A tucked edge stands proud because its warp goes straight

A rapier or air-jet loom cuts every pick and lays both tails back into the next shed, so for the length of the tuck each shed holds two picks. A tucked-in edge asks every weave the odd repeat’s question counted the room for them and found a limit: the pairs lie flat only below two thirds of a plain cloth’s maximum pick density, and below about three fifths of a twill’s. It then said what it could not say, which was what an overfull zone does instead. Bulge, or hold the fell back.

Neither word is quite right, and the reason is not in the room at all. It is in how much warp the zone is given.

An end in the tucked zone comes off the same beam as an end in the body, past the same let-off, and into a cloth the same roller takes up at one speed. So it gets the same length per pick. That is a constraint the jamming count never needed, and it turns out to decide nearly everything: which yarn rises, when the edge first passes the body, and how late the pairs stand up.

Why the fell cannot help #

The earlier account left two reliefs open, and one of them can be closed by an argument with no arithmetic in it.

The second relief was that the doubled zone might hold the fell back, costing pick spacing at the edge. A curved fell is real and it is visible at the reed. But it is an offset rather than a density. The fell at the edge may lie behind the fell in the body, and it then stays behind by the same amount: on every later pick both advance together, because the same take-up roller is pulling both, and a straight-gripped cloth has only one speed across its width.

Once that offset has formed, the picks per centimetre at the edge are exactly the body’s. A permanently lower density at the edge would need the edge cloth to travel faster than the body cloth, and nothing between the fell and the roller lets it. So holding the fell back buys one transient and no room, and the question narrows to what happens at the body’s pick spacing.

One feed closes the crossing #

Peirce’s geometry of a crossing famously has one equation too few. It does not decide how the crimp divides between warp and weft, and the ordinary answer is to measure the division and take it as given. The tucked zone does not need that. Its division is already fixed, because its warp length has been set by the body next door.

An end passing over a shed’s pair lies straight across it, so the second pick is a flat in the end’s path rather than another crossing. Between two crossings, a weave of $k$ picks to an interlacing has $k$ sheds to pass, each $w$ wide, so what remains of the cloth for the end to climb in is

$$s = k\,p - (k\,w - d),$$

with $p$ the pick spacing and $d$ the yarn’s diameter. Flat pairs have $w = 2d$. Peirce’s crossing then gives the path length over that span at a climb angle $\theta$, and the requirement is that this path plus its flat equal the length the body used:

$$\frac{s - D\sin\theta}{\cos\theta} + D\,\theta + (k\,w - d) = L_{\text{body}}, \qquad D = d_1 + d_2.$$

Everything else follows from solving that for $\theta$. Two checks are available, and neither was fitted. Give each shed one pick rather than two and the solution must return the body’s own warp height, and it does, to nine figures. And the zone’s span must reach $D$ at exactly the density the jamming count named, $(P+I)/(2P+I)$, in every weave, though the two calculations are separate and neither calls on the other.

The zone’s end goes straighter #

A thick edge invites the picture of warp being forced up over a pile of weft. It is the other way round.

The reason is that a flat uses length at no cost in crimp. A body end spends its surplus over the plain spacing between crossings, by rising. Part of a zone end’s length is spoken for by the run across the pair, where it gets no rise at all, and the same length over the same number of crossings then has to be climbed in a shorter run. Rising through a shorter run costs more length per unit of height. So on a fixed length of warp the height it can afford is smaller.

At seventy per cent of the plain maximum the end loses twenty-two per cent of its rise. In a 2/2 twill at sixty-five per cent it loses thirty-eight. Crimp heights add up to one combined diameter, $h_1 + h_2 = D$, so whatever the warp gives up the weft must take: the zone is a place where the weft is doing the bending, and the crowns on its faces are increasingly pairs of weft rather than warp.

That is what the proud edge is made of. A cloth is thinnest when its two systems’ crowns are level, at $h_1 = h_2$, and thicker on either side of that. So a zone gets thicker once its weft crowns stand higher than the crowns of the body beside it.

It is proud well before it is full #

Three densities have to be kept apart, and on every weave they come in the same order.

So the count’s limit is neither where the complaint begins nor where the pairs stand up. It is a jamming calculation in which every end has to pass between the pairs on end, which is the steepest climb a crossing allows. But the zone’s ends have gone the other way, so at that density they do not need the room the count reserved for them.

At sixty-five per cent, a density that is entirely ordinary, a plain zone is ten per cent thicker than its body and a 2/2 twill zone nineteen. Both pairs are still lying flat and both are already proud.

This corrects something the earlier essay implied by placing clothing against the limit: that a shirting’s tucked edge is fine until about two thirds of maximum. On one feed, it has been thickening since fifty-five, and the two-thirds mark is not an event in the cloth at all. It is a real limit on a different state: on a cloth whose ends are forced to stand on end, which this one’s are not.

A pair stands up late, and then fast #

Past the last column the end has reached its steepest available climb and still has length left. The span is too short for it. Something has to make the span longer, and the only thing that can is the pair itself, by tilting so that its two picks sit partly over one another. A pair tilted through $\varphi$ is $d(1 + \cos\varphi)$ wide and $d\sin\varphi$ taller.

The onset is abrupt, because a tilt shortens a pair by a cosine, which hardly moves at first. Getting a small amount of extra span out of a nearly flat pair needs a large angle, and the height goes as the sine. So a plain zone at eighty per cent is twelve per cent over its body with flat pairs, and at eighty-five, twenty-five degrees up, it is thirty-one per cent over. The jump in the thickness curve is geometry rather than any change in the yarn.

A 2/2 twill reaches this sooner and goes further: thirty-four per cent over at seventy per cent of maximum with pairs at eighteen degrees, and about forty-eight per cent over by eighty. Those high numbers want a warning, given below, because round yarns are the least favourable assumption here.

The body’s own crimp moves the first date and not the last #

One number in this is assumed rather than computed: how the body divides its crimp between warp and weft, taken as one to one above. That is the same missing equation, inherited from the body, and it matters in a way that can be read off.

So whether a given cloth’s edge is proud at all, at moderate density, depends on how that cloth divides its crimp. In plain weave at sixty per cent it is seven per cent over its body with a crimp ratio of a half and nine per cent under with a ratio of two. A warp-faced body, such as a poplin, has height to lose. Its warp stands high and straightening it levels the crowns, so its tucked edge can come out slightly thin. A weft-faced body has no such margin. This gives a mill a way to predict which of its cloths will complain before it weaves them, and the input is a measurement it can already make on a sample of the body.

The tilt depends far less on the division because at that point the geometry has run out of warp height whichever way the crimp started.

What the trade’s remedies buy #

The familiar complaint has familiar remedies, and the arithmetic can price two of them and explain why it cannot price the others.

Finer selvedge ends work, and they work twice. They lower the crown directly, by having less diameter on the outside, and they lessen the end’s share of the combined diameter, so the climb needs less. In the twill at seventy per cent they do something else as well: they take the pairs back to lying flat. The trade’s instinct here is exactly right and larger than it might have looked.

A plain edge weave barely helps with thickness. That is worth knowing, because a plain column is what the tucked turns need to be held, where a shuttle’s edge could use two of the weave’s own columns. The plain zone in a twill body no longer tilts its pairs, which is an improvement of some kind, but its ends are short of length: they must make twice as many crossings as a body end on the same yarn, and so have almost no rise to spend on each. It stands twenty-nine per cent over rather than thirty-four. The firmness a plain edge buys is paid for at the thickness it was already going to have.

Fewer selvedge ends cannot be priced here at all. Taking ends out of the zone widens the room the weft has to bend in, but with the ends spaced as they are that room is not what is limiting it, so the model’s answer is unchanged. It would be a mistake to read that as “no effect.” Fewer ends reduce the cloth in the ridge, and they give the weft less to press on, and neither of those is in a calculation about a single crossing.

A shorter tuck does not enter the thickness at all. It narrows the zone without lowering it. That sounds as if it should not help, and the next section says why it does.

What a proud edge does on the roll #

A thick edge is a nuisance in the cloth, but in a cloth that has been wound up it is something more. Wound on a core of radius $r_0$, a length $L$ of cloth of thickness $t$ fills

$$r^2 = r_0^2 + \frac{L\,t}{\pi},$$

and the edge of the cloth, being thicker, climbs faster. For a large roll the ratio of radii goes to the square root of the ratio of thicknesses, so the edge asks for extra length on each turn in proportion to half its thickness excess, independent of how much cloth has been wound.

At a hundred metres and a ten per cent excess the ridge is 4.7 mm high and the edge wants about 29 mm more cloth per turn than the body beside it. Four per cent is a large strain for a cloth to find on one edge. Cloth at low load extends mostly by spending the crimp it has, and on one feed the zone’s warp has exactly the body’s crimp: 5.5 per cent in plain weave at sixty-five per cent of maximum, most of which this would use. So the edges of the roll take the winding tension and the body rides inside them on a smaller diameter, loose, while two narrow rims carry the load. It is the same trouble a seam that stands proud has on a garment, wound round on itself a hundred times.

The numbers overstate it for one good reason: a roll is under pressure and the proud yarn is what gets squeezed. A flattened thread gives up thickness under load, and every thickness here is uncompressed, so the ridge is an upper bound. But the shape of the result holds, and it says why a modest thickness excess is a commercial problem at all. It does not fade with length and it arrives early: at ten metres on a fifty millimetre core a ten per cent excess is already asking for 1.5 per cent.

It also says why a short tuck helps when the thickness says it should not. A narrow rim is much easier for a roll to flatten, and it takes up less of the width. Neither of those changes the height of the ridge, and only the first of them is about mechanics. How far the rim’s narrowness goes is not computed.

How the numbers were made #

Each cloth is 20 tex cotton at the packing used throughout the sett calculations, with its ends at sixty-five per cent of their maximum and its picks swept. Maximum densities are the jamming count’s, by weave. A weave enters through one number, its picks per interlacing: one for plain, two for a 2/2 twill, two and a half for a five-end satin and four for an eight-end. Those are averages for a weave’s ends, so the calculation describes a typical crossing rather than a particular one.

The body is solved by Peirce’s equations at a stated crimp ratio, floats entering as flats. The zone is then solved with no ratio, by bisection on the climb angle against the body’s own warp length per pick. If the steepest climb leaves length over, the zone’s tilt is solved by bisection in the same way.

Relations are tested rather than read off, each against something it was not fitted to: one pick to a shed returns the body; the span reaches $D$ exactly at the jamming count’s limit, in four weaves; flat zones’ ends are straighter than the body’s at every sampled density; and tilting begins above the count’s limit in three weaves at three crimp ratios. The roll’s large limit is checked against its closed form. A pick density outside its range, a weave with less than a pick to a crossing, a shed holding three picks and a roll with no core are all refused.

What this leaves out #

Yarns are round, and this is the largest omission. A pair under beat-up flattens rather than tilts, and a yarn’s flattening is cheap in the range these cloths sit in. Flattened picks have a smaller height and a larger width, and the width makes the shortage of span worse, so flattening would trade some tilt for an earlier straightening. The tilt onsets and the thirty to fifty per cent thicknesses above them are therefore the likeliest numbers to be overstated. The ordering of the three columns, which rests on the end’s length and on where its flat goes, should survive it, but that has not been shown.

The two picks are taken as one yarn twice. A real tail has been cut and is under almost no tension, and it lies more loosely than its neighbour, which arrived under the weft’s tension. Nothing here distinguishes them. A loose tail is the first thing that would tilt.

The ends are inextensible. They are under working tension in the loom, and a difference in need between zone and body could be met in part by stretch. But the closure gives the two the same length, so the need differs only past the tilt, where the pair takes it rather than the yarn. If the tilt were resisted, the end would go slack rather than stretch. The model cannot tell a slack edge from a tilted one, since both are ways of having too much end.

The beat-up is not priced. A reed that has to close two picks a shed against the body’s single ones meets a larger force at the edge, and the force a loom has available is what limits how close a cloth can be picked at all. Nothing here says whether the reed can reach the states drawn.

A zone has neighbours. It is about a centimetre wide, and the ends at its inner side are tied into a body at a different thickness and a different warp height, so there is a transition that these uniform numbers ignore. Whether a rim ten or twelve ends across reaches its full state is not asked here.

The picture cannot show a thickness that nobody has checked. Each section is computed and looks convincing, because round yarns in tidy arcs always look convincing. No mill’s micrometer reading of a tucked edge stands behind any number on this page, and the sections are what the model says rather than a view of cloth.

Who worked out which part #

Peirce’s crossing geometry is from 1937 and the jamming count of picks and interlacings goes back to Ashenhurst in the nineteenth century. Tuck-in selvedges have been standard since shuttleless looms spread, and the thick edge, with shorter tucks and finer selvedge ends as remedies, is familiar loom-shop knowledge. The roll’s radius is an elementary area balance, and rolls with hard or proud edges are a familiar fault in the winding of any web.

What is put together here is the closure of the crossing by one warp feed; the finding that the zone’s end straightens and that its weft takes the height; the separation of the three densities that a jamming count runs together; the argument that a held-back fell cannot relieve the density; and a price for the remedies against that closure.

Still open: what a flattened pair does at the same feed #

Everything above has a round yarn in it because that is the case with a closed form, and flattening has been awkward for this kind of calculation before. A flattened pick is wider and lower, so it lengthens the end’s flat and shortens its span, and it would tilt less readily, since a racetrack section is harder to stand on edge. Solving the same closure for the pair’s flattening and its tilt together would say whether a real twill’s edge goes on thickening in the steep way drawn here or spreads and stays nearly level, and the check on it is a micrometer on a tucked edge at a stated density, which any weaving shed could supply.