A heddle eye lets the kink through

The account of why a leno loom carries an easer found the length and left one question standing over it. In the crossed shed a leno’s crossing end — the end the draft cannot even describe — is lifted at its doup and held down at its back standard a few shafts behind, and between those two heddle eyes it climbs a full shed height in a few centimetres. On an ordinary broad loom that lengthens the end by 70.2 millimetres over 1,200 of free warp, 5.85 per cent, and 63.8 of those millimetres are the one steep leg between the eyes.

The 5.85 per cent was a strain spread over the whole free warp, which is what the end does if it slides through both eyes with no friction. The same account said so, and said what the other extreme would be. If the eyes held completely, the whole 63.8 millimetres would be taken up in the 64 millimetres between them — a strain near a hundred per cent, which cannot happen. The truth, it said, lies between those, and is worse than the frictionless figure.

It is worse. It is not much worse, and the reason it is not is the most useful thing the calculation produces. A heddle eye grips by wrapping, a wrap is a capstan, and a capstan limits a ratio of tensions. On an end whose every span is already stretched by the shed, a ratio of 1.4 is a small thing.

Three spans, two eyes, one yarn #

The crossing end in the crossed shed is three straight spans. From the fell it rises 300 millimetres to the doup’s eye, lifted 50 millimetres. From there it drops through 64 millimetres of run to the back standard’s eye, held 60.7 millimetres down. From there it climbs 836 millimetres back to the back rest.

Yarn can pass from one span to the next only through an eye, and at each eye the end turns through the angle between its two legs: 69 degrees at the doup and 64 at the back standard. A thread wrapped round a surface is held one wrap at a time by the Euler–Eytelwein relation: it slides only when the tension on the pulling side reaches $e^{\mu\theta}$ times the tension on the other. At a coefficient of 0.3 that is 1.44 at the doup and 1.40 at the back standard.

The model is a warp end at a background tension of half a newton, opened into the crossed shed in small steps from a straight warp. The yarn is the 25 tex cotton at 700 turns a metre that put the shed into newtons, at 112.6 newtons per unit strain. At every step each eye either holds or slides at its limit, and the spans are relaxed eye by eye until neither moves. A span given more yarn than its path goes slack rather than into compression.

What holding the kink would take #

The first thing to settle is how much grip would be needed to trap the kink where it forms, because it decides at once how far from that extreme any real eye can be.

If neither eye let a millimetre through, the 63.8 millimetres would be stretch in 64 millimetres of yarn: 112.8 newtons, thirty times the yarn’s breaking load. The fell side, taking only its own share of the shed, would sit at 2.05 newtons and the back span at 0.80.

So holding it needs a ratio of 55 across the doup’s eye and 142 across the back standard’s. Over wraps of 69 and 64 degrees those are coefficients of 3.3 and 4.4. A friction coefficient is already two numbers for a yarn on a yarn, and a yarn on a smooth eye is ordinarily quoted in tenths; nothing that could be threaded through a heddle comes near one, let alone three.

No heddle eye can hold the kink. The frictionless figure is therefore not a lower bound that the truth sits far above. It is the right order, and the question becomes by how much a ratio of 1.4 moves it.

Most of the kink slides in #

At a coefficient of 0.3, the kink’s 63.8 millimetres are found in three places. 12.3 millimetres slide in through the doup’s eye from the fell side. 42.1 slide in through the back standard’s eye from the back. 9.4 are taken as stretch in the span between the eyes.

Even with frictionless eyes some of the kink is stretch between them, 7.1 millimetres, because every span of an end at one tension is at one strain and the middle span’s share of the whole is its share of the length. What friction adds is 2.3 millimetres at a coefficient of 0.3 and 5.4 at 0.6. Those look small beside the 42 that came through the back.

They are not small in the way that matters. A millimetre kept between the eyes is a millimetre of stretch in about 120 millimetres of yarn, and a millimetre slid through is a millimetre spread over nearly a metre. The 2.3 extra millimetres raise the middle span’s strain from 5.85 to 7.97 per cent, which is a third more load on the part of the end the doup is pulling on.

The back standard’s eye passes three times the doup’s #

The two eyes grip almost equally, 1.44 against 1.40, and one passes 42 millimetres where the other passes 12. The difference is not in the eyes at all.

It is in the spans behind them. Yarn slides through an eye until the outer span has stretched far enough for the ratio to fall back to the capstan, and a long span stretches a long way for a small rise in tension. The back span is 836 millimetres and the fell side 300, so the back can give up nearly three millimetres for each one the fell side gives at the same rise.

This is the same arithmetic that makes a shed a strain and not a length: a displacement costs tension in proportion to how short the yarn is that has to absorb it. A leno’s back span is the loom’s long free warp, and the doup’s front span is only the distance to the fell.

A capstan bounds a ratio #

The whole result follows from one property of the capstan relation, and it is the one that is easiest to forget. It fixes the ratio of two tensions and says nothing about their difference.

A wrap of 69 degrees at 0.3 lets the pulling side carry 44 per cent more than the other. If the other side is at half a newton, the eye holds back two tenths of a newton. If it is at seven newtons, it holds back three. The more the shed stretches the whole end, the more force each eye can withhold, and the less that force is as a share of what is already there.

An eye in a crossed shed never meets the case where the grip would count, because the shed never loads the kink without loading the outer spans. The front span has its own 4.1 millimetres of shed extension and the back its own 2.2, and both have received the kink’s pull through their eyes long before the middle has anything to hold against them.

So the eyes concentrate the kink, and the concentration is bounded by the capstan whatever the kink wants. At 0.3 the middle span can be at most 1.44 times the fell side, and it settles at 1.36.

More friction concentrates more, and never much #

The concentration grows with the coefficient, smoothly and without a threshold anywhere.

At 0.1 the middle span carries 7.72 newtons, at 0.3 9.48, at 0.6 12.74. The outer spans fall a little as the middle rises, because they are giving up less yarn to it. Doubling the coefficient from 0.3 to 0.6 adds 34 per cent to the middle, which is roughly what a capstan’s exponent does to a ratio of 1.4 when it doubles, and nothing like the factor of fifteen that holding the kink would take.

The coefficient is the least known number in the calculation. A steel eye on unsized cotton, a nylon-coated doup on a sized warp and a worn eye with a groove in it will not share one, and nothing here measures any of them. What the curve says is that no plausible value changes the answer’s order: the middle span sits between 7 and 13 newtons across every coefficient anybody would put on a heddle eye.

A background tension barely moves it #

The other assumed number is the warp’s background tension, the tension an end carries with the shed closed. Half a newton is a round value for a 25 tex cotton warp, and it is not measured here.

It hardly matters. With no background tension at all the middle span settles at 8.80 newtons; at half a newton, 9.48; at a whole newton, 10.16. A newton of background moves the middle span by 1.36 newtons, which is the background itself passed through the doup’s capstan and nothing more.

That is the ratio property again. The background is a floor under every span, and a floor multiplied by 1.4 is still small against seven newtons of shed.

Nine and a half newtons on a yarn that breaks at under four #

None of this rescues the end from the conclusion the account below it reached, and it makes that conclusion firmer. Through frictionless eyes the crossed shed put 7.08 newtons into a yarn whose breaking load is 3.74, 1.89 times breaking. Through gripping eyes the span the doup pulls on carries 9.48 newtons, 2.53 times breaking.

The reading is linear, and a yarn far along its load–extension curve is softer than its initial modulus, so both figures overstate. They overstate together, and the comparison between them is not affected. What changes is where on the end the danger is: frictionless, every span is equally loaded and a break could be anywhere; gripping, the doup is pulling on the most heavily loaded 64 millimetres of the whole end, which is where leno warp breaks would be expected to cluster.

A leno loom without an easer was already impossible on this loom. With gripping eyes it is impossible in one particular place — and it is the price of the half turn that makes a leno’s pick hard to pull out and its hole impossible to drift.

Moving the back standard back takes the eyes’ share down too #

The frictionless account found the crossed shed least with the back standard seventeen shafts behind the doup, and rising again beyond, because a heddle further back has to be pulled further down.

With gripping eyes the least moves a little further back, to twenty shafts, at 5.69 newtons, and the penalty the eyes add falls on the way: 49 per cent at one shaft, 34 at four, 14 at twenty. Both effects have the same cause. A back standard further back makes the kink a longer, shallower leg, so the end turns through smaller angles at both eyes, and a smaller wrap is a smaller capstan.

A back standard far enough back is both a longer kink and a looser grip on it, which is a second reason, beside the first account’s, to want the back standard as far back as a loom’s harness allows — against the depth a harness can have before its own back shafts pass a strain budget.

An easer behind the harness reaches the kink through an eye #

An easer gives the crossing end length as the crossed shed opens. The usual one is a bar behind the harness over which the doup ends pass, so the length it gives arrives in the back span — between the back standard and the back rest — and has to pass the back standard’s eye to reach the kink at all. The account below asked whether it could, and the calculation says it can, at a price.

Every span first comes down to an ordinary end’s tension at 67.9 millimetres of easer, against 64.4 through frictionless eyes. The difference, 3.5 millimetres, is what the two capstans cost. It is five per cent more easer travel for eyes that each grip by forty per cent, which is the ratio property once more: as the easer relieves the back span, the tension that has to be overcome at the eye falls with it.

The same figure on this reckoning without friction is 64.4 millimetres rather than the account below’s 64.7, because that account subtracted two lengths and this one balances strains, which are extensions over slightly different lengths of yarn. The difference is three tenths of a millimetre and it runs the same way in both reckonings.

It leaves the fell side tightest #

The order in which the spans come down is the part of the easer’s behaviour that friction changes most visibly.

With no easer the middle span is the tightest. As the easer gives length, the back span falls first, because the length arrives there; the middle follows, a capstan’s ratio behind; and the fell side comes down last, because its only supply is the middle span through the doup’s eye, and the doup’s eye passes yarn outward only when the fell side is 1.44 times the middle.

So at the easer length that just clears every span, the fell side is at an ordinary end’s tension and the rest of the end is below it: the middle at 0.79 newtons, the back at 0.56. An easer behind the harness trades a crossing end whose most loaded span was between the eyes for one whose most loaded span is the 300 millimetres nearest the cloth.

An easer in front of the doup, acting between the doup and the fell, is the mirror case. It needs 67.0 millimetres and at that length leaves the span between the eyes tightest, at an ordinary end’s tension, with the fell side at 0.82 and the back at 0.73. It needs a little less, and it would act on the part of the warp closest to where the cloth is formed, which is not where a loom builder has room to put anything.

The kink wants its length late #

The part of the easer’s job friction exposes most sharply is not its length but its timing, and it comes from the shape of the kink rather than from the eyes.

The kink is a hypotenuse less its run. While the climb between the eyes is small, a hypotenuse grows as the square of the climb, so the extra length is tiny at first and arrives mostly as the shed nears full. The easer’s job at each moment is to hand over what the end needs then, and no more.

At half the shed the end needs 18.6 millimetres and a proportional easer has handed over 34.0, which is 15.3 millimetres of slack in an end that is meant to be rising cleanly on its doup. At a quarter open the end needs almost nothing and the proportional easer has given seventeen.

The easer’s length is also not the only length the doup end is short. The crossing’s own extra yarn accumulates at every pick and is paid by a second, faster beam, and nothing about that changes with friction at the eyes, because it is a length along the cloth rather than along the loom.

A slack end in a rising shed is not a tension problem and the model does not say what it does. It says the easer’s motion has to lag the shed: nothing for the first quarter, a quarter of its travel by half the shed, the last half of its travel in the last third of the shed’s rise. An easing motion timed from the shed’s own cam and scaled to its stroke would give its length too early at every point.

Closing the shed traps nothing #

A gripping eye raises the possibility of a ratchet. The kink pulled 54 millimetres in through two eyes as the shed opened; as it closes, those millimetres have to slide back out through the same eyes, and a capstan grips a thread going either way.

It traps nothing, and the reason is geometric. As the shed closes the warp straightens and the wrap at each eye goes to nought, and a capstan with no wrap is a ratio of one. The model, closed step by step after opening, returns every span to the background tension of half a newton to within a thousandth of a newton. Whatever the eyes do in the crossed shed, they give it all back by the time the shed is level, and the next pick starts from a warp that is not carrying the last one.

What was computed, and on what #

Every number here is on the ordinary broad loom the shed essays priced: fell to reed 90 millimetres, first shaft 300, shafts at 16-millimetre centres, back rest 1,200, and 30 millimetres of clear opening at the reed, with each heddle eye displaced from the straight warp line in proportion to its own distance from the fell. The back standard is four shafts behind the doup except where it is swept.

The crossing end is three straight spans between point eyes. The shed is opened in 120 steps; at each, yarn slides through an eye until the pulling side is the capstan’s multiple of the other, the eyes are settled alternately until neither moves, and a span’s tension is the background plus the yarn’s modulus times its strain, with no span in compression. An easer adds its length to one span in proportion to the shed at each step. The length an easer needs is found by bisection on the easer, and the length at a partial opening by the same bisection on a smaller shed with the full shed’s ordinary-end tension as its target.

The model was confirmed to give every span the frictionless tension when the coefficient is nought; to make the middle span the tightest, and tighter than frictionless, when it is 0.3; to carry no more than each eye’s capstan across it; to conserve yarn, the length kept between the eyes being exactly the middle span’s stretch; to raise the middle span’s tension at every step of a sweep in the coefficient; to agree within two per cent when the shed is opened in four times as many steps; to leave the fell side tightest at the easer length that clears every span from behind, and the fell side below the middle from in front; to need more easer through gripping eyes than frictionless ones on either side; to want under three fifths of its easer at half the shed; and to return every span to the background tension when the shed is closed again.

Where the eyes are still idealised #

The eyes are points. A real heddle eye has a length and a doup carries its end round the partner over a few millimetres of heddle. Both spread the wrap over a short arc and change the capstan by the change in angle, which is small.

The coefficient is assumed. Nothing here measures yarn on a heddle eye, and a sized warp, a coated doup and a worn eye would each differ. The sweep from nought to 0.6 is how the calculation carries that ignorance, and the answer’s order does not change across it.

The yarn is linear. Tensions of seven to thirteen newtons on a yarn that breaks at under four are far beyond the region where one modulus describes it, and no loom runs a leno at those tensions — that is the point of an easer. The no-easer figures are comparisons, and the easer figures, which end near a newton, are in the region where the modulus is a fair description.

The motion is quasi-static. A crossed shed opens in a fraction of a second and a warp end has mass and a let-off that responds with a lag. Nothing dynamic is in the model, and the slack an early easer produces is exactly the kind of thing a dynamic model would be needed to follow.

Still open: what a slack crossing end does in a rising shed #

The timing result ends at a question the tension model cannot reach. An easer that gives its length in proportion to the shed leaves the crossing end fifteen millimetres slack at half the opening, and the account here says only that the end has more length than it can use. Whether that slack lets the doup end hang below the shed line and catch the shuttle, whether it snarls on the doup’s loop, or whether the end’s own weight and the let-off simply take it up, depends on the end’s behaviour with no tension in it — a matter of bending and mass over a few centimetres of yarn, which the snarl threshold begins to describe for a twisted yarn hanging free and nothing here applies to a warp end in a shed.

A slow-motion record of a crossing end through one pick, with a proportional easer and with a lagging one, would show which, and the calculation above says exactly where in the pick to look: between a quarter and a half of the crossed shed’s rise.

Who described the harness, and what is new here #

The doup, the back standard and the easer are old weaving hardware described in manuals for leno and gauze, and friction at heddle eyes is known to every weaver who has fought a sticky warp. The capstan relation is Euler’s. Putting the relation at both eyes of a crossing end in the crossed shed, finding that no plausible coefficient can hold the kink between them, measuring what gripping adds, which span an easer leaves tightest from each side, and that the end wants its easer length late in the shed, was done here.