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首页> 外文期刊>Journal of Applied Mechanics: Transactions of the ASME >Marginal Instability and Intermittency in Stochastic Systems - Part II: Systems With Rapid Random Variations in Parameters
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Marginal Instability and Intermittency in Stochastic Systems - Part II: Systems With Rapid Random Variations in Parameters

机译：随机系统中的边际不稳定性和间歇性-第二部分：参数具有快速随机变化的系统

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Dynamic systems with lumped parameters, which experience random temporal variations, are considered. The variations "smear" the boundary between the system's states, which are dynamically stable and unstable in the classical sense. The system's response within such a "twilight zone" of marginal instability is found to be of an intermittent nature, with alternating periods of zero (or almost-zero) response and rare short outbreaks. As long as it may be impractical to preclude completely such outbreaks for a designed system, subject to highly uncertain dynamic loads, the corresponding system's response should be analyzed. Results of such analyses are presented for cases of slow and rapid (broadband) parameter variations in Papers I and II, respectively. The former case has been studied in Paper I (2008, "Marginal Instability and Intermittency in Stochastic Systems-Part I: Systems With Slow Random Variations of Parameters," ASME J. Appl. Mech., 75(4), pp. 041002) for a linear model of the system using a parabolic approximation for the variations in the vicinity of their peaks (so-called Slepian model) together with Krylov-Bogoliubov averaging for the transient response. This resulted in a solution for the probability density function (PDF) of the response, which was of an intermittent nature indeed due to the specific algorithm of its generation. In the present paper (Paper II), rapid broadband parameter variations are considered, which can be described by the theory of Markov processes. The system is assumed to operate beyond its stochastic instability threshold-although only slightly-and its nonlinear model is used accordingly. The analysis is based on the solution of the Fokker-Planck-Kolmogorov partial differential equation for the relevant stationary PDF of the response. Several such PDFs are analyzed; they are found to have integrable singularities at the origin, indicating an intermittent nature of the response. Asymptotic analysis is performed for the first-passage problem for such response processes with highly singular PDFs, resulting in explicit formulas for an expected time interval between outbreaks in the intermittent response.
机译：考虑具有集总参数的动态系统，这些系统会经历随机的时间变化。变化“涂抹”了系统状态之间的边界，在经典意义上，这些状态是动态稳定和不稳定的。发现系统在这种边缘不稳定的“暮光区”内的响应具有间歇性，响应周期为零（或几乎为零），并且出现了短暂的爆发。只要在高度不确定的动态负载下完全阻止设计的系统爆发此类暴动是不切实际的，则应分析相应系统的响应。这些分析的结果分别在论文I和II中介绍了慢速和快速（宽带）参数变化的情况。论文I（2008年，“随机系统中的边际不稳定性和间歇性-第I部分：参数缓慢随机变化的系统”，ASME J. Appl。Mech。，75（4），第041002页）已对前一种情况进行了研究。对于系统的线性模型，使用抛物线逼近其峰附近的变化（所谓的Slepian模型），并结合Krylov-Bogoliubov求平均以得到瞬态响应。这导致了响应概率密度函数（PDF）的解决方案，由于其生成的特定算法，该解决方案实际上具有间歇性。在本文（论文二）中，考虑了宽带参数的快速变化，这可以用马尔可夫过程的理论来描述。假定该系统在随机不稳定阈值之上运行（尽管只是很小），并据此使用其非线性模型。该分析基于Fokker-Planck-Kolmogorov偏微分方程对相应平稳PDF响应的解。分析了几种这样的PDF。发现它们在原点具有可积分的奇异性，表明响应是间歇性的。对于具有高度奇异PDF的此类响应过程，对第一遍问题执行渐近分析，从而得出明确的公式，用于确定间歇响应爆发之间的预期时间间隔。

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