Effect of Bending Rigidity on the Capstan Equation

Abstract

A tensioned fiber or yam in contact with a circular body is analyzed. In the model, a fiber is considered as a linear elastic and inextensible beam. An exact mathematical model is derived and the analytical solution is obtained. Both the beam-body contact and two noncontact regions are analyzed. From the equilibrium equation of force and bending moment, the derived model has three compatible ordinary differential equations, and one algebraic equation from the Euler beam theory with four unknown parameters. From the solutions, the relationship between incoming and outgoing tension is obtained, and we call this the modified capstan equation. The results show the correction connecting the capstan equation and its applications. For example, in a typical case, the capstan equation underestimates the effect of bending rigidity, which renders the only physically possible situation of tension in a real capstan to be the equilibrium of inclined tension. Moreover, the sensitivity of the tension ratio on the variation of the inclined angle is extensive. For instance, the difference in the tension ratio between the classical and modified results may be the maximum 71% value for a mere 100 variation of the inclined angle direction. Also, it is directly applicable to fiber contact with a body whose surface has a noncircular convex profile. Several further topics are suggested for discussion.