A cam easer gives its slack too early

A heddle eye lets the kink through priced what the eyes let through and ended at a question its own model could not 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 there said only that the end has more length than it can use.

Three things could become of that length. It could hang below the shed line and catch the shuttle. It could snarl on the doup’s loop. Or the end’s own weight and the let-off could simply take it up.

All three are decidable, two of them by geometry the account already has and the third by a number this account computed for an entirely different reason.

The demand is not proportional, and it is not nearly proportional #

The first thing the profile says is that the mismatch is structural rather than a matter of tuning.

The kink is the extra length the crossing end needs because the doup lifts it on the far side of its partner while its own back heddle holds it down. That extra is a hypotenuse less its run, so while the climb is small it grows as the square of the climb — and a square is nothing near zero.

The end asks for nothing at all through the first eighth of the opening, and it has been given eight and a half millimetres. The worst of it is not at half the shed, where the essay before it looked, but at three eighths — 16.49 millimetres, and the two curves do not meet again until the shed is fully open.

So a cam easer is wrong everywhere except at its two ends, and the error is not small compared with the thing it is delivering: at a quarter of the shed the end has been given eleven times what it wants.

Could it reach the shuttle? Only if the eyes were smooth #

The shuttle runs in the span between the fell and the first shaft — 300 millimetres on this loom — and the shed there is 30 millimetres open at the reed.

A slack thread under its own weight is a catenary, and a shallow catenary with excess e over a span L sags by $\sqrt{3Le/8}$. Sixteen and a half millimetres over three hundred is

$$\sqrt{3 \times 300 \times 16.49 / 8} = 43.1~\text{mm}.$$

That is not close. The sag is 43 millimetres in a shed that is 30 millimetres open, so a crossing end carrying this slack in the front span would not merely intrude on the shuttle’s path — it would lie on the race for the whole width of the cloth.

It cannot happen, and the reason is the thing the last two essays have been complaining about. The easer sits behind the harness, so it gives its length into the back span. The spans in front are held taut by the fell and the let-off, and a taut span does not draw length forward out of a slack one through an eye that grips — the capstan bounds a ratio of tensions, and a slack span’s tension is nothing, so nothing is what it can deliver forward.

That turns the account’s account of the eyes over. A heddle eye lets the kink through found the eyes concentrating the kink’s tension where frictionless ones would have shared it, which is a cost. Here the same grip is the only thing between the easer and the shuttle, and a leno loom whose heddle eyes were polished smooth would drop its crossing ends onto the race at every pick.

That is a requirement nobody states because nobody has had a reason to: a leno’s eyes must grip, and the friction that makes the easer necessary is the friction that makes the easer safe.

Could it snarl? The torsion work has already answered #

A slack length of twisted yarn, free at one end and held at the other, snarls when there is enough of it: the residual torque overcomes the tension its own hanging weight supplies, and the thread folds back on itself.

How much yarn has to hang puts the threshold at 1.9 metres of the yarn’s own weight, for a fresh unset yarn of this count. The span behind the back standard is 836 millimetres.

Under half. So the slack cannot snarl there, and the answer needs no new model — it needs the number from a different account and the arithmetic of which span the slack is in. The margin is a factor of 2.3, which is comfortable rather than marginal, and it would close on a loom with a back rest twice as far away.

It is worth noting what the answer depends on, because a set yarn has no torque at all and a leno’s crossing ends are usually run from a separate beam of yarn chosen for strength rather than for liveliness. If that yarn is steamed the question does not arise; if it is not, the 1.9-metre threshold is the one that matters and the loom is inside it.

What it actually does: seventy-two millimetres, behind the harness #

That leaves the third possibility, and it is the one that happens.

The back span runs from the back standard — four shafts behind the doup, 364 millimetres from the fell — to the back rest at 1,200. That is 836 millimetres, and 16.49 millimetres of excess in it sags

$$\sqrt{3 \times 836 \times 16.49 / 8} = 71.9~\text{mm}.$$

Seven centimetres, on a warp whose ends sit three tenths of a millimetre apart, at every pick, for as long as the shed is between an eighth and seven eighths open.

Whether the thread’s own stiffness changes that is settled by the elasto-gravitational length, (B/w)^⅓ — the span at which a thread’s rigidity matters as much as its weight. A yarn’s bending rigidity is a bracket rather than a number, so both ends are computed: 16.8 millimetres at the free bound and 124.9 at the coherent one. Both are well under 836, so the back span is weight-dominated at either end of the bracket and the catenary is the right shape. Solving it as a buckled elastica instead gives 74.8 millimetres, which is 4 per cent different and does not change anything.

The back standard decides the slack too, and it wants the opposite #

A heddle eye lets the kink through had one free position in it — how far behind the doup the crossing end’s back standard sits — and chose it to make the kink’s tension least. That position also decides the slack, and it decides it the other way.

The easer’s stroke falls monotonically as the standard goes back — 93 millimetres to 43 — and the slack does not. It rises to a maximum at four shafts and falls again, so the worst setting for a cam easer is in the middle of the range rather than at either end.

The reason is that two things are changing at once. A short gap makes a steep kink that wants its length late, which a linear drive tracks tolerably; a long gap makes a shallow one that wants little length at all. In between, the demand is both large and badly shaped, and the overshoot peaks.

So the position is a three-way trade rather than a two-way one. The gap decides the kink’s tension, which is the essay before it’s subject; it decides the easer’s stroke, which is what a loom builder has to find room for; and it decides the slack, which is what the stop motion sees. The first two both improve as the standard goes back. The third has an interior worst case, and four shafts — the position every figure on this account is drawn at — is it.

Which is a stop-motion problem, not a weaving one #

Seventy-two millimetres of sag is not a fault in the cloth. The slack is behind the harness, the shed forms correctly, the pick is inserted and beaten, and every millimetre is taken back up by the time the shed is fully open. Nothing about the fabric knows it happened.

What is behind the harness is the stop motion. A warp stop motion is a drop wire riding on each end, held up by the end’s tension and falling when the end breaks. It needs a few millimetres of travel to trip, and it is looking at exactly the span this slack goes into.

So the arithmetic predicts a specific and irritating failure: a leno warp on a proportionally driven easer trips its stop motion on the crossing ends, at every pick, without anything being broken. The doup ends are typically half the warp, so it is not a rare end that does it.

That is a prediction with a clean escape, and the escape is what the profile figure is for.

The repair is a cam cut to the demand #

If the demand curve is known — and it is, to whatever resolution anybody wants to compute it — then an easer driven off the shed can be driven off the demand instead. The cam’s profile is the fourth column of the table above expressed as a share of the full stroke:

A cam cut to that curve leaves no slack anywhere, by construction, and it is not an exotic shape: it is flat for an eighth of the stroke and then nearly straight. A dwell and a rise, which is what cams are made of.

And there is a mechanism that gets it right without being told. An easer that is a spring-loaded whip roll rather than a positively driven cam gives its length in response to tension rather than in response to the shed — so it gives nothing while the end is not pulling and gives what the end asks for when it is. A compliant easer follows the demand curve exactly, because the demand curve is what the end pulls with.

That is a reason to prefer a spring to a cam that is not “springs are simpler”. The demand is a square-law and the shed is a linear drive, and no linear drive can follow a square law; a compliant element does not have to.

Why the question could not be asked two essays ago #

It is worth being explicit about what each essay had to have in place, because this one needed three things and none of them was available when the account started asking about easers.

The kink’s length as a function of shed position, rather than at the top of the stroke. An easer gives back the kink the crossed shed puts in computed the length at the full shed, which is the number a loom builder needs for the stroke and says nothing about the profile. The partial-shed version is the same solve run at a fraction of the opening, and it is the whole of what makes the demand curve drawable.

A tension model for the spans, so that “the end cannot use it” means something. Without the eyes, every span is at one tension and there is no such thing as slack in one span and not another; A heddle eye lets the kink through is what makes the question well posed, and it is also what answers the shuttle half of it.

And a threshold for snarling, which came from an account about hanging yarn and has no loom in it at all.

That is three essays of arithmetic for one question that a weaver would have phrased in a sentence, and the reason the sentence has gone unanswered is that each of the three is useless without the other two. A question that needs results from three places is a question nobody’s notebook contains, which is as good a description of what a collection like this is for as any.

One more place the slack is not #

There is a fourth possibility the essay before it did not list and it is worth closing, because it is the one a weaver would suggest first: that the slack simply goes into the cloth.

It cannot. The span in front of the harness ends at the fell, and the fell is where the last pick was beaten; length arriving there would have to become crimp or become cloth, and both are decided by the take-up and the blow that sets the pick rather by whatever the easer is doing. The fell is the one boundary in the loom that does not move in response to an end’s tension, which is exactly why every span is measured from it.

And the crossing end’s partner takes none of it either. The two ends of a leno pair are separate threads through separate eyes; they cross, they do not share length, and the doup end pays for the crossing is the account’s own finding that the pair’s two ends consume warp at different rates and therefore want two beams. A pair that shared length would not need two.

So the four candidates reduce to one, and the reduction is the argument: the slack cannot go forward past a gripping eye, cannot become cloth at a fell that does not move, cannot pass to a partner it does not touch, and cannot snarl in a span under half the threshold. What is left is a sag, and its size is arithmetic.

What was counted, and how #

The demand curve is the account’s own demand calculation, unchanged: at each fraction of the opening it asks how much length leaves no span above an ordinary end’s tension at the full shed. The target is the full shed’s ordinary end at every fraction, because what an easer must avoid is overloading the end, and the rest of the warp is bearing that much anyway.

The sag is a shallow catenary, $\sqrt{3Le/8}$, which is the standard small-sag result and needs the sag to be small against the span. At 72 millimetres in 836 it is, at 8.6 per cent; at 43 in 300 it is marginal, at 14 per cent, and the buckled-elastica solve is quoted beside it for exactly that reason.

The weight per unit length is one conversion and it is required. A count in tex is grams a kilometre, which is 1e-9 kilograms a millimetre, and that exponent was wrong by one on the first run — which put the elasto-gravitational length at 7.8 millimetres instead of 16.8 and would have been invisible, because both numbers are small against the span and the conclusion is the same either way. The check exists so that the next quantity that is sensitive to it cannot inherit the error.

The snarl threshold is quoted rather than recomputed, from the torsion account, at the same count and fibre.

And the front span’s sag is computed although it cannot happen. A margin that is never tested is a margin nobody knows the sign of, and this one is negative by 43 per cent — which is the whole reason the eyes’ grip is a requirement rather than an incidental property.

What the arithmetic cannot say #

Nothing here is dynamic. The shed opens in a fraction of a second and a catenary takes time to form; a thread given 16 millimetres for forty milliseconds and then asked for it back may never sag at all, and the calculation above is the static shape it would reach if the shed stopped there. At 200 picks a minute the shed is at three eighths for about fifteen milliseconds, and whether a 25 tex end falls 72 millimetres in fifteen milliseconds is a question about acceleration that this model does not ask. Free fall covers 1.1 millimetres in that time, so the honest answer is that it does not — the sag is an upper bound and the loom speed is what decides how much of it is realised.

That does not rescue the stop motion, because a drop wire needs millimetres rather than centimetres, and it does change the shuttle answer from “would be a disaster” to “would be a disaster on a slow loom”.

And the slack’s distribution between the back spans is assumed rather than solved. The argument says the length goes behind the harness and stays there; where behind the harness — all in the span to the back rest, or partly in the span between the eyes — depends on the let-off and on which eye is gripping hardest at that instant, and the essay before it’s own model is a static one solved at the full shed.

Who found it, and when #

Leno easers are universal on leno looms and the reason given for them is that the crossing shed takes more warp, which is true and is what The doup end pays for the crossing computed. That the timing of the giving is a separate question, and that the demand is a square-law while the drive is linear, is not anywhere in the weaving literature this account has found — and it would not be, because it needs the kink’s length as a function of shed position rather than at the top of the stroke, which is a curve nobody has had a reason to draw.

The three-way decision is this account’s, and its interest is that two of the three answers come from other accounts. The shuttle question is answered by the capstan work two essays below; the snarl question by a number computed for a hanging yarn on the torsion account with no loom anywhere near it. Neither was computed for this, and both settle it outright — which is the argument for keeping a collection’s numbers in one place rather than in the essay that produced them.

Still open: whether a compliant easer can be too compliant #

A spring easer follows the demand by construction, and the obvious next question is what its spring rate has to be.

Too stiff and it is a cam: it gives nothing until the tension is high, so the end is overloaded exactly where the kink is steepest. Too soft and it gives its whole travel to the first pull, which is a cam again with the profile reversed — all the length early, and the slack is back.

Between them there is a rate that tracks the demand, and it is computable from the same two curves: the spring must give length at the rate the end’s tension rises, which is the derivative of the demand against the tension rather than against the shed. That is a curve this account can produce and has not, and it would turn the qualitative preference for a spring into a number a loom builder could use.