确定性周期与随机激励联合作用下非线性系统非平稳响应的统计线性化方法

Non-stationary response of non-linear systems subjected to combined periodic and non-stationary stochastic excitation via the statistical linearization method

提出一种用于求解确定性周期与非平稳随机激励联合作用下,单自由度非线性系统非平稳响应的统计线性化方法。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非线性微分方程。利用统计线性化方法将非平稳随机激励作用下的非线性随机动力方程化为等效线性方程,得到关于线性随机响应二阶矩的李雅普诺夫方程。联立李雅普诺夫方程与谐波激励作用下的确定性微分方程,并利用数值算法对其进行求解。以蒙特卡洛模拟验证了此方法的适用性和精度。

A non-stationary statistical linearization method is proposed for determining the stochastic response of non-linear singledegree-of-freedom systems,subjected to the combined periodic and non-stationary stochastic excitation.Specifically,assuming the system response as a sum of a periodic and of a zero-mean stochastic component,leads two equivalent coupled differential equations,governing the deterministic and the stochastic component,respectively.The derived stochastic sub-equation under non-stationary zero-mean stochastic excitation is cast into an equivalent linear equation,by resorting to the non-stationary statistical linearization method.The related Lyapunov equation governing the second moment of the linear stochastic response,and the deterministic sub-equation governing the periodic response,are solved simultaneously using standard numerical algorithms.Pertinent Monte Carlo simulation demonstrates the applicability and accuracy of the proposed semi-analytical method.