改进的基于雅可比椭圆函数的随机平均法

Extended Elliptic Function-Based Stochastic Averaging Technique

改进了基于雅可比椭圆函数的随机平均法,用于预测高斯白噪声激励下硬弹簧及软弹簧系统的随机响应.引入包含雅可比椭圆正弦函数、余弦函数及delta函数的雅可比椭圆函数变换,导出关于响应幅值和相位的随机微分方程.应用随机平均原理,将响应幅值近似为Markov扩散过程,建立其平均的It随机微分方程.响应幅值的稳态概率密度由相应的简化Fokker-Planck-Kolmogorov方程解出,进而得到系统位移和速度的稳态概率密度.以Duffing-Van der Pol振子为例,研究了硬弹簧及软弹簧情形下的随机响应,通过与Monte-Carlo数值模拟结果比较证实了本文方法的可行性及精度.

The stochastic averaging technique based on Jacobi elliptic functions was generalized to evaluate the random responses of nonlinear system with hardening and softening stiffness to Gaussian white noises. By introducing the Jacobi elliptic functions transformation including Jacobi elliptic sine, cosine and delta functions, the stochastic differential equations with respect to the system amplitude and phase were derived. With the application of the stochastic averaging principle, the amplitude response was approximated as a Markov diffusion process and the associated averaged It6 stochastic differential equation was obtained. Solving the corresponding reduced Fokker-Planck-Kolmogorov equation yields the stationary probability density of the amplitude response, from which the stationary probability densities of the displacement and velocity were derived. Numerical results for a Duffing-Van der Pol oscillator with hardening and softening stiffness were given to verify the applicability and accuracy of the proposed procedure by comparing with the results from Monte-Carlo simulations.