Dynamical analysis of an SDOF quasi-linear system with jump noises and multitime-delayed feedback forces

IF 3.4 Q1 ENGINEERING, MECHANICAL 国际机械系统动力学学报(英文) Pub Date : 2022-07-13 DOI:10.1002/msd2.12043
Wantao Jia, Mingxia Luo, Mengli Hao, Yong Xu
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引用次数: 1

Abstract

In this paper, stochastic dynamics of a single degree‐of‐freedom quasi‐linear system with multitime delays and Poisson white noises are investigated using an approximate procedure based on the stochastic averaging method. The simplified equations, including the averaged stochastic differential equation and the averaged generalized Fokker–Planck–Kolmogorov equation, are obtained to calculate the probability density functions (PDFs) to explore stationary responses. The expression of the Lyapunov exponent is presented to examine the asymptotic stochastic Lyapunov stability. An illustrative example of a quasi‐linear oscillator with two Poisson white noises controlled by two time‐delayed feedback forces is worked out to demonstrate the validity of the proposed method. The approximate stationary PDFs of stochastic responses and asymptotic stochastic stability are demonstrated numerically and theoretically. The results show that the Gaussian white noise has a stronger influence on the dynamics than the Poisson white noise with a small mean arrival rate. Moreover, the influence of the time delay and noise parameters on stochastic dynamics is investigated. It is found that the PDFs under the Poisson white noise approach those under Gaussian white noise as the mean arrival rate increases. The time delay can induce stochastic P‐bifurcation of the system. It is also found that the increase of time delay and the mean arrival rates of the Poisson white noises will broaden the unstable parameter region. The comparison between numerical and theoretical results shows the effectiveness of the proposed method.

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具有跳变噪声和多时滞反馈力的SDOF拟线性系统动力学分析
本文采用基于随机平均法的近似过程,研究了具有多时滞和泊松白噪声的单自由度拟线性系统的随机动力学问题。得到了简化方程,包括平均随机微分方程和平均广义Fokker-Planck-Kolmogorov方程,用于计算概率密度函数(PDFs)以探索平稳响应。给出了Lyapunov指数的表达式,用以检验渐近随机Lyapunov稳定性。最后以两个时滞反馈力控制两个泊松白噪声的准线性振荡器为例,验证了所提方法的有效性。从数值和理论上证明了随机响应的近似平稳PDFs和渐近随机稳定性。结果表明,高斯白噪声比平均到达率较小的泊松白噪声对动力学的影响更大。此外,还研究了时滞和噪声参数对随机动力学的影响。随着平均到达率的增加,泊松白噪声下的pdf接近高斯白噪声下的pdf。时滞会引起系统的随机p分岔。研究还发现,延时和泊松白噪声平均到达率的增加会使不稳定参数区域变宽。数值结果与理论结果的比较表明了所提方法的有效性。
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