the deterministic ordinary differential equations are solved simultaneously using standard numerical algorithms.Perti⁃nent Monte Carlo simulations demonstrate the applicability and accuracy of the proposed method.