Dynamic modeling of a sliding ring on an elastic rod with incremental potential formulation
Weicheng Huang, Peifei Xu, Zhaowei Liu
TL;DR
This work addresses the nonlinear dynamics of a sliding ring on an elastic 1D rod by introducing a reduced-order model that treats the ring as a single point and enforces non-deviation via a $C^2$ continuous barrier energy derived from the incremental potential framework. The method couples discrete differential geometry for slender structures with a ring-to-rod contact formulation and a maximal-dissipation-based friction model, solved with an implicit Euler scheme. Key contributions include a ring-to-rod barrier functional, a corresponding tangential friction formulation, and a variationally consistent solver that remains stable under large deformations. The approach is validated against analytical solutions and demonstrated on complex scenarios such as hanging clothes and tether-net closure, illustrating its potential for realistic visual effects and engineering design in cable-based systems and space debris capture.
Abstract
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.