A leno's slackener only has to arrive

In a bottom-doup leno, the loop of the doup lifts the partner end on every crossed pick, and that lift puts a premium on the partner’s tension. A slackener answers it by letting the doup’s standard down. At the full shed the arithmetic was clean: each millimetre of drop takes a millimetre off the loop’s lift, and when drop equals rise the loop is clear and the partner is exactly the end it would have been with no loop at all.

All of that was read at the full shed, on the assumption that the drop grew in step with the lift. A real slackener is a cam or a weighted lever with timing of its own, and the doup rises on its own. So the essay ended by asking how closely a slackener’s motion has to track the doup’s. The parallel looked discouraging. A cam easer gives its slack too early had found the crossing end’s easer sensitive to exactly this kind of thing, with its demand far from proportional to the shed.

The answer for the partner is that it hardly has to track at all. It has to arrive, by a stated point, and the reason is one curve that only goes up.

Two schedules in place of one #

The calculation is the same three straight spans as before: from the fell to the loop, from the loop to the heddle, and from the heddle to the back rest, with two turns that grip by capstan, in 25 tex cotton on the ordinary broad loom of the other leno essays. The one change is to stop assuming that the loop’s lift and the slackener’s drop are one number. At shed fraction $s$ the loop’s height above the partner’s straight line is

$$\max\bigl(0,\; R s - \text{drop}(s)\bigr),$$

with $R$ the loop’s rise at the top, four millimetres throughout unless stated, and $\text{drop}(s)$ now a schedule of its own. Three schedules are tried. In step has $\text{drop}(s) = Rs$, and the loop is never in contact. Late does nothing until a fraction $l$ of the shed and then runs its drop evenly up to the top. Early reaches its full drop at $1 - l$ and stays there.

The in-step schedule has to reproduce the untouched partner at every fraction, and it does, to a hundred-thousandth of a newton, with no rubbing at all. That is not a surprise, but it is the check that the new schedule is the same calculation as the old one.

A late slackener runs high and arrives clear #

The middle curve is the surprise. A slackener that waits until half the shed does nothing for half the opening, and the partner follows the touched curve upward, with the loop pressing and the front span taking the extra. Then the drop begins, and the partner comes down off that curve and joins the clear one exactly at the top.

Friction might have been expected to stop that. The earlier essays established that a turn holds what it is given: yarn drawn through a loop does not simply come back. But the two turns are holding a difference, not a length, and when the loop falls the short span slackens and the capstans let yarn move back through until each turn is again inside its limit. So the final state at the top does not remember how it got there, at least as far as the front span’s tension goes. Over a wide range of schedules the top is 1.0696 newtons every time.

The peak has nowhere to be but the top #

This is why the timing barely matters to the peak. Both curves, touched and clear, rise steadily with the shed, so a partner’s greatest tension along any schedule is either at the top, on the clear curve, or at the moment the slackener starts, on the touched curve.

Those two are equal at one fraction. The partner with no slackener reaches the clear partner’s top at 0.748 of the shed, at a rise of four millimetres, and that is the limit, with no fitting in it:

• Start before 0.748, and on the touched curve the partner never gets as high as the clear top. The peak is unchanged at 1.07 newtons.
• Start after it, and the partner has already passed 1.07 by the time the slackener moves. The peak is the touched tension at that moment: 1.14 newtons at 0.8, 1.30 at 0.9, 1.39 at 0.95.

Both statements are checked. A start 0.05 before the limit leaves the peak at the top at 1.07, and one 0.1 after it puts the peak at the touched value to within a hundredth of a newton, and arrives at 1.07 at the top regardless.

So the working tension the slackener was brought in to protect is safe at any start up to three quarters of the way. The ordinary end on the same shaft carries about 1.02 newtons, and even with a clear loop the partner stays about five per cent above it, as it did before.

How late depends on how much it lifts #

The limit moves with the rise, and the rise is a quantity nobody has measured. It depends on how the doup is built and how taut it is held, which is why the earlier essay swept it.

The more the loop lifts, the faster the touched curve climbs, and the sooner it reaches the clear partner’s top: 0.867 of the shed at two millimetres, 0.748 at four, 0.647 at six, 0.564 at eight. So a slackener with half the shed to spare covers any rise up to about eight millimetres, which is well past anything that would let a shuttle through. A designer unsure of the rise can simply start before half the shed.

What lateness costs is rubbing #

If the peak were all that mattered, a slackener could be built as late as possible. The cost is at the loop, in a quantity the earlier essay stated and did not develop: the yarn sliding through it under load.

The figure has an odd feature that is worth the trouble of explaining. A slackener that starts later than about 0.58 of the shed rubs the loop more than having no slackener at all, and it does so with the partner’s peak still clear.

The reason is that a loop with no slackener only ever draws yarn one way. With no drop, the rising loop takes about nine tenths of a millimetre through from the front span to the short one, and the rub is that distance at that force: 0.121 newton-millimetres over an opening. A late slackener draws yarn through the loop during its wait, and then, when the drop begins, it draws yarn back the other way, for about two and a half millimetres of sliding in all. The yarn is rubbed twice, for a benefit at only one end of the shed.

So there is a choice that the tension alone does not show. In step, or anywhere early, there is no rub at all. Up to about 0.58 there is less rub than the unslackened loop gives and a far better peak. Between 0.58 and 0.748 the peak is still right and the rub is worse. Beyond 0.748 both are worse than they need to be, and the peak is still better than none. Turning that into yarn wear needs a wear rate per newton-millimetre, which is not known here, so the figure stops at the work done. A doup end’s life is what would price it, and a mill with two slackener settings could read it off a broken-end count.

Early costs a little shed and nothing else #

An early slackener has the opposite profile. It drops ahead of the lift, so the loop stands clear of the partner throughout, and for the partner it is identical to the in-step schedule: no contact, no rub, and the clear tension at every fraction. There is nothing to pay on that end.

What it does pay is at the reed. The whole shed drops together while the loop still touches, because the two lines fall by nearly the same amount, which is why a slackener in step narrowed nothing. An early drop has no loop in contact, so the top line falls with the doup while the bottom line waits for the lift to catch up. The shed is smaller at that fraction, by 0.59 millimetres out of a fifteen-millimetre opening at half the shed, if the drop arrived by then.

No shuttle is in the shed at that fraction. A shuttle enters near the top of the opening, and by nine tenths every early schedule has lost less than half a per cent. So the cost of being early is close to nothing, and it falls where it does no harm. That is a real asymmetry: for this partner, early is free and late is not.

The contrast with the crossing end is instructive. A cam easer gives its slack too early found that on the crossing end’s own easer, the early slack goes somewhere and has to be caught, since an end given length before it needs it runs slack and can hang. A slackener’s early drop on the partner gives no length to anything: it merely stops a loop touching a yarn. So what is an error on one end of a leno pair is a safe margin on the other, and it is on the same pick, on the same shaft.

A weight falls only so fast #

The late limit assumes a slackener can run its drop in whatever time remains. A cam can do that, within reason. A weighted lever cannot, since it falls no faster than gravity, and a leno easer should be a light weight already counted what happens to such bars at speed.

There is a helpful effect first. A shed driven as a half cosine slows near its top, so a late start has more time left than its fraction suggests: at three quarters of the opening it has one third of the time left, not a quarter. At 200 picks a minute with the shed opening over a third of a pick, the opening takes a tenth of a second, so 33 milliseconds remain. A four-millimetre drop in free fall takes 28.6.

So at 200 picks a minute a weight could start at 0.81, and the partner only requires 0.748, so gravity has time to spare. The two limits cross at 234 picks a minute. Above that speed a free-falling slackener has to be released earlier than the partner asks for, and at 400 picks a minute it must leave by 0.39 of the shed, which is the same as saying it can no longer be called late at all. That figure is the ceiling, since friction in the lever and the stiffness of the cords would both slow it.

A limit that does not care about the rise #

The result that is worth keeping is that this speed is almost a constant. A larger rise needs a longer fall, by the square root of the distance, so it takes more time; and it brings the latest start earlier, so it has more time. The two nearly cancel, from 228 to 236 picks a minute across an eightfold change in the rise.

That is useful because the rise is the number nobody has. Most results in the leno essays carry it as a sweep, and this one hardly moves. A weighted slackener’s speed limit, in free fall, on this loom’s shed profile, is about 230 picks a minute, whatever the doup. It does move with the share of the pick the shed opens in, directly, so a loom opening its shed over a quarter of a pick has a limit nearer 175 and one opening over half a pick nearer 350.

So the reason a modern leno motion is positively driven need not be anything about tracking. It may simply be that the loom is fast, and that past a little over two hundred picks a minute a weight cannot do the job however it is timed.

What was computed, and what it was held to #

The partner was solved at every step of an opening of sixty steps, with each eye and loop held or slipping at its capstan limit, for eleven to thirteen starts and at thirteen fractions of the shed. A schedule in step had to match the untouched partner at three fractions and do no rubbing. The latest start had to be found between 0.7 and 0.8 at a rise of four millimetres. A start before it had to leave the peak at the clear value at the top, one after it had to put the peak at the touched tension and still arrive clear, and an early schedule had to do no rubbing at all. The rub had to rise with each later start from a fifth of the shed to seven tenths, staying below no slackener at the first and above it at the last. At three quarters of a half-cosine shed, a third of the time had to be left, and the limiting speed had to separate arrival in time from arrival too late on either side by one per cent. A start outside the shed, an early arrival before the shed begins, no rise to take off, a loom not running and a slackener with no drop are refused.

What the model cannot see #

The rubbing on the way back is left out, and the reason is worth stating. The calculation can be run through the closing of the shed as well, and it was. Through the closing the rub jumps by about 0.15 newton-millimetres at a start near 0.72, and the place of the jump moves with the step size. Two gripping turns have more than one state they can hold, and the relaxation picks one of them. The opening is continuous and steady under a fourfold change in the step, so it is what the figure quotes. The closing’s rub is real and it is not in the figure, so every rub here is a part of the total.

The drop is even from its start to the top. A real cam has its own shape, and a real weight accelerates, which would move the rub without moving the peak, since the peak depends only on when the slackener starts. Whether a shaped drop could get the same rub at a later start is a question this does not ask.

The crossing end has been left on one side. The slackener also lowers the doup’s eye, and that shortens the crossing end’s kink. The earlier essay read that at the full shed. Whether a late drop leaves a similar margin on the crossing end has not been computed here, and since the doup end pays for the crossing, it may be the stricter of the two.

The partner is rigidly held at its eye, the loop is a point, and the loop’s lift is taken as proportional to the shed. Each of these was assumed in the earlier essay and each carries over. So does the one that matters most, the rise itself, whose effect on the limit is now at least known.

A shuttle is assumed to want only the top of the shed. A rapier enters earlier in a different geometry, and on such a loom an early slackener’s few per cent could fall where something is.

Who worked out which part #

Leno motions with easing and slackening mechanisms are old and many, and the design of a doup’s standard is a practical art documented in loom manuals. The capstan is Euler’s. What holds a pick in and a leno’s hole cannot drift say why anybody puts up with all this.

Separating the slackener’s schedule from the lift, finding that the peak is unmoved until one crossing, showing that lateness is paid in rubbing with a change of sign against having no slackener at all, and finding a speed limit for a falling weight that barely depends on the rise are computed here. Nothing about them has been measured.

Still open: what the crossing end asks of the same drop #

One drop acts on both ends of the pair, and only the partner’s half has been timed. The crossing end’s tension also rises with the shed and is eased by the same fall of the same eye, so the same kind of crossing should exist there: a fraction beyond which a late drop leaves that end above the tension the easer was set for. If that fraction is earlier than the partner’s, it sets the limit for both, and the three quarters here is a generous figure. If it is later, the two ends can be timed together without any compromise.

The calculation is the crossing end’s eyes with the eye’s height on its own schedule, and it needs the easer set first, since a slackener and an easer share that end’s length. One weight under a leno pair ends on one end has already found that the pair’s two ends do not divide what they are given evenly, so the answer may not be a single fraction at all.