Abstract To improve the analysis accuracy of dynamic reliability of a nonlinear system, an equivalent nonlinear system method is presented. In this method, general nonlinear systems are converted to equivalent Duffing nonlinear systems according to the minimum mean square error criterion, whose exact analytical solution of the random steady-state responses can be determined by a FokkerPlanckKolmogorov equation. Then the exact results of stochastic responses are used to analyze structural dynamic reliability. To use the equivalent nonlinear system method to analyze structural dynamic reliability is not only convenient for calculation but highly accurate. In addition, the presented equivalent nonlinear system has a parameter e, which controls the degree of nonlinearity. Thus, it is easy to obtain the corresponding analysis results from converting the original system to equivalent nonlinear systems with different degrees of nonlinearity by changing the value of e. Especially, when e is zero, the equivalent nonlinear system will degenerate to the linear system, and leading to the analysis results of the equivalent linearization method. An example shows that the analysis results of the proposed equivalent nonlinear system method are reliable, and the calculation appears to be more accurate than that of the equivalent linearization method.

中文翻译：

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基于Duffing系统的等效非线性系统法非线性结构动力可靠性分析

摘要 为提高非线性系统动态可靠性分析的准确性，提出了一种等效非线性系统方法。该方法根据最小均方误差准则将一般非线性系统转化为等效的Duffing非线性系统，其随机稳态响应的精确解析解可由FokkerPlanckKolmogorov方程确定。然后将随机响应的精确结果用于分析结构动力可靠性。用等效非线性系统法分析结构动力可靠度，不仅计算方便，而且精度高。此外，所提出的等效非线性系统有一个参数 e，它控制非线性程度。因此，通过改变e的值，将原系统转换为不同非线性程度的等效非线性系统，很容易得到相应的分析结果。特别是当e为零时，等效非线性系统会退化为线性系统，导致等效线性化方法的分析结果。一个算例表明，所提出的等效非线性系统方法的分析结果是可靠的，并且计算似乎比等效线性化方法的计算更准确。