Coils and Rings

When cutting rings from metal wire, jewelers and chainmail artists typically coil their wire around a cylindrical form, called a mandrel, then cut through one side of the coil to obtain rings. The cutting can be done with clippers, a jeweler's hand saw, or a high-speed rotary blade.

I've long wondered about the geometric relationship between one turn of the coil and a ring. I've finally begun to "do the math." This is a report on my first observations and calculations.

Abstract

I computed the arc length of a single turn of each of two helices: the one where the wire touches the mandrel (the "inner" arc length), and the one formed by the points on the wire furthest from the mandrel (the "outer").

So when the coil is cut and the ring flattened, the "inner" arc length becomes the inner diameter (ID) of the ring, and the "outer" becomes the outside diameter (OD). (For now I'm ignoring the width of the cut itself.) The question at issue is whether it's possible for the flattened ring to be a perfect torus.

It turns out that the computed outer arc length falls just short of what a true toroidal ring would require, given the computed inner arc length as ID. But the effect is tiny: I looked at the chainmail-relevant sizes of 20ga rings and the shortfall is one or two hundredths of a millimeter in the worst cases (the chunkiest rings).

Argument

Arc Length of Inner Helix

We take the center of a cylindical mandrel as the z axis, and parametrize our right-circular helices in Cartesian co-ordinates as

$x(t)=r\cos (t)$
$y(t)=r\sin (t)$
$z(t)=at$

For the inner helix (the locus of contact between the wire and the mandrel), $r=\frac{d_m}{2}$, where $d_m$ is the diameter of the mandrel.

In one turn, $t$ runs from $0$ to $2\pi$ (radians). And in one turn, the inner helix rises by twice the radius of the wire. So, we have $z(2\pi )=2\pi a=d_w$, where $d_w$ is the diameter of the wire. So $a=\frac{d_w}{2\pi }$.

We can now compute the arc length of our inner helix as $\int_0^{2\pi } \sqrt{}(\frac{d x(t)}{d t}^2+\frac{d y(t)}{d t}^2+\frac{d z(t)}{d t}^2)\,d t=2\pi \sqrt{}(r^2+a^2)=2\pi \sqrt{}(\left(\frac{d_m}{2}\right)^2+\left(\frac{d_w}{2\pi }\right)^2)$.

The ratio of $d_m$ to $d_w$ is called the aspect ratio of the ring. The (positive real) value of the aspect ratio determines the similarity class of the (presumably toroidal) ring. In these investigations, it turns out that the aspect ratio most often appears in the denominator. So I'll use the reciprocal instead, which I denote by $\tau$ (tau).

Eliminating $d_w$ in favor of $\tau d_m$, we arrive at the following expression for the inner arc length:
$2\pi \sqrt{}(\left(\frac{d_m}{2}\right)^2+\left(\frac{\tau d_m}{2\pi }\right)^2)=\pi d_m\sqrt{}(1+\left(\frac{\tau }{\pi }\right)^2)$

Arc Length of Outer Helix

For the outer helix, our basic formula for the arc length holds, with no change in the parameter $a$, but with $r=d_w+\frac{d_m}{2}=d_m(\tau +\frac{1}{2})$.

The outer arc length is therefore given by:

$2\pi \sqrt{}((d_m(\tau +\frac{1}{2}))^2+\left(\frac{\tau d_m}{2\pi }\right)^2)=\pi d_m\sqrt{}(1+4\tau +4\tau ^2+\left(\frac{\tau }{\pi }\right)^2)$

Fitting Inner and Outer Diameters

Using the third-order Taylor expansion for $g(x)=\sqrt{}(1+x)$ around $0$, namely:

$g(x)\approx 1+\left(\frac{x}{2}\right)-\left(\frac{x^2}{8}\right)+\left(\frac{x^3}{16}\right)$

and keeping terms up to order 3 in $\tau$, we can make the following approximations:

Inner arc length: $\pi d_m\sqrt{}(1+\left(\frac{\tau }{\pi }\right)^2)\approx \pi d_m(1+\frac{\left(\frac{\tau }{\pi }\right)^2}{2})$

Outer arc length: $\pi d_m\sqrt{}(1+4\tau +4\tau ^2+\left(\frac{\tau }{\pi }\right)^2)\approx \pi d_m(1+2\tau +\frac{\left(\frac{\tau }{\pi }\right)^2}{2}-\left(\frac{\tau ^3}{\pi ^2}\right))$

To second order in $\tau$, these terms differ by $2\tau \pi d_m=2\pi d_w$, which is precisely the expected difference between the inner and outer circumferences of a torus. (The difference between ID and OD is twice the diameter of the wire.)

It is the third-order term, $\pi d_m\left(\frac{\tau ^3}{\pi ^2}\right)$, which approximates the shortfall of the outer arc length, relative to what would be required for a perfect torus.

Data

In progress. I will provide links to spreadsheets illustrating the results.

Next Steps

Width of Saw Cut

As I mention above, this analysis ignores the width of the saw cut (kerf). I'd like to refine the geometric model, adding kerf as a parameter. Note that the shortfall of the outer arc length won't be affected: both ID and OD are diminished by the same amount, leaving their difference unchanged. But there are no doubt other effects where the size of the kerf plays a significant role.

Coil "Spring-Back"

A more drastic idealization in the current model is that it ignores the mechanical springiness of the wire, which has a significant effect on the final size and shape of the rings.

In one common process for forming rings, for example, the artist coils wire tightly around a metal mandrel. S/he then removes the mandrel and inserts a wooden core for cutting. When the metal mandrel is removed, the coil immediately unwinds a bit, increasing its diameter. The magnitude of this effect depends on both the geometry and the material properties of the metal, but in typical cases it is large enough to be plainly visible.

It would be very interesting to estimate the effect of this "spring-back," based on published material parameters for commonly used metals. Such an analysis would be useful in predicting the size of the finished rings from the sizes of the mandrel and wire.

Note that the analysis of ID and OD given above remains valid, if we simply take $d_m$ to be the inner diameter of the coil at the time of cutting, rather than literally the diameter of the mandrel.