Capstan equation
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The capstan equation or belt friction equation, also known as Euler-Eytelwein's formula (after
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and Johann Albert Eytelwein), relates the hold-force to the load-force if a flexible line is wound around a cylinder (a
bollard A bollard is a sturdy, short, vertical post. The term originally referred to a post on a ship or quay used principally for mooring boats. It now also refers to posts installed to control road traffic and posts designed to prevent automotive v ...
, a
winch A winch is a mechanical device that is used to pull in (wind up) or let out (wind out) or otherwise adjust the tension of a rope or wire rope (also called "cable" or "wire cable"). In its simplest form, it consists of a spool (or drum) attache ...
or a capstan). Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small ''holding'' force exerted on one side can carry a much larger ''loading'' force on the other side; this is the principle by which a capstan-type device operates. A holding capstan is a ratchet device that can turn only in one direction; once a load is pulled into place in that direction, it can be held with a much smaller force. A powered capstan, also called a winch, rotates so that the applied tension is multiplied by the friction between rope and capstan. On a
tall ship A tall ship is a large, traditionally- rigged sailing vessel. Popular modern tall ship rigs include topsail schooners, brigantines, brigs and barques. "Tall ship" can also be defined more specifically by an organization, such as for a race or fe ...
a holding capstan and a powered capstan are used in tandem so that a small force can be used to raise a heavy sail and then the rope can be easily removed from the powered capstan and tied off. In
rock climbing Rock climbing is a sport in which participants climb up, across, or down natural rock formations. The goal is to reach the summit of a formation or the endpoint of a usually pre-defined route without falling. Rock climbing is a physically and ...
this effect allows a lighter person to hold (
belay Belaying is a variety of techniques climbers use to create friction within a climbing system, particularly on a climbing rope, so that a falling climber does not fall very far. A climbing partner typically applies tension at the other end of t ...
) a heavier person when top-roping, and also produces
rope drag A rope is a group of yarns, plies, fibres, or strands that are twisted or braided together into a larger and stronger form. Ropes have tensile strength and so can be used for dragging and lifting. Rope is thicker and stronger than similarly c ...
during
lead climbing Lead climbing is a climbing style, predominantly used in rock climbing. In a roped party one climber has to take the lead while the other climbers follow. The ''lead climber'' wears a harness attached to a climbing rope, which in turn is connected ...
. The formula is :T_\text = T_\text\ e^ ~, where T_\text is the applied tension on the line, T_\text is the resulting force exerted at the other side of the capstan, \mu is the
coefficient of friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
between the rope and capstan materials, and \phi is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle \phi =2\pi\,). Several assumptions must be true for the formula to be valid: # The rope is on the verge of full sliding, i.e. T_\text is the maximum load that one can hold. Smaller loads can be held as well, resulting in a smaller ''effective'' contact angle \phi. # It is important that the line is not rigid, in which case significant force would be lost in the bending of the line tightly around the cylinder. (The equation must be modified for this case.) For instance a
Bowden cable A Bowden cable ( ) is a type of flexible cable used to transmit mechanical force or energy by the movement of an inner cable relative to a hollow outer cable housing. The housing is generally of composite construction, consisting of an inne ...
is to some extent rigid and doesn't obey the principles of the capstan equation. # The line is non-
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
. It can be observed that the force gain increases
exponentially Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
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''. The table below lists values of the factor e^ \, 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 The tail is the section at the rear end of certain kinds of animals’ bodies; in general, the term refers to a distinct, flexible appendage to the torso. It is the part of the body that corresponds roughly to the sacrum and coccyx in mammals, r ...
(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 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. The cables necessary to support this weight, as well as the capstan's ability to withstand the crushing force of those cables, are separate considerations.


Derivation

The applied tension T_(\varphi) is a function of the total angle subtended by the rope on the capstan. On the verge of slipping, this is also the frictional force, which is by definition \mu times the normal force R(\varphi). By simple geometry, the additional normal force \delta R(\varphi) = R(\varphi + \delta\varphi) - R(\varphi) when increasing the angle by a small angle \delta\varphi is well approximated by \delta R(\varphi)\approx T_\mathrm(\varphi) \sin (\delta \varphi) \approx T_\mathrm(\varphi)\delta\varphi. Combining these and considering infinitesimally small \delta \varphi yields the differential equation \frac = \mu T_\mathrm(\varphi),\qquad T_\mathrm(0) = T_\mathrm, whose solution is T_\mathrm(\varphi) = T_\mathrm e^


Generalizations


Generalization of the capstan equation for a V-belt

The belt friction equation for a
v-belt A belt is a loop of flexible material used to link two or more rotating shafts mechanically, most often parallel. Belts may be used as a source of motion, to transmit power efficiently or to track relative movement. Belts are looped over pulley ...
is: : T_\text = T_\text^ where \alpha is the angle (in radians) between the two flat sides of the pulley that the v-belt presses against. A flat belt has an effective angle of \alpha = \pi. The material of a
V-belt A belt is a loop of flexible material used to link two or more rotating shafts mechanically, most often parallel. Belts may be used as a source of motion, to transmit power efficiently or to track relative movement. Belts are looped over pulley ...
or multi-V
serpentine belt A serpentine belt (or drive belt) is a single, continuous belt used to drive multiple peripheral devices in an automotive engine, such as an alternator, power steering pump, water pump, air conditioning compressor, air pump, etc. The belt may al ...
tends to wedge into the mating groove in a pulley as the load increases, improving torque transmission. For the same power transmission, a V-belt requires less tension than a flat belt, increasing bearing life.


Generalization of the capstan equation for a rope lying on an arbitrary orthotropic surface

If a rope is lying in equilibrium under tangential forces on a rough orthotropic surface then all three following conditions are satisfied: # No separation – normal reaction N is positive for all points of the rope curve: #: N=-k_nT>0, where k_n is a normal curvature of the rope curve. # Dragging coefficient of friction \mu_g and angle \alpha are satisfying the following criteria for all points of the curve #: -\mu_g<\tan \alpha <+\mu_g # Limit values of the tangential forces: #: The forces at both ends of the rope T and T_0 are satisfying the following inequality #: T_0 e^\le T \le T_0 e^ #: with \omega = \mu_\tau \sqrt=\mu_\tau k \sqrt, #: where k_gis a
geodesic curvature In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's ...
of the rope curve, k is a curvature of a rope curve, \mu_\tauis a coefficient of friction in the tangential direction. #: If \omega = const then T_0 e^\le T \le T_0 e^. This generalization has been obtained by Konyukhov.


See also

*
Belt friction Belt friction is a term describing the friction forces between a belt and a surface, such as a belt wrapped around a bollard. When a force applies a tension to one end of a belt or rope wrapped around a curved surface, the frictional force between ...
*
Frictional contact mechanics Contact mechanics is the study of the deformation of solids that touch each other at one or more points. This can be divided into compressive and adhesive forces in the direction perpendicular to the interface, and frictional forces in the tangen ...
*
Torque amplifier A torque amplifier is a mechanical device that amplifies the torque of a rotating shaft without affecting its rotational speed. It is mechanically related to the capstan seen on ships. Its most widely known use is in power steering on automobiles. ...
, a device that exploits the capstan effect


References


Further reading

* Arne Kihlberg, Kompendium i Mekanik för E1, del II, Göteborg 1980, 60–62.


External links


Capstan equation calculator
{{DEFAULTSORT:Capstan Equation Equations of physics Winches