In the present work, a truss model inspired by von Mises and Bergan trusses is considered. The resulting structure is a phenomenological model that represents the behavior of many engineering structures and may exhibit in-plane snap-through or pitchfork unstable bifurcation and/or lateral buckling, leading two a multiwell potential function. The topology of the resulting potential function generates a complex bifurcation scenario under harmonic forcing with multiple attractors. The first aim of this work is to investigate the nonlinear vibrations of the proposed model. The interaction of the bifurcation phenomena leads to motions in two-, four- and six-dimensional phase-space. The use of different integrity measures is essential for assessing the robustness of the driven structure to unpredictable finite disturbances of a both static and dynamic nature in such cases. However of the various numerical tools for global dynamic analysis, few are well suited for higher-dimensional systems. In particular the evaluation of integrity measures of multidimensional system requires significant computational resources, being a rather time-consuming procedure. Thus the second aim of the paper is to propose numerical methodologies for evaluating the dynamic integrity measures of multidimensional systems by a Monte Carlo approach. Dynamic integrity analysis of the resulting multidimensional system is performed using three integrity measures, global and local integrity measures (GIM and LIM, respectively) and the integrity factor. Three algorithms based on the Monte Carlo method are proposed to estimate these measures by means of sampling initial conditions. We demonstrate that the proposed approach has a major advantage in comparison to classical methods, since basins of attraction are not needed. Computationally expensive procedures, such as simple cell mapping, are not required. Therefore, the proposed approach has the potential to estimative the dynamic integrity of high dimensional systems with less computational effort, as shown by the obtained results.

中文翻译：

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多稳定桁架的非线性振荡和分岔以及通过蒙特卡罗方法进行的动态完整性评估

在目前的工作中，考虑了受 von Mises 和 Bergan 桁架启发的桁架模型。由此产生的结构是一个现象学模型，它代表了许多工程结构的行为，并可能表现出平面内突触或干草叉不稳定分叉和/或横向屈曲，导致两个多孔势函数。所得势函数的拓扑结构在具有多个吸引子的谐波强迫下生成复杂的分叉场景。这项工作的第一个目的是研究所提出模型的非线性振动。分叉现象的相互作用导致二维、四维和六维相空间中的运动。在这种情况下，使用不同的完整性措施对于评估受驱动结构对静态和动态性质的不可预测的有限扰动的稳健性至关重要。然而，在用于全局动力学分析的各种数值工具中，很少有适合高维系统的。特别是多维系统完整性度量的评估需要大量的计算资源，这是一个相当耗时的过程。因此，本文的第二个目的是提出通过蒙特卡罗方法评估多维系统的动态完整性度量的数值方法。使用三种完整性度量、全局和局部完整性度量（分别为 GIM 和 LIM）和完整性因子对生成的多维系统进行动态完整性分析。提出了三种基于蒙特卡罗方法的算法来通过采样初始条件来估计这些度量。我们证明了所提出的方法与经典方法相比具有主要优势，因为不需要吸引力盆地。不需要计算昂贵的过程，例如简单的单元映射。因此，所提出的方法有可能以较少的计算工作量来估计高维系统的动态完整性，如所获得的结果所示。例如简单的单元格映射，不是必需的。因此，所提出的方法有可能以较少的计算工作量来估计高维系统的动态完整性，如所获得的结果所示。例如简单的单元格映射，不是必需的。因此，所提出的方法有可能以较少的计算工作量来估计高维系统的动态完整性，如所获得的结果所示。