非对称二次随机系统的稳态响应

Steady-state Response of Asymmetric Quadratic Stochastic System

基于雅可比椭圆函数傅里叶级数展开,提出一个新的雅可比椭圆函数的随机平均法,研究高斯色噪声激励下非对称二次随机系统响应的稳态概率密度.首先,引入雅可比椭圆函数变换,并基于傅里叶级数展开的随机平均法得到系统响应幅值的平均伊藤随机微分方程;其次,建立与之相对应的福克-普朗克-柯尔莫哥洛夫(FPK)方程,进而求出该FPK方程的稳态概率密度函数;最后,通过一个典型的例子,结合蒙特卡洛数值模拟验证了方法的有效性,并详细讨论了噪声强度、相关时间和线性刚度对系统稳态响应的影响.

Based on the Fourier series expansion of Jacobi elliptic function,a new stochastic average method of Jacobi elliptic function is proposed to study the stationary probability density of the response of an asymmetric quadratic stochastic system excited by Gaussian color noise.Firstly,the Jacobi elliptic function transformation is introduced,and the average Ito stochastic differential equation of the system response amplitude is obtained based on the random average method of Fourier series expansion.Then,the corresponding Fokker-Planck-Kolmogorov(FPK)equation is established,and the stationary probability density function of the FPK equation is derived.Finally,the effectiveness of the method is verified by Monte Carlo numerical simulation through a typical example,and the influence of noise intensity,correlation time and linear stiffness on the steady-state response of the system is discussed in detail.