Capstan equation

The capstan equation or belt friction equation, also known as Eytelwein's formula^[1]^[2], relates the hold-force to the load-force if a flexible line is wound around a cylinder (a bollard, a winch or a capstan)^[3] ^[4]. 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 with so-called top-roping, a lighter person can hold (belay) a heavier person due to this effect.

The formula is:

$T_\text{load} = T_\text{hold}\ e^{ \mu \phi} \,$

where $T_\text{load}$ is the applied tension on the line, $T_\text{hold}$ is the resulting force exerted at the other side of the capstan, $\mu$ is the coefficient of friction 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. $T_\text{load}$ 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 is to some extent rigid and doesn't obey the principles of the Capstan equation.
The line is non-elastic.

It can be observed that the force gain grows 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. In the table below are listed $e^{ \mu \phi} \,$ for different numbers of turns and different values of $\mu\,$

$\begin{array}{|c|r|r|r|r|r|r|r|} & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7\\ \hline 1&1.9&3.5&6.6&12&23&43&81\\ 2&3.5&12&43&152&535&1881&6611\\ 3&6.6&43&286&1881&12392&81612&437503\\ 4&12&152&1881&23228&286751&3540026&43702631\\ 5&23&535&12392&286751&6635624&153552935&3553321281\\ \end{array}$

(No. turns vertical, coefficient of friction $\mu$ horizontal)

From the table it is evident why one seldom sees a sheet (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", 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" 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

$\varphi=\frac{s}{r}\qquad{d}\varphi=\frac{ds}{r}\qquad\frac{d\varphi}{ds}=\frac{1}{r}$ (1), (2), (3)

Let $\mathbf{s}_{u}$ and $\mathbf{n}_{u}$ denote unit vectors;

$\mathbf{s}_{u}=-\sin\varphi\cdot{\mathbf{x}}_{u}%2B\cos\varphi\cdot{\mathbf{y}}_{u}$ (4)

$\mathbf{n}_{u}=\cos\varphi\cdot{\mathbf{x}}_{u}%2B\sin\varphi\cdot{\mathbf{y}}_{u}$ (5)

Then from (5)

$\frac{d}{d\varphi}{\mathbf{s}}_u = -\cos\varphi\cdot{\mathbf{x}}_u -\sin\varphi\cdot{\mathbf{y}}_u=-{\mathbf{n}}_u$ (6)

$\frac{d}{ds}=\frac{d}{d\varphi}\cdot\frac{d\varphi}{ds}=\frac{d}{d\varphi}\cdot\frac{1}{r}=\frac{1}{r}\cdot\frac{d}{d\varphi}$ (7)

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

$\frac{d}{ds}{\mathbf{s}}_u=\frac{1}{r}\cdot\frac{d}{d\varphi}{\mathbf{s}}_u=-\frac{1}{r}\cdot{\mathbf{n}}_u.$ (8)

2. Forces on cordage in general

Now, let's study a piece of cord in general, subject to an arbitrary force. Let $s\,$ denote the length of the cord and let the force per unit length be $\mathbf{q}(s)$ . Consider a short piece $\Delta{s}\,$ of the cord and introduce the cross-sectional force $\mathbf{T}(s)\,$ . Balancing the forces, we get

$\mathbf{T}(s%2B\Delta{s})-\mathbf{T}(s)%2B\mathbf{q}(s)\cdot\Delta{s}=0,$ (9)

$\frac{\mathbf{T}(s%2B\Delta{s})-\mathbf{T}(s)}{\Delta{s}}=-\mathbf{q}(s).$ (10)

Letting $\Delta{s}\rightarrow0$ , we conclude that

$\frac{d}{ds}\mathbf{T}(s)=-\mathbf{q}(s).$ (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 $s\,$ be the length of the line from a point A where the line makes contact with the cylinder. At the point $s\,$ on the short piece $\Delta{s}\,$ of the line acts a force from the cylinder that can be subdivided into a tangential component $t\Delta{s}\,$ (friction) and a normal component $n\Delta{s}\,$ . That is to say that