  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.
