A thread is gripped where it turns

The rung that priced a thread’s hold did it in one line. A thread gripped over a length of cloth is held at every crossing it makes; the resistance at each is μ times the normal force; sum over the crossings in a gripped length and the pull-out force is proportional to that length, while the thread’s own breaking load is not. The two curves cross, and the crossing point — a few millimetres to a couple of centimetres — is a fraying width, a seam allowance and a tuft’s anchorage all at once.

It has two things wrong with it and the rung below wrote down both. The float correction is missing: counting every intersection as a grip is right for a plain weave and overstates a satin’s hold. And the friction along a gripped thread is not a sum of independent contacts, because the thread’s own tension presses it into the contacts ahead of it.

Where the grips are, and it is a matrix question #

A thread presses on the thread it crosses where it turns. Over a float it lies on the surface of the cloth, touching nothing above and passing over threads it is not wrapped around, and it presses on nothing.

That is a property of the draft, exactly, with no geometry in it at all. The interlacings of a weave are the places a thread changes from being under to being over, the matrix counts them without ambiguity, and the count divided by the repeat’s length in cloth is the turns per millimetre.

What the matrix does not say is how hard each grip bites. That is the wrap angle, it comes from the crimp, and the crimp comes from a geometry with a yarn diameter and a sett in it. A weave decides where the grips are; only a geometry decides how hard each one holds — which is this site’s standing division of labour, applied to a quantity it had not been applied to before.

The tension compounds #

A rope round a bollard obeys the capstan equation: the normal force is generated by the tension itself, so friction multiplies rather than adds and Tₒᵤₜ = Tᵢₙ·exp(μβ). A thread in cloth is not that, and the reason is worth being careful about, because taking the capstan equation straight would give an absurd answer.

A free end has no tension. And exp(μβ) times nothing is nothing, so a pure capstan says a thread can never be pulled out of cloth at all, which is false.

What resolves it is that the cloth supplies a normal force of its own, from the crossing thread’s tension and from its own compression, whether the withdrawn thread is tensioned or not. So there are two terms:

$$\frac{dT}{ds} = \mu n_0 + \mu T \frac{d\beta}{ds}$$

a floor that does not depend on the tension and a capstan term that does. Integrating along a gripped length L, with wraps of angle β arriving at a rate ρ per unit length,

$$T(L) = \frac{n_0}{\rho\beta}\left(exp(\mu\rho\beta L) - 1\right)$$

which is exponential in the gripped length rather than proportional to it.

The previous answer is the tangent at the origin #

Expand the exponential and the leading behaviour is μ·n₀·L, which is exactly a sum of independent contacts. The previous rung is not wrong; it is the first term.

That is checked as a rate rather than against a tolerance, because asserting that the two agree at some short length would be an assertion about the length chosen. What is claimed is that the departure is proportional to the gripped length as that length goes to zero, so it is measured at four lengths each half the last and required to halve with them. It does: 5.80 per cent, 2.85, 1.41, 0.70, 0.35, with the ratio converging on two from above.

Which shortens every crossover on the site #

The crossover length under the sum is a division: the breaking load over the resistance per millimetre. Under the capstan it is a logarithm,

$$L^* = \frac{1}{\mu\rho\beta}\ln\!\left(1 + z\right), \qquad z = \frac{T_{\text{break}}\,\beta}{n_0}$$

and the correction is a single factor, ln(1 + z)/z, applied to whatever the old arithmetic said. For a sheeting it is 0.269. A plain cotton frays about two millimetres in rather than the seven and a half the sum predicted, and two millimetres is what a cut edge of plain cotton actually does.

And leaves one previous result exactly intact #

The rung below asserted, to twelve figures, that μ·L* is constant — that a slippier cloth needs a proportionally longer grip before its thread breaks rather than slides, with no residue.

That looked like an artefact of a linear model, and it is not. The friction coefficient cancels out of z entirely: n₀ carries a factor of μ and so does the exponent, so what is left is a pure geometric quantity, and μ survives only in the factor outside the logarithm. The product μ·L* is still exactly constant, to twelve figures, in a model that changes the answer by a factor of four.

A result that lives through a correction to the model it was derived in is a different kind of result from one that does not, and this collection has not had many of them.

What it changes about the three trade rules #

The previous rung’s best line was that fraying, seam slippage and tuft anchorage are three trade rules of thumb with one number under them, and that number is the crossover length. That survives, and all three move together by the same factor.

A cut edge frays until the exposed threads are long enough to break rather than slide. Under the old arithmetic a plain cotton sheeting frayed 7.4 mm in, which is a centimetre and is more than anybody sees. Under this one it frays 2.0 mm, and a raw edge of plain cotton loses about two millimetres before it stabilises.

A seam slips when the thread pulled across the seam line comes out of its allowance rather than breaking. The previous rung found an ordinary allowance divides this site’s table rather than settling it — four cloths break and four slip — and the correction shortens every crossover by between a factor of two and a half and a factor of five and a half. Which cloths fall on which side of a stated allowance therefore changes, and it changes in the direction of more of them holding.

A tuft comes away when its anchorage is shorter than the crossover, and the anchorage is fixed by the fastening rather than by the cloth. So the tuft case is the one where the correction does not merely rescale an answer: it moves a fixed length across a boundary that has moved.

The general point is that the three rules share a number, the number has been divided by about four, and the shared-ness is what makes that a single correction rather than three.

What was counted, and how #

The turns per millimetre come from the matrix: interlacings per thread per repeat, divided by the repeat’s length in cloth. A plain weave gives 2.6 per millimetre in a sheeting and a five-end satin 1.0, and both are counts rather than estimates.

The wrap angle is taken as 2θ, the turn from rising at the weave angle to falling at it, which is the total angle of contact a thread makes with the thread it crosses. That is the one place in the argument where a geometry enters, and it enters through the crimp.

The closed form for the shortening is asserted rather than admired: the computed ratio of the two crossovers is required to equal ln(1 + z)/z to one part in a million million, at three weaves and six friction coefficients, because both sides are exact expressions of the same quantity and anything looser would mean the implementation had taken a route through an approximation. The constancy of μ·L* is asserted over the same grid.

Two guards earn their place. A gripped length of nothing is refused rather than returning zero, and a negative resistance per unit length is refused rather than producing a negative pull-out force — which would look exactly like a thread being pushed out of a cloth and would be arithmetically unremarkable.

What the exponential does to a specification’s margin #

Replacing a proportional law with an exponential one does more than divide every answer by four. It changes how a length responds to the quantities that set it, and the response is what a specification’s margin is made of.

Under the sum, the crossover is a division: halve the friction and the length doubles, exactly. Under the capstan it is a logarithm of one plus a group, and a logarithm is a very flat function of its argument. Doubling z — which is what doubling the thread’s breaking load does — lengthens the crossover by rather less than doubling it, and the effect weakens as z grows.

So the two models disagree most about the strong-yarned cloths. The duck’s correction factor is 0.178 and the batiste’s is 0.436, and the reason is that the duck’s z is much the larger: a coarse strong yarn in a firm cloth is exactly the case where the old arithmetic was most optimistic, and it is also the case a specification is most likely to be written about.

That inverts a useful intuition about margins. A designer choosing a stronger yarn to make a seam hold better is buying a crossover that grows logarithmically, so the second doubling of strength buys much less than the first — while the friction, which sits outside the logarithm, still buys in proportion. Under this model a finish is worth more than a yarn, for the purpose of making a thread hard to withdraw, and under the old one they were interchangeable.

The one quantity that has not moved is the turn rate, which is also outside the logarithm. So the ordering by weave is exactly as it was, the ordering by friction is exactly as it was, and only the ordering by yarn strength has compressed — which is a small and specific correction rather than a wholesale one, and it is the kind that would be invisible to anybody comparing two cloths of the same yarn.

It is worth noting which way that cuts for the trade’s own experience. Most comparisons a mill makes are between constructions in one yarn, so the compression is exactly the effect nobody would have noticed — and the comparison it does change is the one made between a light cloth and a heavy one, which is where the trade’s rules of thumb are least trusted anyway.

Which of the two factors a mill can actually change #

The grip is a product of two things and they are altered by completely different people, which is worth setting out because the model makes the division exact.

The turns per millimetre are the designer’s. They come off the matrix and the setts, they are fixed the moment a cloth is specified, and nothing done to the fabric afterwards changes them. A mill cannot make a satin grip like a plain weave.

The friction is the finisher’s. It is the one input here that is altered after the construction is fixed, it is altered routinely and in one direction — softeners lower it, and milling raises it — and every length in this rung is inversely proportional to it.

So a cloth’s resistance to fraying and seam slippage is set half at the drawing board and half at the very last stage of manufacture, by two people who are not in the same conversation. A construction chosen to grip well can be softened until it does not, and no measurement taken before the finishing would show it.

The contact force is the third factor and belongs to neither: it comes from the yarn, the sett and the geometry, and is the same for a given cloth however it is finished. That makes it the stable part of the answer, and it is why the crossover’s dependence on the weave is exact while its absolute size is not.

Where the model stops #

The wrap angle is a plain weave’s, in every weave. A satin’s crimp is genuinely smaller than a plain weave’s at the same construction, so its turns are genuinely gentler and its grip weaker still. This model does not have a per-weave geometry — every Peirce argument here is a plain-weave argument — so the satin’s disadvantage is understated, and the direction of the error is known while its size is not. It is the largest single thing this rung does not do.

The contact force is a tensioned cloth’s. n₀ comes from a stated thread tension, so every crossover here is a crossover in cloth that is being pulled. A relaxed cloth’s contact force is a different and smaller number, and a cut edge on a shirt in a wardrobe is relaxed.

Past the crossover the curve is fiction. The exponential goes on rising and the thread has broken, so any pull-out force quoted at a length beyond L* is arithmetic about a thread that is no longer in the cloth. The figures are drawn short of it deliberately.

One coefficient, and a pull-out is a slide. What resists the first movement is static friction and what resists the movement itself is kinetic, and they are different numbers. This rung uses one.

The generalisation #

When contacts along a path are coupled by the very quantity being transmitted, they compound rather than add — and the sum is the first term of the compounding.

That is the capstan in general, and the reason it appears here in an unfamiliar form is the floor. A pure capstan has nothing at all at zero tension, which makes it useless for anything with a free end; adding a tension-independent term makes it a linear differential equation with a bounded solution and turns the multiplicative story back into an additive one at short lengths. The same shape governs a knot, a rope through a fairlead, a belt on a pulley with a pretensioner, a fibre pulled from a matrix and a thread through the eye of a needle.

The second lesson is about corrections that were expected to change an ordering. This one was expected to make a satin worse relative to a plain weave and does not — the ordering survives exactly — and what it changes instead is the scale, by a factor of four across every weave equally. A correction that moves everything by the same factor is invisible in every comparison and decisive in every absolute number, which is the opposite of the failure mode most corrections have.

Who found it, and when #

The capstan equation is Euler’s, from 1762, and applies to any flexible line wrapped round a rough surface. Its use for yarn pull-out is standard in the composites and textile-mechanics literature, where a fibre or a yarn embedded in a matrix or a fabric is treated with exactly the two-term equation above and the exponential build-up is well known.

The observation that the grips are the interlacings rather than the crossings is older than any of it and belongs to weaving practice: every weaver knows a satin frays and a plain weave does not, and knows why.

What is this site’s is putting the two together at the level of a specific draft — taking the turn count off the matrix, the wrap angle off the geometry, and insisting on the separation between them — and the finding that the friction coefficient cancels out of the correction’s size.

Where the ladder goes next #

The next question is what the correction does to the comparison between weaves, and the answer is nothing at all, which was not expected.

Sideways, the shortened crossover is what decides how far a cut edge frays and when a seam slips, and the friction it is proportional to turns out to be two numbers rather than one.