The generalized capstan equation: contact mechanics between an elastic rod and a frictional rigid cylinder

The Euler-Eytelwein equation for the classic capstan problem is often used to model the mechanics of entangled filaments in frictional contact, such as in the cases of ropes wound around poles or belts driven by pulleys. This modeling framework predicts an exponential tension growth along a filament of vanishing thickness and zero bending stiffness, in frictional contact with a rigid cylinder. However, in many physical settings, the sliding filament is neither infinitely thin nor perfectly flexible, thus violating the capstan equation’s underlying assumptions. In this talk, we present an enhanced capstan model based on Kirchhoff-rod theory, which considers both the thickness and elasticity of the sliding filament. Contrary to existing adaptations of the capstan equation, we assume that the extent of the contact region between the filament and the rigid cylinder is unknown a priori. Our results highlight the role of elasticity in setting the contact geometry. In turn, this nontrivial geometry strongly influences the force transmission along the sliding filament. Specifically, we investigate custom-made elastomeric rods, as well as engineering driving belts. Throughout, we validate our modeling assumptions through a combination of FEM and precision model experiments.