联合激励下分数阶非线性系统非平稳响应的半解析方法  

A semi⁃analytical method for non⁃stationary response determination of nonlinear systems subjected to combined excitation

作　　者：孔凡[1,2] 廖海君 韩仁杰 张义 洪旭 KONG Fan;LIAO Hai‑jun;HAN Ren‑jie;ZHANG Yi;HONG Xu(School of Civil Engineering&Architecture,Wuhan University of Technology,Wuhan 430070,China;College of Civil Engineering,Hefei University of Technology,Hefei 230009,China;College of Civil Engineering,Tongji University,Shanghai 200092,China;China Construction Third Bureau First Engineering Co.,Ltd.,Wuhan 430040,China)

机构地区：[1]武汉理工大学土木工程与建筑学院,湖北武汉430070 [2]合肥工业大学土木与水利工程学院,安徽合肥230009 [3]同济大学土木工程学院,上海200092 [4]中建三局第一建设工程有限责任公司,湖北武汉430040

出　　处：《振动工程学报》2024年第8期1339-1348,共10页Journal of Vibration Engineering

基　　金：国家自然科学基金面上项目(52078399);中央高校基本科研业务费专项资金项目(JZ2023HGTA0194)。

摘　　要：确定性和随机激励联合作用下的非线性动力系统具有特殊的动力响应特征。本文提出一种用于计算联合激励下含分数阶阻尼的非线性系统非平稳响应的半解析方法。将系统响应表示为确定性响应和零均值随机响应之和,则原分数阶非线性运动微分方程可等效地化为分数阶确定性微分方程和随机子微分方程的组合。利用时变谐波平衡法处理非线性确定性微分方程,利用统计线性化处理非线性随机子微分方程。对于后者,结合Prony‐SS算法和Laplace变换得到其分数阶等效线性方程的半解析解。联立得到的相关耦合方程,通过数值算法迭代求解响应未知量。蒙特卡罗模拟验证了此方法的适用性和精度。The nonlinear dynamic systems exhibit particular behaviors when subjected to combined deterministic and stochastic excitation.A semi-analytical method for calculating the nonstationary response of a fractional nonlinear oscillator subjected to com‐bined excitation is proposed.Representing the system response as a sum of a deterministic component and zero-mean stochastic component leads to two equivalent sub-equations for the differential equation of motion.The time-varying harmonic balance method is used for the nonstationary solution of the deterministic differential sub-equation,while the statistical linearization method is utilized for obtaining an equivalent linear substitution for the stochastic sub-equation.A semi-analytical solution of the equivalent linear equation is obtained by the Prony-SS and Laplace transform technique.The unknown deterministic/stochastic response components are obtained by solving the derived nonlinear algebraic equations simultaneously.Monte Carlo simulations demonstrate the applicability and accuracy of this method.

关 键 词：统计线性化 时变谐波平衡法 分数阶导数 非线性系统 Prony‐SS算法 

分 类 号：O324[理学—一般力学与力学基础] O322[理学—力学]