Capstan equation

Capstan equation

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Encyclopedia
The capstan equation or belt friction equation, also known as Eytelwein's formula, relates the hold-force to the load-force if a flexible line is wound around a cylinder (a bollard
Bollard
A bollard is a short vertical post. Originally it meant a post used on a ship or a quay, principally for mooring. The word now also describes a variety of structures to control or direct road traffic, such as posts arranged in a line to obstruct the passage of motor vehicles...

, a winch
Winch
A winch is a mechanical device that is used to pull in or let out or otherwise adjust the "tension" of a rope or wire rope . In its simplest form it consists of a spool and attached hand crank. In larger forms, winches stand at the heart of machines as diverse as tow trucks, steam shovels and...

or a capstan
Capstan
Capstan may refer to:*Capstan , a rotating machine used to control or apply force to another element*Capstan , rotating spindles used to move recording tape through the mechanism of a tape recorder...

)
.
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. For instance in rock climbing
Rock climbing
Rock climbing also lightly called 'The Gravity Game', is a sport in which participants climb up, down or across natural rock formations or artificial rock walls. The goal is to reach the summit of a formation or the endpoint of a pre-defined route without falling...

with so-called top-roping, a lighter person can hold (belay) a heavier person due to this effect.

The formula is:

where is the applied tension on the line, is the resulting force exerted at the other side of the capstan, is the coefficient of friction between the rope and capstan materials, and is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle ).

Several assumptions must be true for the formula to be valid:
1. The rope is on the verge of full sliding, i.e. is the maximum load that one can hold. Smaller loads can be held as well, resulting in a smaller effective contact angle .
2. 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
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...

is to some extent rigid and doesn't obey the principles of the Capstan equation.
3. The line is non-elastic
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

.

It can be observed that the force gain grows exponentially
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...

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. In the table below are listed for different numbers of turns and different values of

(No. turns vertical, coefficient of friction horizontal)

From the table it is evident why one seldom sees a sheet
Sheet (sailing)
In sailing, a sheet is a line used to control the movable corner of a sail.- Fore-and-aft rigs:Fore-and-aft rigs comprise the vast majority of sailing vessels in use today, including effectively all dinghies and yachts. The sheet on a fore-and-aft sail controls the angle of the sail to the wind,...

(a sailing rope) wound more than three turns around a winch. The force gain would be extreme and would be counter-productive since there is risk for a riding turn, meaning that the sheet will not run out when one lets go of the hold end. A modern winch is not exactly circular if seen from above, it usually has what can be best described as "inverted fluting
Fluting
Fluting may refer to:*Fluting *Fluting *Fluting *Fluting...

", this is to increase the friction, but only for tangential slip, the rope coils can still slip unobstructed axially avoiding riding turns.

For instance, the factor 153552935 means, in theory, that a newborn baby would be capable of holding the weight of two "USS Nimitz
Nimitz
Nimitz can refer to:* Fleet Admiral Chester W. Nimitz USN, * Nimitz class aircraft carrier of US Navy aircraft carrier* USS Nimitz , the lead ship of the above class* Nimitz High School, Houston, Texas...

" supercarriers (97 000 ton each, but for the baby it would be only a little more than 1 kg).

Proof of the capstan equation

1. Circular coordinates
(1), (2), (3)

Let and denote unit vectors;
(4)

(5)

Then from (5)
(6)

(7)

From (6) and (7), it follows that
(8)

2. Forces on cordage in general

Now, let's study a piece of cord in general, subject to an arbitrary force. Let denote the length of the cord and let the force per unit length be . Consider a short piece of the cord and introduce the cross-sectional force .
Balancing the forces, we get
(9)

(10)

Letting , we conclude that
(11)

3. A line around a capstan

A line is wound around a cylinder(a bollard or a capstan). In this case the curvature of the line is circular which makes the problem easier. Let be the length of the line from a point A where the line makes contact with the cylinder. At the point on the short piece of the line acts a force from the cylinder that can be subdivided into a tangential component (friction) and a normal component . That is to say that
(12)

With the cross-sectional force (which is tangential) we get
(13)

From (11), (12) and (13), it follows that
(14)

Derivative of a product and (8) imply that

(15)

Identifying components in (15), we get
(16)

and
(17)

Dividing (16) by (17), we get
(18)

From (18) and reciprocal of (2), we get
(19)

From (18) and (19) it follows that
(20)

Let (21) be the coefficient of friction (no slip). Then
(22) :

(23)

Integration of (23) yields
(24)

(25)

(26)

Finally,