In this article, nonlinear forced response of dynamical systems is studied using numerical continuation methods. Several methods are available to calculate nonlinear normal modes. Along with the existing analytical methods, recently, numerical methods, especially the pseudo-arclength continuation method, have attracted many researchers. The pseudo-arclength continuation method is a very powerful method which is capable of handling strongly nonlinear systems. However, as mentioned in recently published article reviews, the computational cost of the method has limited its application. In this research, an updating formula is embedded in the pseudo-arclength continuation algorithm to reduce the computational cost. This modified method is called the efficient path-following method. The assumptions and basis of the efficient path-following method algorithm are same as those presented in other references, but none of them have exploited the updating formula of the efficient path-following method to study the forced response of nonlinear dynamical systems. To investigate the capabilities of the method, forced response of a single-degree-of-freedom Duffing system is computed. It is seen that the efficient path-following method has decreased the computational time significantly up to 70%. The results are in very good conformance with those obtained in other references, which shows the accuracy of this method. To study the ability of the efficient path-following method to handle the multi-degree-of-freedom system, a four-degree-of-freedom nonlinear system is considered, and stable and unstable branches of the solution are computed. It is observed that as the nonlinearity of the system gets stronger, the updating formula becomes more effective.

中文翻译：

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非线性系统强迫振动的有效路径跟踪法

在本文中，使用数值连续方法研究动力系统的非线性强迫响应。有几种方法可用于计算非线性法线模式。与现有的分析方法一起，近年来，数值方法，特别是伪弧长连续法，吸引了许多研究者。伪弧长延续方法是一种非常强大的方法，能够处理强非线性系统。但是，如最近发表的文章评论中所述，该方法的计算成本限制了其应用。在这项研究中，在伪弧长延续算法中嵌入了一个更新公式，以降低计算成本。这种修改的方法称为有效路径跟踪方法。有效路径跟随方法算法的假设和基础与其他参考文献中的假设和基础相同，但是没有一个人利用有效路径跟随方法的更新公式来研究非线性动力系统的强迫响应。为了研究该方法的功能，计算了单自由度Duffing系统的强制响应。可以看出，有效的路径跟踪方法已将计算时间显着减少了多达70％。结果与其他参考文献中的结果非常吻合，表明了该方法的准确性。为了研究有效的路径跟随方法处理多自由度系统的能力，考虑了四自由度非线性系统，以及解的稳定和不稳定分支。可以看出，随着系统的非线性变得越来越强，更新公式变得更加有效。