The half of the beat-up that is all zone

The rung that computed the beat-up force got a result that disagrees with the whole practical literature of weaving and is not wrong.

The reed drives a new pick to the fell against the warp’s resistance, and the warp resists because a closer pick spacing needs more crimp and more crimp needs more thread. Write the virtual work down and the length of the beat-up zone — the depth of the region at the fell where the picks are still settling — cancels exactly, leaving the warp tension times the crimp’s elasticity with respect to the pick spacing. The cancellation is not a piece of algebra taken on trust: the same force is computed from finite zones of 2, 20 and 200 picks and the three agree with the closed form to five decimal places.

Every practical account of beat-up treats the zone as a quantity that has to be measured. That rung recorded, in its own list of what it had not done, that the frictional resistance of the cloth itself — the picks already woven sliding against the warp — was not modelled at all.

It is modelled here, and the two halves settle their disagreement by turning out to be about different things.

Why the frictional half cannot cancel #

The elastic half cancels its zone because it is a gradient. The reed advances the fell by a small distance, every pick in the zone moves a little closer to its neighbour, each of them demands a little more warp length, and the total extra length demanded per unit advance is the same whether the demand is spread over two picks or two hundred. Nothing about how the settling is distributed survives.

Friction is not a gradient. A warp end crossing a pick that is being moved resists with μ times the force pressing there, and that resistance does not depend on how far the pick moves. Two picks sliding cost twice one pick sliding. Two hundred cost two hundred times.

$$F_{\text{friction}} = \mu \cdot N \cdot (\text{picks sliding})$$

per warp end, with N the contact force at a crossing. So the disagreement resolves cleanly: the zone is absent from the elastic force and is the whole of the frictional one, and the practical literature and this site’s arithmetic are each right about their own half.

The number is uncomfortably large #

At one sliding pick the frictional half is 1,133 newtons per metre of reed and the elastic half is 573. So even the smallest possible zone — one pick moving — makes friction two thirds of the whole force, and this site’s exact closed-form result is the minority term.

At twenty sliding picks the frictional half is 22,662 N/m and the total is over twenty-three thousand, which is more than an order of magnitude past anything a loom is built to apply.

That is a problem, and it is the interesting part of the rung rather than a defect. The fell region a weaver can see and measure is ten to fifty picks deep. If every pick in it were sliding against the warp, weaving would need forces nobody applies.

Reading it backwards #

Loom builders quote beat-up forces because they have to size a sley and a crankshaft against them, and for ordinary cotton weaving the figures are of the order of several hundred newtons per metre of reed, rising into the low thousands for a heavy or a very densely set cloth. That is a measurement, it was not taken for this purpose, and it is a ceiling.

Taking the ceiling seriously and solving for the count gives about one pick, and less than one at the bottom of the reported range.

The conclusion is not that the fell region is one pick deep. It is that the depth of the fell region and the number of picks still sliding are two different quantities, and the literature calls them by one name.

A pick two courses back from the fell can still be settling — its crimp still developing, its spacing still closing — without sliding along the warp. Settling is a change of shape; sliding is a relative displacement at a contact. The virtual-work argument prices the first and needs no zone; the friction argument prices the second and is nothing but zone; and the fell region is deep in the first sense and shallow in the second.

Which puts the previous rung’s ceiling in the right direction #

The rung that turned the beat-up force into a limit on pick density counted how much of a take-up gear’s catalogue a force ceiling cuts off — 548 of 4,825 selectable densities weavable at 200 N per metre, 3,244 at 500, and past 1,000 the binding limit stops being the machine and becomes the geometry.

Every one of those counts was computed from the elastic half alone. So each was a lower bound on the force and an upper bound on the catalogue, and the direction of the correction is now known: with friction in it the ceiling arrives sooner and fewer densities are weavable.

How much sooner depends on the count of sliding picks, which is exactly the quantity nobody can measure well. So the honest form of the corrected result is not a new number but a bracket, and the bracket’s width is a factor of three.

The split does not move with the warp tension #

The obvious lever a weaver has at the fell is the warp tension, and the obvious question is whether raising it changes the balance between the two halves. It does not, and the reason is worth following because it removes the one adjustment anybody would try.

The elastic half is the warp tension times the crimp’s elasticity, so it is proportional to the tension outright.

The frictional half is μ times the contact force at a crossing, and the contact force is the warp tension resolved through the weave angle — the same tension arriving and leaving, so the normal force is proportional to the tension as well.

Divide one by the other and the tension cancels. The ratio of the frictional half to the elastic half is μ times a function of the cloth’s geometry, and nothing else, so a weaver who doubles the warp tension doubles both halves and changes the split not at all. At one sliding pick the frictional half is two thirds of the total at every tension a loom can apply.

That has three consequences and they run in an unhelpful direction.

Tension buys a denser cloth and not a cheaper beat-up. Raising the tension raises the elastic force available at a given fell displacement, which is why a tighter warp weaves a closer cloth — but it raises the frictional cost in exact proportion, so the fraction of the loom’s effort going into heat and warp abrasion is fixed by the construction before the weaver touches anything.

The only lever left is μ, which is what sizing is. A size that halves the friction coefficient halves the frictional half at every tension, and it is the only input in the ratio anybody can change during weaving. That is a much sharper account of why sizing is indispensable than the usual one about protecting the yarn from abrasion — the abrasion is not a side effect to be protected against, it is two thirds of the force the loom is applying, and sizing is the only thing that reduces it.

And a denser cloth is worse in both terms at once. A closer sett raises the weave angle, which raises the contact force, which raises the frictional half; and it raises the crimp’s elasticity, which raises the elastic half. Both climb together with no compensating term, which is why a heavy construction runs into a force ceiling rather than approaching one asymptotically.

The energy reading is the same statement in another currency. About two thirds of the work done at the fell is dissipated rather than stored, at every warp tension, so the split between what the let-off gets back and what becomes heat in the cloth is a property of the construction and the size — not of how the loom is set. A weaver adjusting tension is moving both numbers up the same line and cannot move along it.

What it does to the picture of the fell #

The distinction between settling and sliding is worth drawing out, because it changes what a fell measurement is a measurement of.

A weaver identifies the beat-up zone by looking: there is a region behind the reed where the pick spacing is visibly wider than it will finally be, and the depth of that region is what gets quoted. That region is real and its depth is a genuine measurement. What it measures is how far back the pick spacing is still closing, which is a statement about geometry.

The quantity the frictional half needs is how far back a pick is still moving relative to the warp it crosses. A pick can have its spacing closed and still be under a warp that is itself creeping forward; more usefully, a pick can have stopped sliding while its crimp continues to develop, because crimp develops by the warp bending around it rather than by either thread translating past the other.

So the two depths measure different processes and there is no reason for them to agree. This rung cannot measure the second — nothing here can — and what it does instead is bound it, which is the only honest thing available.

What was counted, and how #

The two claims are asserted against each other rather than separately, and that pairing is the whole design of the check. At counts of 1, 10 and 100 sliding picks, the elastic half must be identical to twelve figures and the frictional half must be exactly proportional to the count. Either assertion alone would be weak; together they say that the zone left one term and entered the other, which is the finding.

A third assertion says that even one sliding pick outweighs the whole elastic half, which is the uncomfortable result and the one most likely to be quietly lost in a later edit.

The inversion runs over the friction range and the reported force range together, because both are ranges and a single number assembled out of two would be a fiction. What comes back is that the count is at most 1.23 across the whole grid, and that at the bottom of the reported range it is zero — the elastic half alone is above 400 N/m for a sheeting at 26 picks. That is not a failure of the model. It says the low end of the reported range belongs to lighter and more openly set cloths than this one, which is worth knowing and is the kind of thing a figure quoted to one significant figure hides.

The warp tension itself is not fitted: it comes from the site’s own yarn tensile model at a stated strain, and the contact force from the weave angle Peirce’s geometry solves for at the cloth’s construction.

What it says about where a loom’s power goes #

The split has a practical reading that the elastic half alone did not support.

If friction is two thirds of the beat-up force at one sliding pick, then most of the work a loom does at the fell is dissipated rather than stored. The elastic half is recovered — the warp stretches and relaxes, and the energy goes back into the let-off — while the frictional half becomes heat in the cloth and wear on the warp.

That reframes two things weavers know. Warp abrasion at the fell is not a side effect of the beat-up; it is the beat-up’s dominant term, which is why sizing exists, why an unsized warp is unweavable at any useful density, and why the back shaft works hardest. And the reason a loom’s power consumption rises so steeply with pick density is not that the elastic term is climbing — that term is well behaved until the cloth approaches jamming — but that the frictional term is multiplied by both the picks per centimetre and the force at each crossing, and both rise together.

Neither is computed here and both follow from the split rather than from any number in it. What the rung supplies is the ratio: at one sliding pick the frictional half is about twice the elastic one, at two it is four times, and every practical consequence of that scales with a quantity nobody can measure directly.

Where the model stops #

Static friction throughout, and a beat-up is a blow. The reed arrives at the fell at speed, the contact is impulsive, and what resists an impulsive slide is not the same coefficient that resists a slow one. The two coefficients this site now carries are static and kinetic, both quasi-static; neither is a dynamic one.

The contact force is the finished cloth’s. The pick at the fell is not in the state the finished cloth is in — the warp is under weaving tension and the weft is not, so the crimp at the fell divides quite differently from the crimp a centimetre back. Using the finished cloth’s weave angle overstates the turn the warp makes at a pick that has only just arrived, and therefore overstates the normal force. The direction of that error is known and its size is not; it is the largest single reason the frictional half here should be read as an upper bound at a given sliding count.

No lubrication. Warp is sized before weaving precisely to make it slide better, and sizing changes μ substantially. Every number here is for a cloth of unsized cotton on unsized cotton.

And nothing about the reed’s own friction. The reed’s dents slide along the warp too, and the warp is threaded through them under tension. That is a second frictional term with its own geometry and is not modelled at all.

The generalisation #

A cancellation is a property of a mechanism, not of a quantity, and adding a second mechanism can bring back what the first one removed.

The zone cancelled because the elastic work is a gradient of a potential, and potentials do not care how a displacement is distributed. It came back because friction is a dissipation, and dissipations care about nothing else. Both statements are about the kind of term, and either could have been written down before any arithmetic was done.

The useful form is a warning about elegant results. A quantity that drops out of an analysis has dropped out of that analysis, and the temptation is to report it as having dropped out of the problem. Here the difference matters a great deal: the cancelling term is the small one, and the term that keeps the zone is two thirds of the force.

There is a second lesson about names. The literature’s “beat-up zone” was doing two jobs, and nothing signalled it, because the two quantities are measured in the same units on the same piece of cloth and both are depths at the fell. It took a model in which one of them cancels to notice that the other does not.

Who found it, and when #

The virtual-work argument for the beat-up force is standard and old; the observation that the zone cancels from it is this site’s, from the rung below, and is checked there rather than asserted.

The frictional resistance of the fell region is treated in the weaving literature empirically, as part of what a fell displacement measurement reports, and is not usually separated from the elastic part at all — which is why the disagreement this rung resolves could persist: nobody had a model in which the two halves were separate enough to disagree.

Reported beat-up forces come from loom design practice rather than from fabric measurement, and their use here as a ceiling on a count is entirely parasitic — the same move the yarn stiffness ladder makes with a cantilever test.

Where the ladder goes next #

Sideways, the friction that dominates here is the same quantity that decides how far a cut edge frays and how a seam slips, and that rung’s account of both turns out to be the first term of a series.

The other half of the friction story is that there are two coefficients rather than one, which changes what a resting state is and matters most where a cloth is being agitated rather than pushed.

Along this ladder, the missing piece is a dynamic coefficient and an impulsive contact model, and neither is in prospect.