Dynamic substructuring (DS) is a research field that has gained a great deal of attention in both science and industry. The aim of DS techniques is to provide engineers in structural vibrations and sound practical solutions for analyzing the dynamic behavior of complex systems. This paper addresses the singularity problem that occurs when flexible joints are implemented as substructures into the Lagrange Multiplier Frequency-Based Substructuring (LM-FBS) coupling process. For illustration, we use rubber bushings from an automotive application. Considering the rubber isolators to exhibit hysteretic damping, we assume that only the property of the dynamic stiffness of material is given. To avoid singularity appearing in the admittance when inverting the impedance of a massless joint, we compare three different approaches to include rubber bushings in the framework of LM-FBS. One method consists in including the dynamic stiffness of material directly in the space of the interface constraints and add it to the assembled interface flexibility of the LM-FBS equation. This corresponds to a relaxation of the interface compatibility condition. In the second method, the rubber bushing is treated as a substructure by adding small masses to the equation of the joint. As a result, we obtain a nonsingular total dynamic stiffness matrix that can be included in the coupling process. The third method describes a novel extension of the LM-FBS approach, based on a solution for singular problems. If the applied forces are self-equilibrated with respect to the rigid body modes, a solution for the singular dynamic stiffness matrix exists. The methods are outlined, both mathematically and conceptually, based on a notation commonly used in LM-FBS. They facilitate the integration of connecting elements together with experimental or numerical determined system dynamics of substructures in order to predict the assembled system behavior.

中文翻译：

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将连接元素包含在基于频率的子结构中的不同方法的比较

动态子结构化 (DS) 是一个在科学和工业界都受到广泛关注的研究领域。DS 技术的目的是为工程师提供结构振动和合理的实用解决方案，以分析复杂系统的动态行为。本文解决了在将柔性接头作为子结构实施到基于拉格朗日乘子频率的子结构 (LM-FBS) 耦合过程中时出现的奇异性问题。为了说明，我们使用来自汽车应用的橡胶衬套。考虑到橡胶隔振器表现出滞后阻尼，我们假设只给出了材料的动态刚度特性。为了避免在反转无质量关节的阻抗时导纳中出现奇异性，我们比较了在 LM-FBS 框架中包含橡胶衬套的三种不同方法。一种方法是将材料的动态刚度直接包含在界面约束空间中，并将其添加到 LM-FBS 方程的组装界面柔性中。这对应于界面兼容性条件的放宽。在第二种方法中，通过在接头方程中添加小质量，将橡胶衬套视为子结构。因此，我们获得了一个非奇异的总动态刚度矩阵，可以包含在耦合过程中。第三种方法描述了 LM-FBS 方法的新扩展，基于奇异问题的解决方案。如果施加的力相对于刚体模式自平衡，则存在奇异动态刚度矩阵的解。基于 LM-FBS 中常用的符号，在数学和概念上概述了这些方法。它们促进了连接元件与子结构的实验或数值确定的系统动力学的集成，以预测组装系统的行为。