Transient response analysis of a system with nonlinear stiffness and nonlinear damping excited by Gaussian white noise based on complex fractional moments
Transient response analysis of a system with nonlinear stiffness and nonlinear damping excited by Gaussian white noise based on complex fractional moments
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DOI：
10.1007/s00707-022-03264-w
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发表时间：
2022-06
期刊：
影响因子：
2.7
通讯作者：
Daizoh Itoh;T. Tsuchida
中科院分区：
文献类型：
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作者：
Daizoh Itoh;T. Tsuchida
提出了一种基于复分数阶矩的分析方法，用于求解高斯白噪声激励下具有刚度和阻尼非线性的动力系统的暂态响应概率密度。CFM是近年来发展起来的一种新的统计矩，它与概率密度函数的Mellin变换有关。在本方法中，首先应用等效线性化技术确定系统的等效固有频率，它是响应幅值的函数。然后，利用等效固有频率和随机平均法，导出了关于响应幅值的随机微分方程和相应的Fokker-Planck方程。Fokker-Planck方程的Mellin变换得到了振幅CFM的控制方程。控制方程由联立的线性常微分方程组给出。最后，对由上述方程得到的CFM进行逆Mellin变换，得到响应概率密度函数。在数值算例中，考虑了三类非线性随机系统。结果表明，该方法得到的瞬时响应概率密度与相应的蒙特卡罗模拟结果(包括尾区)吻合较好。
An analytical method based on complex fractional moments (CFMs) is presented to obtain the transient response probability density of a dynamic system with nonlinearity in both stiffness and damping excited by Gaussian white noise. The CFM is a new kind of statistical moment developed in recent years and is related to the Mellin transform of a probability density function. In the present method, first, the equivalent linearization technique is applied to determine the equivalent natural frequency of the system, which is a function of the response amplitude. Then, using the equivalent natural frequency and the stochastic averaging procedure, the stochastic differential equation with respect to the response amplitude and the associated Fokker–Planck equation are derived. The Mellin transform of the Fokker–Planck equation yields the governing equations of the amplitude CFMs. The governing equations are given by simultaneous linear ordinary differential equations. Finally, the inverse Mellin transform of the CFMs obtained from the above equations leads to the response probability density function. In numerical examples, three types of nonlinear stochastic systems are considered. It is shown that the results of the transient response probability densities obtained by the present method agree well with the corresponding Monte Carlo simulation results, including their tail region.