Mechanical interactions between rigid rings and flexible cables find broad application in both daily life (hanging clothes) and engineering systems (closing a tether-net). A reduced-order method for the dynamic analysis of sliding rings on a deformable one-dimensional (1D) rod-like object is proposed. In contrast to the conventional approach of discretizing joint rings into multiple nodes and edges for contact detection and numerical simulation, a single point is used to reduce the order of the model. To ensure that the sliding ring and flexible rod do not deviate from their desired positions, a new barrier function is formulated using the incremental potential theory. Subsequently, the interaction between tangent frictional forces is obtained through a delayed dissipative approach. The proposed barrier functional and the associated frictional functional are C2 continuous, hence the nonlinear elastodynamic system can be solved variationally by an implicit time-stepping scheme. The numerical framework is initially applied to simple examples where the analytical solutions are available for validation. Then, multiple complex practical engineering examples are considered to showcase the effectiveness of the proposed method. The simplified ring-to-rod interaction model has the capacity to enhance the realism of visual effects in image animations, while simultaneously facilitating the optimization of designs for space debris removal systems.

翻译：刚性环与柔性缆索之间的机械交互广泛存在于日常生活（如晾衣）和工程系统（如收拢绳网）中。本文提出了一种面向可变形一维杆状物体上滑环动力学分析的低阶方法。不同于传统方法将连接环离散为多个节点和边进行接触检测与数值模拟，本研究采用单点来降低模型阶次。为确保滑环与柔性杆不偏离预期位置，基于增量势理论构建了新的障碍函数。随后，通过延迟耗散方法推导出切向摩擦力之间的相互作用。所提出的障碍泛函和摩擦泛函具有C2连续性，因此可通过隐式时间步进方案对非线性弹性动力系统进行变分求解。该数值框架首先应用于已知解析解的简单算例进行验证，进而通过多个复杂工程实例验证了所提方法的有效性。这种简化的环-杆相互作用模型既能增强图像动画中视觉效果的逼真度，又能促进空间碎片清除系统的设计优化。