几类滞迟系统在泊松白噪声及其滤波过程激励下的随机响应

In the last twenty years, there're many works carried out for dynamical problem of strong nonlinear systems subjected to Poisson white noise, at home and abroad. But little work has been done on research of random response prediction of hysteretic systems under excitations of Poisson white noise and its filtered processes. As new intelligent materials and energy dissipating elements with more complex properties of hysteresis have been brought into use, it is necessary to study the problem of random response prediction of these complex hysteretic systems under excitations of Poisson white noise and its filtered processes in order to insure that theoretical research can meet the engineering demands. In the present project, statistics of stationary response and transient response of Duhem hysteretic system and Preisach hysteretic system under excitations of Poisson white noise and linear/nonlinear filtered Poisson processes are studied to find out the effects of parameters of filtering systems and those of hysteretic systems on statistics of random response. The procedures of research are as follows. Firstly, the equivalent nonlinear system without hysteresis is generated from the simultaneous equations of filtering system and hysteretic system and the corresponding averaged generalized Fokker-Planck-Kolmogorov equation can be derived by using the stochastic averaging method. Then, an approximate solution is presented by using the exponential polynomial closure method to solve the averaged equation. Finally, by substituting the approximate solution into the Kolmogorov-Feller integro-differential equation to calculate the actual value of residual and modifying the approximate solution, a new approximate solution of statistics of response which meets the demand of error is obtained.

近二十年，国内外对强非线性系统在泊松白噪声激励下的动力学问题开展了大量工作，但针对滞迟系统在泊松白噪声及其滤波过程激励下随机响应预测的研究还很少，而具有更复杂滞迟特性的新型智能材料和耗能构件已经开始应用，为使理论研究紧跟工程需求，有必要开展泊松白噪声及其滤波过程激励下复杂滞迟系统随机响应问题的研究。本项目以Duhem和Preisach滞迟系统为研究对象，分析其在泊松白噪声和线性/非线性滤过的泊松过程激励下的稳态和瞬态响应统计量，以便确定滤波参数、滞迟系统参数对响应统计量的影响。研究方法如下：从滞迟系统和滤波系统的联立方程出发推导非滞迟的等价非线性系统，以随机平均法建立平均广义Fokker-Planck-Kolmogorov方程，用指数多项式截断法求此平均方程的近似解，再代入Kolmogorov-Feller积分微分方程计算实际误差余项并加以修正，最后获得满足误差要求的响应统计量近似解。

项目“几类滞迟系统在泊松白噪声及其滤波过程激励下的随机响应”历经三年的主要工作，围绕几类滞迟系统在非高斯随机激励作用下的响应分析问题，在下述三方面主要研究内容上，获得了一系列具有科学意义的研究成果。.一，非高斯随机激励作用下滞迟系统等效非线性化方法研究。滞迟系统势能函数会受系统状态演化的影响，一般无法直接用随机平均法等解析方法进行分析。项目团队通过推广高斯随机激励问题的等效非线性化方法，实现诸如泊松白噪声随机激励作用下Coleman-Hodgdon铁磁体滞迟模型等非高斯随机激励问题的等效非线性化分析，通过蒙特卡洛模拟验证了等效非线性系统的预测精度。该成果对应于本项目拟解决的第一项关键科学问题，可以为现有针对高斯随机激励问题的解析方法打通到非高斯随机激励问题的通道。.二、基于泊松白噪声输入生成有界噪声的非线性滤波系统研究。有界噪声当前一般由高斯白噪声作为初始激励滤波生成，无法模拟强脉冲等激励特性。为此，项目团队利用泊松白噪声来生成有界噪声，通过数值仿真分析了一系列激励参数变化对自复位摇摆结构等含滞迟的非线性系统的响应和动力可靠性的影响。该成果对应于本项目拟解决的第二项关键科学问题，为分析非高斯随机激励特性对非线性系统响应和可靠性的影响提供了技术支持。.三、基于随机平均法的滞迟系统随机响应分析。在处理泊松白噪声激励作用下常见非线性系统随机响应分析方面，随机平均法已获得了成功推广。而为贴近工程动力模型，项目团队推广了随机平均法的使用范围，对诸如Yar-Hammond双线性滞迟系统、Duhem滞迟系统、含双线性滞迟项或Bouc-Wen滞迟项的碰撞振动系统等问题，进行随机响应分析，建立一套基于随机平均方法的求解流程。该成果对应于本项目拟解决的第三项关键科学问题，将随机平均法的应用推广到大多数工程应用中必然面对的含滞迟项的非线性系统响应分析中。