This second part of a two part paper uses concepts from graph theory to obtain a deeper understanding of the mathematical foundations of multibody dynamics. The first part (Jain in Graph theoretic foundations of multibody dynamics. Part I. Structural properties, 2010) established the block-weighted adjacency (BWA) matrix structure of spatial operators associated with serial- and tree-topology multibody system dynamics, and introduced the notions of spatial kernel operators (SKO) and spatial propagation operators (SPO). This paper builds upon these connections to show that key analytical results and computational algorithms are a direct consequence of these structural properties and require minimal assumptions about the specific nature of the underlying multibody system. We formalize this notion by introducing the notion of SKO models for general tree-topology multibody systems. We show that key analytical results, including mass-matrix factorization, inversion, and decomposition hold for all SKO models. It is also shown that key low-order scatter/gather recursive computational algorithms follow directly from these abstract-level analytical results. Application examples to illustrate the concrete application of these general results are provided. The paper also describes a general recipe for developing SKO models. The abstract nature of SKO models allows for the application of these techniques to a very broad class of multibody systems.

中文翻译：

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多体动力学的图论基础第二部分：分析和算法。

两部分论文的第二部分使用图论中的概念来更深入地了解多体动力学的数学基础。第一部分（多体动力学的图论基础中的 Jain。第一部分。结构属性，2010）建立了与串行和树拓扑多体系统动力学相关的空间算子的块加权邻接（BWA）矩阵结构，并介绍了空间核算子（SKO）和空间传播算子（SPO）的概念。本文以这些联系为基础，表明关键的分析结果和计算算法是这些结构特性的直接结果，并且需要对底层多体系统的具体性质进行最少的假设。我们通过引入一般树拓扑多体系统的SKO 模型概念来形式化这个概念。我们展示了关键分析结果，包括质量矩阵分解、反演和分解，适用于所有 SKO 模型。它还表明，关键的低阶分散/聚集递归计算算法直接来自这些抽象级分析结果。提供了应用示例来说明这些一般结果的具体应用。本文还描述了开发 SKO 模型的一般方法。SKO 模型的抽象性质允许将这些技术应用于非常广泛的多体系统。