One weight under a leno pair ends on one end

A leno easer should be a light weight found that the bar which gives a crossing end its slack ought to be a dead weight matched to the warp’s own tension, and it left a question about the other end of each pair. The two ends of a leno pair come off a loom in different amounts — the doup end pays for the crossing found the crossing end runs nearly a quarter further than its partner — and on a loom where both come off one beam, the obvious economy is one easing bar under both. Whether a single weighted bar can carry the pair, or whether each pair needs its ends separated, was the open question.

It needs posing with care, because the end that crosses and the end threaded through the doup are the same end. The other end of the pair is the partner: the end the crossing end winds about, threaded through one ordinary heddle and lifted on no pick at all. What the question is really asking is what that end does on a bar it does not need.

The answer is that it hands the bar’s whole weight to the crossing end. A weight set so that the bar rests on its stop with the shed closed is shared between the two ends at rest and borne by one of them at the top of the shed.

The partner is an ordinary end lying down #

In a plain gauze the crossing end is lifted on every pick: by its own back heddle in the open shed, and by the doup in the crossed shed, on the far side of its partner. The partner is lifted on none. It lies in the bottom of every shed, and on a loom whose shed closes to the centre between picks it is pulled down from the centre line to the bottom line once a pick, exactly as any other end in the bottom of the shed is.

So the partner’s path is an ordinary end’s path. The shed is an extension priced that path as two straight legs meeting at a heddle eye, and the back shaft works hardest found the price rising towards the back of the harness, because a back shaft has to move its eye further to make the same opening at the reed. The partner pays exactly that price at whichever shaft its heddle hangs on.

The one thing it adds is that its eye grips. A thread is gripped where it turns established the capstan: yarn slides round a turn of angle $\theta$ only when the tension on one side is $e^{\mu\theta}$ times the tension on the other. The partner turns once, at its eye, through the small angle between its two legs, and so carries a little more in its front span than in its back.

On the doup’s own shaft the partner’s front span carries 1.07 newtons at the full shed against 1.02 with its eye sliding freely, and its back span 1.00. Twelve shafts further back the front span carries 1.66. The grip adds five per cent to the frictionless tension at every position, to the first decimal place, and that is not a property of partners. Every ordinary end on the same shaft passes through the same kind of eye at the same angle and carries the same five per cent.

That is the first result, and it is a reassurance. The partner needs nothing an ordinary end does not need. It is not eased on a plain loom and nothing about being half of a leno pair asks for it. The 1.02 newtons the crossing end has been held to in every easing calculation is the strictest number in the chart — an ordinary end on the doup’s shaft with a frictionless eye — and the partner, gripping, sits five per cent above it on that shaft and more on any shaft behind.

A weight the closed shed shares #

An easing bar behind the harness acts on the back span of every end that rides it, the length between the end’s last heddle and the back rest. It sits against a stop until the ends pull on it harder than its weight, and then it moves and gives length, until their pull falls back to the weight. It answers to the sum of the back spans it carries, and it gives every one of them the same length.

The first decision for a bar under both ends is its weight. With the shed closed, both ends sit at the warp’s resting tension, half a newton each in the case worked throughout, so a bar that is to rest on its stop with the shed closed — and not let warp off before anything has happened — must weigh at least a newton a pair. That is the matched weight, the shared bar’s equivalent of the crossing end’s own bar weighted at its own resting half newton.

At the closed shed the matched weight is shared equally. As the crossed shed opens, both back spans start to rise, the bar moves, and both ends are given length. The hero figure shows what follows. At an eighth of the shed the crossing end’s back span carries 0.57 newtons and the partner’s 0.43; at a quarter, 0.76 and 0.24. By three eighths of the shed the partner carries nothing at all, and from there to the full shed the crossing end’s back span carries the whole newton.

The reason is in the two ends’ different appetites for length. The crossing end needs about seventy millimetres eased to it by the full shed; an easer gives back the kink the crossed shed puts in found that length in the steep climb between the doup and the back heddle. The partner needs none, and can take only as much as relieves its own stretch before it goes slack.

The partner can take eleven millimetres #

How much is that? An ordinary end on the doup’s shaft is stretched by its shed and by the warp’s resting tension together, and handing its back span length relieves both. At the full shed its tension reaches nought when the bar has given 10.9 millimetres, and less while the shed is still opening: 5.4 at an eighth, 6.1 at three eighths, because the shed has stretched it less. Past that the partner has more yarn between the fell and the back rest than its path, and it hangs.

So the two ends of the pair need from the bar about seventy millimetres and at most eleven, and the bar can give them only one length. Every millimetre the shared bar gives above the shaded region is slack in the partner. The shared bar’s curve leaves the region at three eighths of the shed and goes on to 63 millimetres at the full shed; at every fraction it gives less than the crossing end’s own bar would, and from three eighths on it gives less than the crossing end needs.

The shortfall at the top is 5.8 millimetres against the crossing end’s own bar. That sounds small beside seventy, and it would be, if the crossing end were a single span. It is three spans joined through two gripping eyes, and the last few millimetres of easing are exactly where a heddle eye lets the kink through found the end most sensitive.

The crossing end is left at twice its rest #

Once the partner is slack the bar’s whole weight hangs on the crossing end’s back span, and that span carries one newton where the crossing end’s own bar would hold it at half of one. The other spans follow it up through their eyes.

On its own bar the crossing end’s tightest span climbs gently to 0.93 newtons at the full shed, under the working tension with room to spare. On the shared bar it crosses the working tension between a quarter and three eighths of the shed, reaches 1.20 newtons at three eighths, as the partner goes slack, and climbs to 1.43 newtons at the full shed, forty per cent over the working tension. The partner’s front span, meanwhile, falls from its half newton to nothing and stays there.

The weight did exactly what the light-weight essay said a weight does: it held the span it answers to at its preload. The trouble is the preload. A bar per crossing end is preloaded at that end’s rest, which is what keeps the kink under the working tension. A bar per pair is preloaded at the pair’s rest, which is twice one end’s, and once one end of the pair has gone slack the other is being held at twice its rest. The weight that was right for two ends at the closed shed is the wrong weight for one end at the open one, and the shed decides which of those it is facing.

No weight serves both ends #

The matched weight was chosen so the bar would not let off warp with the shed closed. A lighter weight would be allowed to let off a little at rest, in effect running both ends of every pair at a lower tension, and perhaps that buys the crossing end its easing.

It does buy the crossing end its easing. At half a newton a pair the crossing end’s tightest span comes down to 0.93 newtons, the same as on its own bar, because once the partner is slack the bar is then holding the crossing end’s back span at its own rest. But the partner is then 57.9 millimetres slack. At 1.1 newtons a pair the crossing end is at 1.54 and the partner 50.9 slack. Across the whole range the trade is a little over a millimetre of the partner’s slack for every tenth of a newton the crossing end is relieved, and the curve never comes within fifty millimetres of the corner where the partner is taut and the crossing end is under its working tension.

That is not a property of weights. The bar gives both ends the same length, the crossing end needs about seventy millimetres of it, and the partner can absorb eleven, so a bar that eases the crossing end hangs the partner by the difference whatever holds it up. A spring makes it worse: a bar on four newtons a metre an end overloads the crossing end to 1.74 newtons and still leaves the partner 49 millimetres slack. Nor does it matter where the partner’s heddle hangs. Moved back eight shafts, the partner can take 14.6 millimetres rather than 10.9, and is 48 slack rather than 52. The verdict is kinematic: one bar, one length, two ends that want different lengths.

Where the partner’s slack goes #

A slack partner does not sit where a taut one runs. Its front span, between the fell and its heddle, is held by the cloth and by the eye, and the slack collects behind the harness, in the long back span where the bar gave it.

Fifty-two millimetres of extra yarn over a span of nine hundred is not much as a proportion, about six per cent. As a shape it is a great deal. A thread slack by $e$ over a span $L$ hangs, under its own weight, as a shallow catenary whose sag is $\sqrt{3Le/8}$, and that is 133 millimetres at the middle of the partner’s back span, at every crossed shed, every other pick. A cam easer gives its slack too early computed the same sag for a crossing end given its slack before it could use it, and found it harmless only because the slack stayed behind the harness. Here it stays behind the harness too, and it is nearly twice as deep: 133 millimetres against seventy-two.

Behind the harness is where a loom keeps its lease rods and, on most power looms, its warp stop motion: a drop wire riding every end, held up by the end’s tension, which falls when the end breaks or goes slack and stops the loom. A partner hanging at every crossed shed is an end the stop motion cannot tell from a broken one. And a snarl needs only slack and a twisted yarn; how much yarn has to hang bracketed the threshold between two metres and six hundred of the yarn’s own weight, far longer than this, but an end that goes slack and taut again every other pick is being given the chance a hundred times a minute.

Two requirements, one split of the warp #

A leno loom therefore has two independent reasons to divide its warp along the line between crossing ends and partners, and they arrive from opposite ends of the time scale.

Over a piece, the doup end’s crossing costs it yarn: a crossing end of quarter-millimetre yarn crossing every millimetre runs 22.5 per cent further than its partner, so after a hundred metres of cloth the two have drawn twenty-two metres apart. No single beam can feed that, and the crossing ends need a beam of their own turning faster. Within a single pick, the crossed shed asks the crossing end for seventy millimetres and gives it back at the next pick, and the partner asks for nothing. No single easing bar can give that, and the crossing ends need a bar of their own.

The two divisions are the same division. The ends that must come off a second beam are exactly the ends that must ride the easer, and the partners that stay on the ground beam are exactly the ends that must stay off it. So a loom built for the first need is built for the second at no further cost: the crossing ends come off their own beam, over their own bar, and the partners come off the ground beam beneath it. That is the usual leno arrangement, and the arithmetic says it is not merely convenient. It is the only one in which each device sees only the ends it is for.

The surprise is the direction of the failure. The intuitive worry about sharing a bar is that the crossing end’s large swing would drag the partner about. It is the other way round: the partner’s small need makes it drop out early, and the crossing end is then left alone with a weight sized for two.

What the model assumes #

The crossing end is the eyes model of a heddle eye lets the kink through: three spans through two eyes gripping at a coefficient of 0.3, a 25 tex cotton end on an ordinary broad loom with its back heddle four shafts behind the doup, the shed opened in small steps so that the direction yarn last slid through each eye is remembered. Its tensions are tabulated against the bar’s travel at each eighth of the shed, exactly as the light-weight essay tabulated them.

The partner is the same yarn through one heddle on the doup’s shaft unless stated otherwise, held down by the centre-closed shed of the loom used throughout, its single eye gripping at the same coefficient. With the coefficient set to nought it reproduces the ordinary shed’s tension exactly, which is required of it; with friction its front span is required to be the tighter by no more than its eye’s capstan, at every shaft up to twelve behind the doup.

The shared bar settles where the two back spans sum to its weight, plus its spring rate times its travel where a spring is asked about, solved by bisection at each eighth of the shed. The partner’s slack is the bar’s travel less the length at which its tension first reaches nought, and its sag is the shallow-catenary sag over its back span. The matched weight is required to leave the crossing end’s back span carrying the whole of it by the full shed; every weight in the sweep is required to fail one end or the other; and a bar under the crossing ends alone, with the partners off it, is required to serve both.

What the arithmetic cannot say #

The doup’s loop is taken not to touch the partner. In a bottom-doup leno the loop passes beneath the partner and, in the crossed shed, rises past it carrying the crossing end to the far side. If it rubs the partner on the way, the partner has a second turn and a second capstan, and a third span in front of its heddle. That raises the partner’s tension in the crossed shed by much more than a little — a leno’s slackener eases the partner, not the doup finds nearly two fifths at a four-millimetre lift, through the grip of two turns in series — but it changes none of the conclusions above, which turn on the partner’s slack rather than its tension: the touched partner hangs after twelve millimetres of easing rather than eleven.

And the slack is drawn hanging freely. A real back span passes over lease rods and through drop wires, and a slack end meets them long before it reaches the 133 millimetres a free catenary would sag. The drawing says how much yarn is unaccounted for; it does not say what the rods and wires do with it.

Who worked out which part #

Separate beams and separate easers for doup ends are both ordinary leno practice, and so is the drop-wire stop motion. The capstan is Euler’s and Eytelwein’s. The crossing end’s spans, the demand and the weighted easer are this account’s earlier leno essays.

What is added here is the partner as an ordinary end whose eye adds a constant five per cent at any shaft; the shared bar as a weight the closed shed shares and the open shed hands to one end; the finding that no weight and no spring serves both; and the observation that the split the easer needs is the split the beam already made.

Still open: what the doup’s loop does to the partner #

Everything here keeps the doup’s loop clear of the partner, and in a bottom-doup leno it is not clear: the loop passes under the partner on every pick and climbs past it in the crossed shed. A loop that touches is a third eye on the partner’s path, a few millimetres from its own heddle, with a turn whose angle is set by how far the loop rises past the partner and how tightly the doup’s standard holds it.

The calculation is the eyes model with one more eye, which is the model the crossing end already has. It would say how much the partner is pulled up in the crossed shed, whether the loop’s friction concentrates tension in the short span between the loop and the partner’s heddle as the doup and the back heddle concentrate it in the crossing end’s kink, and whether a loom’s slackener — the device that lets the doup’s standard drop slightly as the crossed shed forms — is relieving the partner as well as the doup.