The shed is an extension

Every mechanical model on this site rests on one assumption, and it is stated everywhere it is used: nothing stretches. The trellis has inextensible threads free to rotate where they cross. The tensile locus is the set of states a cloth can reach at constant thread length. The drape solver throws if any segment has changed length between frames. That assumption is what makes the geometry exact, and it is a good assumption about a fabric.

It is a statement about a fabric that has left the loom.

While the cloth is being made, the warp is stretched — deliberately, by the machine, at every pick. Opening a shed lifts half the ends off the straight line they were lying on, and a path that goes up and comes back down is longer than the line it replaced. The end is under tension, so it takes up the difference by stretching. A loom running at 400 picks a minute does this to every end four hundred times a minute for the length of the piece.

The amount is not a fitted number. It is trigonometry on three distances, and it comes out scale-free.

The one condition that decides the shape #

A shuttle, a rapier or a jet of air has to cross the warp, and it crosses it at the reed. That is where the clear opening has to be, and it is nowhere near the shafts.

The warp sheet runs straight from the fell to the heddle eye, so its height at the reed is the heddle’s lift scaled by how far the reed is along that run. Requiring the same clear opening at the reed from every shaft therefore forces each shaft’s lift to be proportional to its own distance from the fell. A shaft twice as far back has to lift twice as far to make the same hole.

That one sentence is the whole reason a harness is not a set of interchangeable parts. It is not that the back shaft is further from anything. It is that the back shaft has to lift further to make the same hole, and lifting further is the thing that costs.

The exact extension, and the form it takes #

Put the fell at the origin, the back rest at L, and a shaft at a, so b = L − a is what lies behind it. The clean-shed condition gives the lift as h = t·a, where t is the tangent of the half-shed at the fell — the clear opening at the reed divided by twice the reed’s distance from the fell.

The end’s path when the shed is closed is the straight line, of length L. When it is open the path is two straight runs, and the strain is the difference over the original:

ε = [√(a² + h²) + √(b² + h²)] / L − 1

Expanding both roots to leading order in t and collecting terms gives something much more informative than the exact expression:

ε ≈ t²·a / 2(L − a)

Three things fall out of that form and none of them is obvious from the picture.

The shed angle enters squared. Halving the opening at the reed quarters the strain, at every shaft. Nothing else a loom builder controls is that strong.

The strain grows without bound as a shaft approaches the back rest. The denominator is what is left behind the shaft, and it goes to zero. A harness deep enough would tear its own warp with nothing else happening.

And nothing has a unit. The tangent is a length over a length and so is the strain, so the whole expression depends on the shaft’s fraction of the way back and not on how big the loom is. Doubling every dimension of the machine changes nothing — which is asserted rather than remarked, because a scale-free result that quietly depended on a millimetre would be the worst kind of wrong.

The numbers on an ordinary loom #

An ordinary broad loom puts the reed 90 mm in front of the fell, the first shaft at 300 mm, the shafts at 16 mm centres, and the back rest at 1,200 mm, and needs 30 mm of clear opening at the reed for a shuttle. That makes the half-shed tangent 0.167, or a shed angle of 9.5°.

On an eight-shaft harness the front shaft’s ends are strained by 0.460 per cent and the back shaft’s by 0.722 — a ratio of 1.57. On a sixteen-shaft harness the back shaft reaches 1.13 per cent, and on twenty-four it reaches 1.73, which is 3.75 times what the front shaft sees.

Those are strains on top of whatever the warp is already carrying, applied and released at every pick. Cotton breaks somewhere near 7 per cent extension and a warp is already held at a working tension, so a couple of per cent of cyclic strain on top is not a small addition. It is a second term in the same account as why the warp shrinks more than the weft — that essay puts the asymmetry down to the warp being held under tension for the whole of weaving, and this is the cyclic part of the same history rather than a rival explanation. It is also why a weaver puts the ground weave, whose ends lift most often, on the shafts at the front.

The harness that produces the depth is worth drawing beside the shed it opens, because the two pictures are the same measurement seen along two axes.

Where the hyperbola bites #

A hyperbola is the kind of curve worth reading at particular points rather than admired in general, and two of those points can be given as distances from the fell.

The strain goes as a/(L − a), so asking where it reaches twice the front shaft’s value is asking where that ratio doubles. The front shaft sits at 300 mm on a 1,200 mm warp line, so its ratio is a third; twice that is two thirds, and two thirds is reached at 480 mm from the fell. On 16 mm centres that is the twelfth shaft. Three times the front shaft’s strain arrives when the ratio reaches one — that is, when the shaft is exactly halfway along the warp line, at 600 mm — which is the twentieth.

So an ordinary twenty-shaft harness has its back ends taking three times the punishment of its front ends, and the doubling has already happened a little past the middle of the harness rather than at the end of it. This is not a gentle ramp with one bad shaft on the end. It has doubled by halfway.

That is what makes the front of the harness worth so much. The first four shafts of the loom above lie between 0.460 and 0.505 per cent — a spread of a twentieth of what the whole harness covers — so the front is very nearly flat and the back is very nearly vertical. Every decision about which ends go where is a decision about which part of that curve they sit on, and the front four are interchangeable in a way that the back four are not.

The two levers, and which is worth more #

Only two things can be changed once a loom is built: how far the shed opens and how deep the harness goes. They act on the strain differently, and the difference is a square against a hyperbola.

The opening acts on the whole harness at once, and quadratically. Reducing the clear opening at the reed from 30 mm to 21 — a factor of √2 — halves the strain at every shaft simultaneously. No threading changes, no shaft moves, and the back shaft benefits by exactly the same factor as the front.

The depth acts on one shaft at a time, and hyperbolically. Moving a shaft forward helps that shaft and no other, and how much it helps depends entirely on where it started. Pulling the front shaft forward by 16 mm saves it about three hundredths of a per cent; pulling the twenty-fourth forward by the same 16 mm saves it several times as much, because the denominator it is dividing by has shrunk to less than half.

There is a third lever that nobody thinks of as one, and the form above says exactly what it is worth. The shaft pitch does not appear in the strain at all, so packing the shafts more closely moves every shaft but the first forward. Twenty-four shafts on 16 mm centres reach 668 mm and 1.73 per cent; the same twenty-four on 8 mm centres reach 484 mm and a little over nine tenths of one per cent — very nearly a halving, from a change that alters no angle and opens no shed differently. It is the same hyperbola read backwards: shafts that were out at the steep end have been brought back to where the curve is still nearly flat.

What limits the pitch is mechanical rather than geometric. A shaft has to be stiff enough to stay straight across a broad loom, the heddles on neighbouring shafts need clearance to pass one another, and the lifting mechanism has to reach every one of them — none of which this arithmetic knows about, and all of which is why a harness is not simply built as shallow as the algebra would prefer.

What was counted, and how #

Every number above is computed from four distances and one opening, all stated once and used everywhere, so a figure that changes one of them is visibly changing a machine rather than tuning a result.

The strain is computed twice and the two are compared. The exact form is the two hypotenuses over the closed length; the leading-order form is the expression above. Their difference is reported as an error rather than hidden, and it stays under a per cent to about thirty shafts and then grows — which is the honest behaviour of a leading-order expansion and is the reason the exact form is what the figures draw.

Two assertions guard the result and both could fail. The first is that the back shaft strains more than the front, which is what the whole essay is about and which would be false if the clean-shed condition had been implemented the other way round. The second is the scale-free claim, checked by doubling every dimension of the loom and requiring the ratio of back to front strain to agree to twelve decimal places.

And the machinery refuses a harness deep enough to reach past the back rest, because at that point b is negative and the arithmetic would produce a plausible number for a loom that cannot exist.

A note on which distances matter #

Three of the loom’s five distances appear in the answer and two do not, which is worth stating.

The reed’s position and the shed opening enter only through their ratio, which is the half-shed tangent, so a loom with the reed twice as far out and an opening twice as large is the same loom for this purpose. The shaft’s distance and the back rest’s enter as a fraction of one another. And the shaft pitch enters nowhere in the strain itself — only in how many shafts fit inside a stated distance, which is the next rung’s arithmetic rather than this one’s.

Where the model stops #

This is a strain and not a force. How hard the shed pulls depends on the yarn’s tensile behaviour, which this site does not model anywhere — the same gap the tensile ladder records at its own top. What is computed here is how much longer the path is, exactly; what that costs in newtons is outside it.

The warp sheet is treated as straight between fixed points. A real warp runs over a back rest that may itself be sprung, through a let-off that pays out as the beam empties, and the fell moves as the cloth is taken up. Every one of those makes the effective L a variable, and the leading-order form says the answer depends on the shaft’s fraction of the way back, so a moving fell moves every shaft’s strain together.

The end is treated as a line. It has a diameter, it passes through a heddle eye with a shape, and it rubs on the way — which is a different mechanism from stretching and is the subject of the next rung.

And the shed is treated as symmetric. Real looms use a variety of shed forms, some with the lower half stationary, and an asymmetric shed strains the two halves of the warp differently. The arithmetic is the same on each side with a different t; the symmetric case is drawn because it is the one where a single number describes the whole warp.

The generalisation #

Strip out the loom and what is left is a statement about routing an inextensible line through a moving constraint, and the shape of it is general.

A line pinned at two ends and displaced sideways at an interior point lengthens by an amount that is quadratic in the displacement and inversely proportional to the two spans it divides the line into. That is why a guitar string pressed at the twelfth fret goes sharp less than one pressed at the first, why a tensioned cable displaced near its anchorage takes more strain than one displaced at midspan, and why a shaft near the back rest of a loom is the one that breaks ends.

The textile case is unusual only in that the displacement is applied and released thousands of times a minute, and that the constraint which forces it — a clear opening at one particular point, not at the point being displaced — is what makes the displacement grow with position in the first place.

Who found it, and when #

Loom builders have known for a very long time that the back shafts break more ends and that a deeper shed is harder on the warp, and the remedies are all in practice: put the ground weave at the front, stagger the shafts so that the back ones lift a little less, run the smallest shed the insertion will allow. Shuttleless weaving is faster partly because a rapier needs less room than a shuttle, and the warp-strain benefit of that is understood in the trade without being written as an equation.

What is not usually written down is the form. The quadratic in the shed angle and the hyperbola in the shaft’s position are the two facts that decide how the remedies rank against each other — a small reduction in the opening is worth much more than a small reduction in harness depth — and that ranking is a consequence of the expression rather than of anybody’s experience.

The connection to this site’s own machinery is the part worth stating plainly. Everything here computes fabric geometry at constant thread length, and this is the one place where a thread’s length is not constant, on purpose, by machine. That the two coexist without contradiction is because they describe different moments: inextensible is a property of cloth, not of warp.

Where the ladder goes next #

The next rung asks which ends actually take the punishment, and finds that it is a product of two numbers from completely different worlds: how often a weave’s column changes sides, which is a property of a binary matrix, and the strain of the shaft it happens to be on, which is a property of a machine. Neither knows about the other, and the threading that joins them is a decision nobody makes on structural grounds.

After that, the bound: a stated tolerance on warp strain is a stated distance from the fell, and a stated distance is a whole number of shafts. A budget of one per cent buys thirteen on the loom above, which is a physical answer to a question this site has so far only answered combinatorially.

Sideways, what the warp does once it is released from all of this is relaxation, and the state every number on this site is quoted in is the state at the end of exactly this process.