The loom hands the crimp to the weft

The energy has no crimp ratio to give profiled the bending energy of a cloth at its own sett and found almost nothing inside it. Six of the eight cloths in the table here minimise at the end of their feasible interval, with the warp taking the whole crossing height and the weft left dead straight; the other two have wells shallower than the stiffness is known to. The diagnosis was one missing term. A warp on a loom is held at a tension, and a tension does work whenever the length of thread inside the cloth changes. Peirce said in 1937 that the crimp ratio is decided by the tensions a cloth was woven under, and the energy with no tension in it was a model of a cloth woven under none.

The term is now in, and what it produces is not the well the diagnosis expected. It is a switch. A pull of a few hundredths of a newton moves the least state from one end of the interval to the other, the band over which both systems carry real crimp is a factor of two or three wide in tension, and a loom holds its warp at half a newton — ten to fifty times above the switch. On the loom, the warp of every one-count cloth in the table is straight to a hundredth of a per cent.

One more term, and what it costs a thread #

The energy of a cloth at a fixed construction is a function of one variable: how the crossing height divides between warp and weft. Each system’s weave angle follows from its share of the height and its own pitch, and the bending energy is Peirce’s arc energy summed over both:

$$u_{\text{bend}} = \frac{2B_1\theta_1 + 2B_2\theta_2}{D}.$$

The new term is the warp’s tension times the warp’s length. A warp end held at $T_1$ that takes up an extra millimetre inside one crossing has pulled that millimetre off the beam against $T_1$, and the work is $T_1$ times the millimetre. So the quantity the cloth minimises on the loom is

$$u = u_{\text{bend}} + T_1\,l_1 + T_2\,l_2,$$

with $l_1$ the length of warp in one crossing of the Peirce path and $l_2$ the weft’s. Everything in it was already computed. The lengths come off the same path the energy is taken along; the rigidities are the ones a yarn’s stiffness is a bracket supplies; the tension is the one what the shed costs in newtons computes from the loom’s own distances — 0.52 N on each end at the front shaft of an eight-shaft broad loom, rising to 0.81 N at the back.

The two terms pull in opposite directions, and that was the point of adding one. Bending wants the warp bent and the weft straight, because a closely set warp gives the weft short spans that are costly to bend across. Tension wants the warp straight, because every per cent of warp crimp is a per cent of warp length pulled off the beam. Where the two balance is the answer.

The switch, where a well was expected #

The profile at one tension shows what happens. Take the muslin at the tension where its two crimps are equal:

At a fiftieth of a newton the muslin has an interior state, 2.62 per cent below the better end — a well of the depth the sheeting had with no tension at all, and at a split near the middle of its interval. The missing term does make wells.

It makes them only here. Move the tension a factor of three either way and the well slides off one end or the other. The hero figure is the whole story in one plot: the muslin’s warp crimp sits at its no-tension value of 28.5 per cent up to about six thousandths of a newton, falls through the crossover at 0.019 N, and is gone by 0.2. From a warp:weft split of four to one to a split of one to four takes a factor of 3.2 in tension. Across the six cloths whose no-tension state was the boundary, that factor runs from 2.0 on the cheesecloth to 3.2 on the batiste and muslin, and 11 on the duck.

So the tension term does not turn a boundary state into a comfortable interior minimum across a working range. It turns one boundary state into the other, through a narrow band, and the band sits wherever the yarn’s stiffness puts it.

The tension that balances is a stiffness #

Where the band sits is the one thing about it that is exact. For a cloth whose warp and weft are the same count, the two rigidities are equal, $B_1 = B_2 = B$, and the energy on the loom is $B$ times a function of the split plus $T$ times another. Its minimum depends on $T/B$ alone. Double the stiffness and the tension needed to straighten the warp doubles with it, to the precision of any search.

That makes the natural measure of the balance tension a number with no units in it: $\tau = T D^2 / B$, the pull made dimensionless by the crossing height and the rigidity. It is fixed by the construction’s geometry, and computing it at the free bound and at the locked bound — rigidities four hundred times apart — gives the same value to four figures:

The poplin is the exception, and for the reason the argument predicts. Its warp is 15 tex and its weft 20, so the two yarns have different fibre counts and different brackets, and moving both up their brackets by the same logarithmic fraction changes the ratio of their stiffnesses. Its τ drifts from 5.3 to 8.2 across the bracket; the other seven do not drift at all.

This is also why the switch’s position is uncertain by a factor of four hundred in principle and by a factor of three in practice. From first principles the yarn could be anywhere between the free bound and the locked one, and so could the balance tension: 0.0034 to 0.55 N for the batiste, 0.0095 to 3.9 N for the sheeting. But a bending test and a washing test agree that the yarn sits between a quarter and two fifths of the way up its bracket on the logarithmic scale — four to twelve times the free bound — and at that placement the balance tension is between one and twelve hundredths of a newton for every one-count cloth.

The loom’s 0.52 N is above all of it. For the muslin it is 9 to 27 times the balance tension, for the batiste 18 to 45.

Two cloths the loom cannot straighten #

Six of the eight arrive on the loom with a warp straight to a hundredth of a per cent. The other two do not, and each has its own reason.

The sheeting’s warp is stopped by geometry, not by stiffness. It was the cloth with a real interior well even with no tension, and it was the cloth whose feasible interval did not reach the left-hand end: below 0.294 of the crossing height the warp’s share cannot go, because its weft is set too close for the weft to take more of the height. Pull the warp as hard as the loom likes and the least state runs into that wall.

At the loom’s tension the sheeting’s warp carries 5.07 per cent crimp and its weft 31.2, and the warp’s figure is not a balance of forces at all. It is the fraction of the height the weft cannot absorb. A closely set cloth’s loom-state warp crimp is a jamming number, and it would be the same at any warp tension above about a tenth of a newton.

The poplin’s warp is stopped by its own stiffness ratio. It is the warp-dense shirting — 32 ends a centimetre against 22 picks, in a finer warp — and its balance tension is ten times the one-count cloths’: 0.16 N at the lower placement, 0.47 at the upper. That is within a factor of 1.1 to 3.2 of the loom’s pull.

So the poplin is the one cloth in the table the loom leaves near its own balance, and its loom-state split is the one cloth’s split that genuinely depends on where in the bracket its yarn sits: 1.76 per cent warp crimp at the lower placement and 7.95 at the upper, a factor of 4.5 in the answer for a factor of 3 in the stiffness.

What the weft’s pull does, which is Peirce’s statement drawn #

Everything above holds the weft at no tension, and a weft is not held at none. A shuttle drags its pirn against a tension spring and a projectile’s weft is braked; either way, when the reed lays a pick against the fell, the weft is under some pull. The weft’s term $T_2 l_2$ enters the energy exactly as the warp’s does, in the opposite direction.

With both terms in, the least state is interior at every weft tension, and it follows the ratio of the two pulls rather than either one. A weft held at a tenth of the warp’s tension leaves the muslin’s warp 0.38 per cent crimp and its weft 27.2. The two crimps come equal when the weft is pulled at between 0.51 and 0.56 N, against the warp’s 0.52 — close to equal tensions, because the muslin’s two setts, 24 and 22, are close to equal.

That is Peirce’s sentence turned into a curve. The crimp ratio of a cloth at a fixed construction is a function of the ratio of the tensions it was woven under, and a stiffness enters only through where the switch sits. On a loom the ratio of the tensions is enormous for most of the cycle — the warp is held for the whole of weaving and a weft only while it is being laid — so the model’s answer is the one the hero figure gives: warp straight, weft crimped.

Let go, the least state runs the other way #

A piece cut from the loom is held at no tension at all, and the least state at no tension is the one the fixed-sett profile found: the weft straight and the warp holding everything.

Every cloth moves in the same direction, and it is the direction loom-state cloth actually relaxes in. Crimp returning to the warp means warp length taken back into the cloth, which is the cloth getting shorter; crimp leaving the weft is the weft straightening. Why the warp shrinks more traced the asymmetry of relaxation shrinkage to the loom’s history — the warp held under tension for the whole of weaving, the weft for a fraction of a second — and argued it from that history. The energy now gives the same direction as a consequence of one sign: releasing $T_1$ removes the only term that was keeping the warp straight.

It also gives the wrong amount, and the size of the error is informative. The least state after release is the whole width of the interval away: the muslin’s warp would go from nothing to 28.5 per cent crimp and the weft from 33.6 to nothing, which is a length shrinkage of about a fifth and a width gain of a third. No cloth relaxes by that. A loom-state muslin washed and dried loses a few per cent of its length and usually a little of its width too.

What stops it is what the energy still does not contain. A cloth gives back less than it took computed the band a relaxed cloth rests in when its crossings hold by friction: the cloth does not roll down to the least state, it stops as soon as the energy’s slope falls below what the crossings can hold. The tension term fixes the direction of relaxation and the friction band fixes how far, and neither can supply the other.

A loom is a bending test that nobody reads #

The exactness of τ has a consequence worth stating in the other direction. For a one-count cloth, the warp tension at which it comes off the loom with equal crimps is the yarn’s rigidity multiplied by a number the geometry fixes. If a loom were run on one construction at a series of warp tensions and the loom-state crimps measured at each, the tension where they cross would give $B = T^\ast D^2 / \tau$ directly — a third measurement of where in its bracket the yarn sits, alongside the cantilever and the washing band, and one taken on the yarn in the cloth rather than on a strip of it.

The complication is exactly the weft’s pull. A crossover measured on a loom is a crossover of the tension ratio, not of the warp’s tension alone, so the experiment would need the weft’s tension at the fell held and recorded. That is a harder thing to know than the warp’s, and it is why nothing here claims the bracket can be closed this way; only that the arithmetic for doing it is one division.

How the minimum was searched, and what was held fixed #

The energy is the fixed-sett energy of the essay before this one, with $T_1 l_1 + T_2 l_2$ added. The lengths are Peirce’s: a straight run and an arc of radius $D/2$ through the weave angle, per crossing, for each system. The crossing height is split in 1,200 steps; at each step the two weave angles are solved from Peirce’s height relation by bisection, a step with no solution is outside the feasible interval, and the least value of the total over the feasible steps is the state reported.

The balance tension is found by bisection on the logarithm of the tension — sixty halvings between a ten-millionth of a newton and a hundred — on the sign of the warp’s crimp minus the weft’s at the least state.

The scaling with rigidity is required, not observed. For every one-count cloth the balance tension at the locked bound divided by the balance tension at the free bound must equal the ratio of the two rigidities to within two per cent, which is the quantisation of a 1,200-step split; all seven pass to better than that. A harder pull on either system is required never to hand that system more of the crossing height, at every one of the 49 tensions swept. And a negative tension is refused rather than computed, because a thread on a loom can be pulled and cannot be pushed.

The yarn’s place in its bracket is taken from two unrelated measurements, a cantilever bending test and the width of the band relaxed cloth rests in, which agree on 24 to 42 per cent of the way up the bracket on its logarithmic scale. Every loom-state number above is quoted at that placement, and the table’s range is its two ends.

What the switch cannot show #

The model has no fell. A loom-state crimp is formed at the fell of the cloth, in the few millimetres where the reed lays the pick and the shed changes over it, and the geometry there is not a cloth at its own sett in equilibrium. The blow that sets the pick and a pick density is a force budget describe that zone, and neither has been joined to this energy. What is computed here is the state a finished cloth would relax to if it were held at the loom’s tensions, which is the right question for the direction of relaxation and a doubtful one for the loom-state split itself.

Six loom-state warps straight to a hundredth of a per cent is the model overshooting, and it should be read that way. A grey cloth’s warp crimp is routinely measured and is not nought. Either the weft’s pull at the fell is a sizeable fraction of the warp’s for long enough to matter, or the crossings lock the split at the fell before the warp’s tension can finish straightening it — or both. The model cannot tell those apart, and the weft-tension figure above shows how little weft tension it would take to explain a few per cent.

And the stiffness is still a bracket. The table’s balance tensions are quoted over the placement the two tests agree on; outside it the switch could be anywhere from a thousandth of a newton to several newtons, and only the dimensionless τ is free of that.

Who found which half #

Peirce stated the result and did not compute it. His 1937 geometry left the crimp ratio as an input and his text said what decided it — the tensions under which the cloth was woven. Every analysis since has measured the ratio or assumed one.

The energy and both of its failures belong to this account. The crimp ratio is not a measurement proposed bending energy as the fourth equation; the fixed-sett profile found that at a cloth’s own construction it mostly minimises at an edge; and the tension term here puts Peirce’s cause into the same energy and finds that it does not so much fill the interval as cross it. The exact scaling of the balance tension with rigidity, and the loom as a bending test, follow from nothing but the form of the two terms.

The part worth carrying is the shape. A variational model that returned a boundary with one term and the opposite boundary with two has not failed twice. It has said, twice, that the crimp split of a real cloth is held by something that is not a minimum of anything — which is what crimp interchange has always implied by being possible at all.

Still open: whether friction at the fell fixes the loom-state split #

The two terms together give a direction and a switch; the friction band gives a stopping distance. What neither gives is the split a cloth actually leaves the loom with, and the candidate for it is specific.

At the fell, each new crossing is formed while the warp is at its loom tension and the weft at whatever the reed leaves it, and within a few picks the crossing is buried under the next ones and held by the normal force every crossing carries. If the split is frozen at that moment, the loom-state crimp is the least state at the fell’s own tension ratio, and the release figure’s “on the loom” column is its prediction once the weft’s term is set to its value at the fell.

So the calculation that would close this is the weft’s tension at the moment of burial, and the question that would test it is whether two looms weaving one construction at two weft brake settings produce loom-state warp crimps in the order the weft-tension curve above predicts. The curve says the order is steep: a weft braked from a twentieth to a fifth of a newton should take a muslin’s loom-state warp crimp from about a third of a per cent to about three.