The blow that sets the pick

The take-up ladder settles a cloth’s weft density by counting gear teeth. The setts a loom can reach enumerates the pick densities a change-wheel train can select — 4,825 of them between 8 and 40 threads per centimetre — and the reed is not the sett works out how much narrower the reed is than the cloth it makes.

Both are statements about a mechanism that decides where the cloth goes. Neither says anything about the mechanism that decides where the pick goes, which is the reed swinging forward and driving the newly inserted weft against the fell. That is the beat-up, it is a force, and the take-up ladder recorded its absence as a shortfall in as many words: the take-up ladder sets the pick spacing by a gear ratio and says nothing about the blow that drives the fell.

The claim #

The beat-up force per warp end is the warp tension multiplied by the elasticity of the crimp with respect to the pick spacing:

F = −T · p · dc₁/dp

and the length of the beat-up zone does not appear in it.

For an ordinary sheeting at its quoted construction that is 0.20 N per end, which over a 1.5-metre warp of 4,200 ends is 859 N — about 570 newtons per metre of reed. Reported beat-up forces on cotton looms are hundreds of newtons per metre, and nothing above was fitted to make that so.

The argument, which is four lines of virtual work #

Let the beat-up zone be z long and hold cloth at pick spacing p, so it contains z/p modular lengths of warp and z·(1 + c₁) of warp thread per end.

Push the fell forward by dx. The zone keeps its length and its picks are squeezed closer, so the spacing falls by p·dx/z. The warp thread the zone now needs is a different amount, and the difference is

dL = (1 + c₁ − dl₁/dp) dx

The z has cancelled. The reed does work F·dx against the warp tension paying that length out, so per end F = T·(1 + c₁ − dl₁/dp), which rearranges to the form in the claim.

Two things about this deserve more attention than the number it produces.

The zone drops out. Every practical account of beat-up treats the beat-up zone — how many picks back from the fell the compression reaches — as a quantity to be measured, and it is a hard one to measure. It is not in the answer. The machinery does not merely say so: it computes the same force from a finite zone of 2, 20 and 200 picks by differencing the total warp length as the fell advances, and requires the three to agree with the closed form. They agree to five decimal places, which at these magnitudes is the finite-difference step and not the physics.

The force is about the crimp changing, not about the crimp. It is the price of being cloth charged as a rate rather than as a total, and the rate is a property of where on its own geometry the cloth is sitting rather than of how much thread it has taken up. A cloth with a great deal of crimp that is not changing costs nothing to beat, and a cloth approaching jamming costs without bound. That is why the factor in the expression is an elasticity — a proportional change over a proportional change — rather than a crimp.

There is a third thing worth saying, and it is about what kind of quantity the answer is. The beat-up force is not a property of the cloth being made; it is a property of the cloth being made at the spacing it is being made at. The same construction beaten to twenty picks per centimetre and to twenty-six is the same yarn, the same weave and the same reed, and the two forces differ by more than a factor of two. Nothing on a specification sheet distinguishes them, and what comes off the loom is where the difference shows.

The numbers, and how they run #

Four cloths at their own quoted constructions, with the warp held at one per cent strain:

The spread is a factor of twenty, and it is not a spread of tension. The weave decides how closely a cloth may be set, and what it is really deciding here is how steep the climb is at the sett a maker has chosen. The warp tensions differ by a factor of two across those four cloths; the crimp elasticities differ by a factor of nine, and that is where nearly all of it comes from.

The ordering also inverts something that looks obvious. A duck is a heavy canvas in coarse yarn and a sheeting is a lighter cloth in finer yarn, and the sheeting takes the larger force per metre — because it is being woven much closer to its own jamming point, where the elasticity is steep, while the duck at 15 picks per centimetre still has room.

Why the force runs away #

Peirce’s geometry has a hard edge. As the picks are brought closer the warp must take more and more crimp, the crimp height rises towards the full thickness of the cloth, and at some spacing there is no solution at all — the circular sections ask for more room than the spacing leaves.

Approaching that edge, the crimp’s derivative with respect to the spacing grows without bound, and so does the force. A loom beating a cloth close to jamming is fighting an asymptote.

The practical form of it is the thing every weaver knows: the last few picks per centimetre cost far more than all the ones before. For a sheeting the force rises from 139 N per metre at 8 picks per centimetre to 613 at 27.5, and the curve is convex the whole way — the second half of that range costs more than twice what the first half did.

The open end of the range is where the budget stops binding at all, and it is worth seeing beside the other two.

What decides a cloth’s weft density, really #

Putting this beside the take-up ladder changes the answer to a question that ladder appeared to have settled.

A change-wheel take-up selects a pick density from a catalogue of 4,825 values. It does not deliver one. What it does is move the cloth forward by a fixed amount between picks; where the pick actually ends up is decided by the fight between the reed and the fell, and if the reed cannot drive the pick home the fell simply sits further forward and the cloth comes out more openly set than the gear says.

So the weft density is the lesser of two things: what the take-up asks for and what the beat-up can achieve. On an openly set cloth the take-up is binding and the gear is the answer; on a closely set one the force is binding and the gear is a request. Which regime a given cloth is in is a computation, and it is the next rung.

That also explains a piece of loom-room lore that sounds like superstition. Raising the warp tension makes a cloth beat up closer — and the expression says why, since T multiplies the whole force: a harder-held warp lets the reed reach a spacing a softer-held one cannot. The cost is paid in what the shed is already spending of the warp’s strength, and the two are competing for the same budget.

What was counted, and how #

The chain is short and each link is checked.

The state is Peirce’s solution at the cloth’s construction, verified by feeding it back through its own equations; the worst residual is at machine precision.

The derivative is a central difference on the pick spacing, at a step of one part in ten thousand — small enough that the third derivative does not matter and large enough that the bisection inside the Peirce solver, which is exact to about one part in ten trillion, does not dominate it.

The force is computed two ways from the same state: once as T·(1 + c₁ − dl₁/dp) and once as −T·p·dc₁/dp. Those are algebraically the same and are computed by different routes, and the machinery requires them to agree to one part in a million before it will return.

The zone independence is checked rather than argued, as described above: three zone lengths spanning a factor of a hundred, all agreeing with the closed form.

The tension is the one input that is not derived here. It is set by a stated warp strain — one per cent, which for a 25 tex cotton yarn is 1.13 N — and every force on this page is proportional to it, so a reader who prefers a different warp tension can rescale the whole page by one multiplication.

Where the model stops #

The cloth’s own resistance is missing, and it is not small. Beyond stretching the warp, the reed has to push the new pick past the picks already woven, which are gripped by the warp and resist sliding. That is a frictional force, it is the other half of what the trade calls weaving resistance, and this model has none of it. The numbers here are therefore a lower bound on the beat-up force, and the amount by which they are low is exactly the quantity the crossing-force ladder would be needed to supply.

The state at the fell is not the state of the finished cloth. At the fell the warp is under weaving tension and the weft is not, so the crimp divides quite differently there from a centimetre back. The crimp ratio is therefore an input to this computation rather than a result of it, and the sensitivity to it is real.

That input has a hard edge, and it is worth stating because finding it was a defect rather than a design. Past a certain pick density Peirce’s geometry cannot divide the crimp in the ratio asked for at all: the solver returns the closest state it can reach, which is at a different ratio, and the crimp derivative there has the opposite sign. The first version of this computation therefore produced a perfectly plausible negative beat-up force on a muslin above thirty-six picks per centimetre — a reed pulling rather than pushing — and nothing about the number looked wrong. It was caught by the assertion that a sweep must rise. The machinery now checks the ratio it actually reached against the one it was asked for and refuses a state that does not hold it, so the model has three ways of running out rather than two: a force ceiling, a geometric wall, and the crimp-ratio assumption failing. For most cloths in the table the third is the one that arrives first.

Nothing here is dynamic. A beat-up is a blow: the reed arrives with momentum, and what the sley delivers is an impulse rather than a static push. A quasi-static force is the right thing to compute first and it is not the whole of what happens in the two milliseconds the reed is at the fell.

And the warp is treated as a linear spring. It is not, over the range that matters, for the reason the shed rung sets out: a real yarn’s load–extension curve is concave, so a straight-line modulus overstates the tension at large strains and understates the extension available.

A large force doing almost no work #

The force is the headline and it invites a question it does not answer: what does the beat-up cost a loom to run? The answer is a multiplication and it comes out in the opposite direction from what the force suggests.

Work is force times distance, and the distance the beat-up force acts over is short. The reed’s push only becomes hard in the last part of its travel — the fell has to advance by one pick spacing per pick, which for a sheeting at twenty-six picks per centimetre is 385 micrometres. Taking the force at its full value over that distance, which over-estimates it,

the beat-up does about 0.33 joules per pick.

At four hundred picks a minute that is two and a fifth watts. A loom’s motor is rated in kilowatts, so the work that actually goes into driving the picks home is a fraction of one per cent of what the machine draws. Everything else is the sley’s own inertia, the shedding, the take-up, the let-off and the losses.

The same arithmetic in the form a mill would want it: work per pick divided by pick spacing is just the total force, so the beat-up costs 859 joules per metre of cloth woven, or 573 joules per square metre — irrespective of how fast the loom is run, and irrespective of the pick density, since the force rises as the spacing falls and the two very nearly cancel.

Which is why a loom is built the way it is #

That combination — a very large force doing a very small amount of work — is unusual, and it explains the shape of the machine better than the force alone does.

A loom is force-limited, not power-limited. Nearly nine hundred newtons across the reed, delivered in about two milliseconds, from a motor that only has to supply two watts on average. No motor is sized for that; a motor sized for the peak would be enormous and idle for the rest of the cycle.

So the sley is a flywheel. The energy comes from the sley’s own kinetic energy, accumulated over the whole cycle and spent in the instant the reed reaches the fell. A heavy sley is not an engineering compromise forced by strength; it is the store that makes a small average power deliver a large instantaneous force. That is why sley mass is a design variable on a loom and why raising a loom’s speed is a problem about inertia rather than about power.

And it explains why the beat-up is the hardest part to model and the cheapest part to run. Every difficulty in this essay — the asymptote, the geometric wall, the missing friction — is about a force at an instant. None of it is about energy, and a designer who costed a loom on its energy budget would conclude the beat-up was negligible and be wrong about everything that matters.

The comparison worth carrying is with what the same cloth costs elsewhere. Two watts is less than a domestic light. The machine around it weighs a tonne, runs at several kilowatts, and exists almost entirely to arrange for that two watts to arrive as nine hundred newtons at the right eight-thousandth of a second.

A mechanism can be dominated by a quantity that is negligible in the energy accounting, which is the same lesson the zone cancellation teaches from the other end: the terms that decide a machine are not always the terms that appear in its budget.

The generalisation #

The move that makes the zone cancel is worth extracting, because it is not about weaving.

When a distributed deformation is driven by advancing a boundary, the force is a derivative of a stored quantity with respect to the boundary’s position, and any length over which the deformation is spread cancels — provided the deformation is uniform in that length. Squeeze a stack of springs at one end and the force does not depend on how many springs are involved, only on how much each is compressed per unit of advance. Push a wave of compression into a granular bed and the same is true.

That is why an apparently unmeasurable parameter turned out not to be needed. The beat-up zone is real, it can be observed on a loom, and papers have been written measuring it — and it is a property of how the deformation is distributed, not of how much work advancing the boundary takes. Those are different questions and only the second one has a force as its answer.

The corollary is a warning about the model’s own scope: the cancellation holds because the zone was assumed to compress uniformly. A real fell does not, and the correction to that assumption is exactly where the zone would re-enter.

Who found it, and when #

Beat-up mechanics belongs to the loom-instrumentation work of the 1950s and 1960s, and Greenwood and Cowhig’s papers on the position of the cloth fell are the standard reference. That work measures the fell’s position through the weaving cycle, relates it to the warp tension and the take-up, and establishes the thing every subsequent account repeats: that the fell moves, that its position is a balance, and that a loom’s cloth is set by that balance rather than by the gear train.

The virtual-work form here is elementary and is the kind of thing that is either in an unpublished appendix or was never worth writing down separately. What is this site’s is the observation that the zone cancels — and the decision to check that by computing the same force from finite zones spanning a factor of a hundred, rather than trusting the algebra that produced it.

Where the ladder goes next #

The next rung asks what a force ceiling does to the take-up gear’s catalogue: a pick density is a force budget, and only some fraction of the 4,825 settings the machine can select are settings it can reach.

Sideways, the missing frictional half of the weaving resistance is the crossing force applied at the fell rather than in a finished cloth, and the warp tension the whole expression is proportional to is what the shed is already spending.

Further out is the dynamic problem: an impulse rather than a force, a sley with inertia, and a fell that moves during the blow. That needs a mass and a stiffness for the loom itself, which is a machine model rather than a cloth model, and this site has neither.