A leno easer should be a light weight

A cam easer gives its slack too early found that a leno’s crossing end wants its extra length late: the kink’s extra is a hypotenuse less its run, so it grows as the square of the shed’s climb, and an easer driven in proportion to the shed hands over sixteen millimetres before the end can use them. The repair it proposed had two forms. One was a cam cut to the demand. The other was an easer that gives length in answer to tension rather than to the shed — a spring — because such an easer gives nothing while the end is not pulling.

It ended with a worry about the spring. Too stiff, and it gives too little until the tension is already high. Too soft, and it might give its whole travel to the first pull. Somewhere between, a rate should track the demand, and that rate is a number a loom builder could use.

The rate turns out to be nearly nought, for a reason about where the spring sits rather than how stiff it is. And the thing that limits a spring easer at speed is not its rate at all.

An easer that answers to tension #

The picture of the spring easer is simple. An easing bar behind the harness carries the crossing ends over it. A spring or a weight holds the bar against a stop. When an end pulls on the bar harder than its preload, the bar moves and gives length: a millimetre for every $1/k$ newtons of excess, for a spring of rate $k$ per end. A dead weight is the case $k = 0$: the bar gives whatever length it takes to bring the pull back down to the preload.

Such an easer cannot overshoot, which is its whole attraction. It gives length only while its span is pulling above the preload, and it stops giving as soon as the span is back at it. The slack that the proportional cam produced, length handed over before the kink had climbed, cannot happen, because an easer answering to tension has nothing to answer to before the kink climbs.

The question is what it answers to. The easer acts in the back span, between the back standard and the back rest. So what it feels is the back span’s tension, and that is where the difficulty lies.

The span the easer feels hardly feels the kink #

A heddle eye lets the kink through found that the crossing end’s two heddle eyes grip. Yarn passes from one span to the next only when the pulling side is $e^{\mu\theta}$ times the other, the capstan limit a thread is gripped where it turns established. So the three spans of the crossing end — the front span from the fell to the doup, the kink from the doup to the back standard, and the back span — carry different tensions, and the kink, which makes the extra length, carries the most.

With nothing given, the kink carries 9.5 newtons and the back span 6.8. As the easer gives, all three fall, the kink fastest, and all three are under the working tension of 1.02 newtons by the time the easer has given 67.9 millimetres. At that length the back span is at 0.573 newtons, only fifteen per cent above the warp’s resting tension of half a newton.

That is the whole problem in one number. An easer that gives length in answer to its span’s tension has to have given 68 millimetres by the time its span has risen by 0.073 newtons. A spring that does that has a rate of about a newton a metre. The easer is reading a gauge that has barely moved while the thing it is meant to protect has gone from two and a half times the yarn’s breaking load to its working tension.

A spring of a newton a metre, or a weight #

Solving for where the bar settles at each eighth of the shed, and asking which springs keep every span under the working tension at every stage, gives the hero figure.

With the preload at the warp’s resting tension, the stiffest spring that works is 1.07 newtons a metre per end. Raise the preload and the limit falls: a bar that starts moving at 0.52 newtons tolerates 0.78 newtons a metre, one at 0.54 tolerates 0.48, and past about fourteen per cent above the resting tension no spring works at all, because the bar then waits too long before it moves.

The failure of a stiffer spring is all at the top of the stroke. A spring of three newtons a metre an end lets the tightest span reach 1.14 newtons at the full shed, twelve per cent over; ten newtons a metre lets it reach 1.56, half as much again. Below half the shed’s opening every spring does nearly as well as every other, because the kink has not climbed yet. What a spring’s rate costs is length withheld at the moment the end needs the most, and the stiffer the spring, the more it withholds.

A newton a metre per end is an odd number to hand a loom builder. A bar carrying a thousand crossing ends would want a total rate of about a newton per millimetre over a stroke of seventy millimetres, with its preload set to the warp’s own tension to within a tenth. In practice that is a weighted bar: a weight gives constant force over its whole stroke, which is a rate of exactly nought, and it sits in the middle of the window.

A weight follows the demand without being told it #

The dead weight’s curve has the demand’s shape: almost nothing through the first quarter of the opening, then a steep rise to 68.8 millimetres at the full shed. It sits a few millimetres above the dots, which are the least the end needs, because it holds the back span at its resting tension rather than letting it rise to the working one. That excess is harmless. It is not slack — no span goes below its resting tension, so none of it hangs, and the snarl how much yarn has to hang priced cannot begin — and it leaves the tightest span at 0.93 newtons at the top of the stroke, under the working tension with room to spare.

That answers the worry the cam essay ended on. An easer answering to tension cannot be too compliant. A softer spring gives more length, but only while its span is above the preload, and it stops giving once its span is back there. “Its whole travel to the first pull” would need the first pull to stay above the preload for the whole travel, and that can happen only if the end is pulling hard enough to need it.

What can be wrong is the preload. A weight set too heavy waits too long before it moves, and the window closes at a preload fourteen per cent above the warp’s tension. A weight set too light gives before the shed opens, which in effect lets off warp. So the setting a loom builder has to get right is the weight, matched to the warp’s running tension, and the rate hardly matters as long as it is small.

In front of the harness the spring could be four times stiffer #

The dashed curve in the hero figure is the same question asked of an easer in front of the harness, acting in the span between the fell and the doup. That span carries much more of the kink’s tension than the back span does, so an easer there reads a gauge that moves. In front of the harness the stiffest workable spring is 4.8 newtons a metre an end, four and a half times the back’s, and the window stays open past a preload twenty per cent above the resting tension.

It is the right gauge in the wrong place. The cam essay found that an easer’s length can reach the shuttle only if it is given in front of the harness, where the shuttle runs, and that the gripping eyes are what keep an easer behind the harness from dropping its ends onto the race. An easer that could read the kink’s tension properly would have to sit where its length is dangerous. The eyes that protect the shuttle are the same eyes that hide the kink from the easer, and a leno loom has to live with both.

At speed, the bar’s mass decides #

Everything so far is static: the bar settled where it would come to rest if the shed stopped at each eighth. A loom does not stop. At two hundred picks a minute the crossed shed rises in about a tenth of a second, in the part of the cycle between the blow that sets the pick and the next insertion, and a bar with mass has to be accelerated to keep up.

The spring the bar mostly answers to while it moves is not the easer’s. It is the yarn. A bar that lags behind where it would settle leaves the back span stretched, and a stretched span pulls on the bar with the end’s own stiffness, about eighty newtons a metre at the top of the stroke. That is eighty times the easer’s spring. So the bar and the yarn make an oscillator ringing at a frequency set by the yarn’s stiffness and the bar’s mass: 147 hertz for a tenth of a gram an end, 46 for a gram, 15 for ten grams.

The shed rises over a tenth of a second. A bar ringing at 147 hertz follows it within a millimetre; one ringing at 46 lags by up to 2.7 millimetres; one at 15 lags by twelve, overshoots when the shed stops rising and rings round its settled position. Every millimetre of lag is a millimetre the kink is stretched, and the yarn’s stiffness converts it into about a tenth of a newton.

So the loom’s speed limits the bar’s mass, not the spring’s rate. For the tightest span to stay within five per cent of the working tension, the bar can weigh 2.3 grams an end at a hundred picks a minute, 0.58 at two hundred, 0.26 at three hundred and 0.15 at four hundred. The limit falls as the inverse square of the speed, as it must: the shed’s rise time falls as one over the speed, the bar’s natural frequency has to rise in proportion, and a natural frequency goes as one over the square root of the mass.

A steel easing rod a centimetre thick and nearly two metres long weighs about a kilogram. Spread over a thousand crossing ends that is a gram an end, which the figure puts at eleven per cent over the working tension at two hundred picks and nearly a quarter over at three hundred. A weighted easer works on a slow loom and loses to a fast one, and the thing to lighten is the bar.

Where this leaves the cam #

The cam essay offered two repairs and preferred the spring. The arithmetic here keeps both and says which is for which loom.

On a slow loom, a light bar with a weight matched to the warp’s tension follows the demand exactly, by construction and without anybody having to compute the demand. It needs no cam profile, and it adjusts itself to a change in the shed, the yarn or the eyes’ friction, because what it answers to is the tension those things produce.

On a fast loom, the bar’s mass carries it behind the demand and past it, and the only easer that cannot lag is one that is driven. A cam cut to the demand is positively driven and has no lag to lose, because the shedding motion moves it at exactly the rate it moves the shafts. Its disadvantage is the one the spring does not have: it is right for one shed, one yarn and one friction, and wrong by the difference for any other.

That is a real trade and it has a crossover. Where the weighted bar’s overload at speed exceeds what a cam’s mismatch costs when the warp’s tension drifts, a cam is better; below it, a weight. Both costs are now computable, and neither is large at ordinary loom speeds.

What the model assumes #

The crossing end is the eyes model of a heddle eye lets the kink through: three spans joined through two gripping eyes at a friction coefficient of 0.3, a 25 tex cotton end on an ordinary broad loom with its back standard four shafts behind the doup, opened in 120 steps. Its span tensions are tabulated against the easer’s length every two millimetres at each eighth of the shed and read between by straight lines.

The easer answers to the back span’s tension through a spring of stated rate and preload, resting on a stop until the span pulls harder than the preload. The working tension it must hold the end under is an ordinary end’s at the full shed, 1.02 newtons, the same target an easer gives back the kink used.

The bar’s motion is a mass on the easer’s spring, pulled by the back span’s tension as a half-cosine shed rises over a third of a pick, with damping stated as a fraction of critical against the span’s own stiffness. Three tenths is used as the central value, for a bar with some friction in its bearings; undamped, the bar rings harder and its worst tension is ragged in the mass, and heavily damped it lags more. The five-per-cent mass limit is quoted at three tenths.

A dead weight is required to keep every span at or under the working tension; a spring of fifty newtons a metre an end is required to fail; a heavier preload is required to leave room only for a softer spring; the front easer’s limit is required to be several times the back’s; and a gram an end at two hundred picks is required to stay within a few millimetres of where it would settle.

What the arithmetic cannot say #

The bar is one mass for every crossing end, and a real bar is not. It is a rigid rod carrying a thousand ends, and if the ends pull unequally — as they do when the crossed shed forms slightly out of step across the width — the rod tilts rather than translating, and the ends at one side are eased more than at the other. A bar per end is the right model for a rod on a flexible hanger and the wrong one for a stiff rod on two levers.

And the yarn is linear. The end’s stiffness is read through the linear modulus the eyes model carries. A real cotton yarn stiffens as it extends and relaxes under a load held for a while; a tensioned cloth loses its load found the same distribution of barriers setting a floor under a held tension. Both would change the bar’s natural frequency a little, and the relaxation would slowly shift the weight a warp wants over the run of a beam, which is the practical reason a weighted easer has to be reset.

Who worked out which part #

Weighted and spring easers are both old leno practice, and the preference for a positively driven easer on a fast loom is the trade’s own. What the trade has not written down, and this account has not found anywhere, is why a spring behind the harness has to be so soft, which needs the eyes’ grip to be computed; or why the bar’s mass rather than its rate sets the speed limit, which needs the yarn’s stiffness to be seen as the spring the bar is really on.

The capstan is Euler’s and Eytelwein’s. The span tensions are the eyes model of this account’s leno essays, and the demand is an easer gives back the kink’s. What is added here is the easer as a thing that answers to tension, its window in rate and preload, and the inverse-square law that ties the bar’s mass to the loom’s speed.

Still open: whether the doup end wants an easer too #

Everything here is about the crossing end. The doup end pays for the crossing found that the pair’s two ends consume warp at different rates and want two beams, and on a loom where both come off one beam, the doup end’s own tension during the crossed shed decides how much of the difference the let-off has to absorb at every pick.

If the doup end’s tension also swings with the shed, a single weighted bar carrying both ends of each pair would feel the sum of two different swings and give length to the one that did not need it. The calculation is the same table built for the doup end’s path, which runs through the doup’s own eye rather than two heddles; whether the two ends can share a bar, or need a bar each, is what it would settle.