Generalized capstan problem: Bending rigidity, nonlinear friction, and extensibility effect

A theoretical model of the capstan problem including the extensibility and the Poisson's ratio of the rod is established in this study. Several cases were examined to investigate the effects of important parameters on the tension transmission efficiency. As a result, the rod extensibility turned out to enhance the tension ratio, competing with the effects of rod bending rigidity and the frictional behaviors of the system. In case of no frictional modification (n=1, the simple Amonton's law), larger initial strain renders greater tension ratio. This effect becomes more remarkable at high radius ratio. However, the effect of the modified frictional law may oppress the effect of rod extensibility if the initial tension T0 grows larger. The effect of Poisson's ratio also tends to increase the tension ratio. But the amount was almost negligible since the maximum decrease was at most 7.8%. We also calculated the average strain throughout the rod by solving the governing equation and iterating the value of λavg. Calculated strain throughout the rod was up to 2.6 times larger than initial strain. But this contribution leaded to at most 1.078 times larger radius ratio than the initial rod radius. Thus, it is well enough to consider only the extensibility effect on the increase of the tension ratio. We also presented several prerequisites to establish the model. Three major concerns about this model were introduced and clarified.