Capstan equation
The capstan equation[1] or belt friction equation, also known as Euler–Eytelwein formula[2] (after Leonhard Euler and Johann Albert Eytelwein),[3] relates the hold-force to the load-force if a flexible line is wound around a cylinder (a bollard, a winch or a capstan).
[4][1] It also applies for fractions of one turn as occur with rope drives or band brakes.
In rock climbing this effect allows a lighter person to hold (belay) a heavier person when top-roping, and also produces rope drag during lead climbing.
For dynamic applications such as belt drives or brakes the quantity of interest is the force difference between
The formula for this is Several assumptions must be true for the equations to be valid: It can be observed that the force gain increases exponentially with the coefficient of friction, the number of turns around the cylinder, and the angle of contact.
Note that the radius of the cylinder has no influence on the force gain.
based on the number of turns and coefficient of friction μ.
From the table it is evident why one seldom sees a sheet (a rope to the loose side of a sail) wound more than three turns around a winch.
The force gain would be extreme besides being counter-productive since there is risk of a riding turn, result being that the sheet will foul, form a knot and not run out when eased (by slacking grip on the tail (free end)).
It is both ancient and modern practice for anchor capstans and jib winches to be slightly flared out at the base, rather than cylindrical, to prevent the rope (anchor warp or sail sheet) from sliding down.
The rope wound several times around the winch can slip upwards gradually, with little risk of a riding turn, provided it is tailed (loose end is pulled clear), by hand or a self-tailer.
For instance, the factor "153,552,935" (5 turns around a capstan with a coefficient of friction of 0.6) means, in theory, that a newborn baby would be capable of holding (not moving) the weight of two USS Nimitz supercarriers (97,000 tons each, but for the baby it would be only a little more than 1 kg).
The large number of turns around the capstan combined with such a high friction coefficient mean that very little additional force is necessary to hold such heavy weight in place.
is a function of the total angle subtended by the rope on the capstan.
is the angle (in radians) between the two flat sides of the pulley that the v-belt presses against.
The material of a V-belt or multi-V serpentine belt tends to wedge into the mating groove in a pulley as the load increases, improving torque transmission.
[6] For the same power transmission, a V-belt requires less tension than a flat belt, increasing bearing life.
[5] If a rope is lying in equilibrium under tangential forces on a rough orthotropic surface then all three following conditions are satisfied: This generalization has been obtained by Konyukhov.
