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首页> 外文期刊>International journal of non-linear mechanics >Dynamics of a Duffing oscillator with the stiffness modeled as a stochastic process
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Dynamics of a Duffing oscillator with the stiffness modeled as a stochastic process

机译：刚度建模为随机过程的Duffing振荡器的动力学

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The Duffing equation has been extensively studied in the last decades, especially in systems of vibrating micro-beams subjected to electro-mechanical fluctuating fields. However, mechanical systems are often subjected to uncertainties coming from the excitation and/or the design parameters. These uncertainties can then be considered as stationary or non-stationary stochastic processes. In many engineering applications, we consider stationary stochastic processes as they are easier to model and simplify our problems. However, some of them do not present stationarity properties and it is desirable to know what are the consequences of leaving the non-stationarity aside and consider a stationary case. For this purpose, two stochastic processes are applied to a Duffing oscillator and the results are analyzed when its stiffness modeling is based on (i) a stationary stochastic process generated with Langevin's equation; and when it is based on (ii) a non-stationary stochastic process known as Brownian Bridge. A methodology is proposed to modify these stochastic processes in order to establish a lower limit for their support. In fact, this methodology can be applied to stationary and non-stationary stochastic processes. The stochastic results showed that the case with uncertainties in the stiffness coefficient of the non-linear term presented the highest variation on system's response. In addition, the nonstationary case presented a result much closer to the deterministic one for the set of parameters chosen. The smooth variation of the second moment with time might explain it. There are cases in which the assumption of a stationary process might be not appropriate and non-stationarity must be assessed.
机译：在过去的几十年中，对Duffing方程进行了广泛的研究，尤其是在振动微梁系统中受到机电波动场的影响。但是，机械系统经常受到来自激励和/或设计参数的不确定性的影响。这些不确定性可以被认为是平稳或非平稳的随机过程。在许多工程应用中，我们考虑静态随机过程，因为它们更易于建模和简化问题。但是，它们中的一些不具有平稳性，因此希望知道将非平稳性放在一边会带来什么后果并考虑平稳的情况。为此，将两个随机过程应用于Duffing振荡器，并在其刚度建模基于以下条件时分析结果：（i）用Langevin方程生成的平稳随机过程；当它基于（ii）称为布朗桥的非平稳随机过程时。提出了一种方法来修改这些随机过程，以便为它们的支持建立下限。实际上，该方法可以应用于固定和非固定随机过程。随机结果表明，在非线性项的刚度系数不确定的情况下，系统响应变化最大。另外，非平稳情况给出的结果与所选参数集的确定性结果更为接近。第二时刻随时间的平滑变化可以解释这一点。在某些情况下，假设平稳过程可能不合适，因此必须评估非平稳性。

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