Direct visualization of Euler’s belt theory by Brillouin optical correlation-domain reflectometry

Rope-based mechanisms are widely utilized in elevators, cranes, and ship moorings. At the root of these are often Euler’s belt theory, in which a rope is wound around a fixed pulley and frictional force is said to increase exponentially with the winding angle. However, experimental validation of Euler’s belt theory has been limited due to the difficulty in measuring continuous tension attenuation when the rope is wound intermittently or multiple times. Moreover, previous study have reported that intermittent winding reduces the amount of tension attenuation, which the existing theory cannot explain. In this paper, we employed Brillouin optical correlation-domain reflectometry (BOCDR) to directly measure the tension distribution along an optical fiber wound around both a circular pulley and an intermittent pulley, simulating a situation where the end of a synthetic fiber rope was fixed. First, we compared two types of belt theories: the conventional Euler’s belt theory and a modified theory that accounted for the bending stiffness of the belt. Next, we experimentally measured the tension distribution using BOCDR in the optical fiber wound around both pulleys. When the fixed pulley diameter was D and the optical fiber diameter was d, our results showed that both belt theories yielded similar results for a circular pulley with D/d ≥ 35. In addition, we successfully observed the phenomenon that intermittent winding reduces tension damping. Considering these results, we proposed a new hypothesis that the observed reduction in tension damping was not due to intermittent winding, but was due to a small D/d ratio.