A force is what an energy does when a crossing moves

A force is not a separate thing from an energy. It is what the energy does when something moves: displace a point by a hair’s breadth, see how much the stored energy changes, divide one by the other. That is the definition, and everything on this ladder is an application of it.

The awkward part is that it looks like a recipe for differencing, and differencing a minimisation is a poor way to get anything. Each evaluation is a solve; each solve has its own convergence tolerance; and subtracting two nearly equal numbers throws away most of the digits both of them had. A force obtained that way inherits every weakness of the solve and adds one of its own.

The number is already there #

It does not have to be differenced, because the solve is a constrained minimisation and its constraints carry the answer.

The statement being solved is: minimise the bending energy over all shapes, subject to the thread arriving at a stated point. Handle that with Lagrange multipliers and a pair of numbers appears alongside the shape — one for each coordinate of the endpoint condition. Those multipliers are not bookkeeping. A multiplier on a constraint is the derivative of the constrained minimum with respect to that constraint’s value.

Which is exactly the force. The thread’s end is held where it is by something, and the multipliers say how hard.

Why that is not a trick #

The identity is worth a sentence because it looks like one. Setting up the problem as minimise E subject to g equals v, the Lagrangian is E plus λ times (g minus v), and differentiating the whole thing with respect to v at the solution leaves −λ. Everything else cancels, because the shape is already at a stationary point of the Lagrangian and moving it contributes nothing to first order.

So the pair falls out of the same small matrix solve that produces the shape, at no extra cost and with no subtraction of nearly equal numbers. It is exact rather than approximate, and it is available at every configuration the solver is asked for — which is what makes it possible to plot a force against something rather than quote one.

The sign, which is the only place to go wrong #

Once. The multiplier is minus the sensitivity, so the sign flips exactly once between the solve and the force, and a sign error there is invisible: a contact force with the wrong sign is still a plausible number of millinewtons pointing the wrong way.

So the flip happens on one line and nowhere else, and it is checked by differencing the energy anyway — not because differencing is a good method, but because it is an independent one and agreement between two methods is worth more than confidence in either. The two agree to four parts in ten million.

The internal force is constant #

A rod carrying no load between its two ends transmits the same force all along. Nothing is pushing on it in the middle, so nothing can change what it is carrying.

That has a use immediately. The multiplier is the force at the end the solver was asked about, and the force at the other end therefore has the same magnitude. A thread’s span between two crossings is one object, and knowing what it carries anywhere is knowing what it delivers at both ends.

Recovering the same force from the shape #

The strongest check available does not use the multipliers at all.

A rod’s moment balance says that its bending moment plus a linear function of position is constant along it, with the coefficients of that linear function being the components of the internal force. So take the solved curvature, fit a plane to it against position, and read the force off the fitted coefficients. Nothing in that route touches the multipliers, and nothing in the minimisation ever saw the moment balance — all it knew was a quadratic form and two endpoint conditions.

The two forces agree to a tenth of a per cent at the largest basis, and the fit’s own residual halves every time the basis doubles. That second half matters as much as the first: a relation that only nearly holds would agree at one basis size and no better at the next.

What the force at a crossing actually is #

The last step is the one that makes the number usable, and it is a symmetry argument.

At the crest of a thread’s wave the two spans on either side meet, and they are mirror images. So the forces they deliver are mirror images too: the components along the fabric are equal and opposite and cancel, and the components across it are equal and add. The normal load on the crossing is therefore

$$N = 2\,\lvert n_{\perp}\rvert$$

twice the transverse component of the internal force, and nothing else.

The shape of that is not new #

It is worth stopping on, because it is the same shape as the force this collection has been computing since its first essays: a thread under tension T arriving at θ and leaving at −θ presses with 2·T·sin θ.

Twice something transverse, in both cases. In a cloth under load the transverse something comes from the tension turning a corner. In a relaxed one it comes from the thread’s own bending turning the same corner. Nothing else about the crossing changes — not the geometry, not the friction, not the capstan arithmetic that follows from it. One term is replaced and every consequence downstream survives intact.

That is a good sign rather than a coincidence. A new quantity that required every existing result to be rebuilt would be a suspicious quantity.

What is being pressed, and by what #

A caution, because the arithmetic is easy to over-read. The force here is the load one thread puts on another through the contact. It is not a tension in either thread, it is not a pressure over an area, and it does not have a direction chosen by anybody: it points along the line joining the two centre lines, because that is the only direction a smooth contact can push in.

What it is for is friction. A normal load times a coefficient is a resistance to sliding, and sliding is what fraying, seam slippage, tuft withdrawal, fibre shedding and a run in a stocking all are. The whole of the second half of this ladder is that one multiplication.

Everything downstream is untouched #

The point of a replacement that keeps the shape of the thing replaced is that nothing built on it has to be revisited, and it is worth checking rather than assuming.

A thread is held one crossing at a time, the grip in a gripped length goes as the length while the thread’s own breaking load does not, and the two curves cross at a length that decides whether a thread slides out or snaps. Every step of that survives verbatim. The capstan arithmetic survives. The criterion that cannot see friction still cannot see it.

What changes is that all of it becomes available for a fabric nobody is pulling, which is most fabrics most of the time.

A force per stitch is not a force per metre #

One conversion, stated once, because getting it wrong scales an answer by a thousand.

The multiplier is a force at one thread’s end. The load on one crossing is twice its transverse component. The force a fabric exerts along one of its directions is that per stitch, divided by the spacing of the stitches in the perpendicular direction — because a fabric’s edge of a given length carries as many of them as fit across it.

So a stitch pressing with tens of millinewtons and a fabric pulling with tens of newtons per metre are the same statement, and the ratio between them is a spacing of a fraction of a millimetre. Both are quoted here, always with their units, and never mixed.

The bracket comes along too #

Every force here is a bending rigidity divided by a length squared, and a spun yarn’s bending rigidity is a bracket rather than a number — the fibre count over the square of the packing factor, which for an ordinary cotton is a factor of a hundred and thirty.

So a force quoted here is quoted at the free bound and called what it is. The claims that carry weight are ratios: two forces computed the same way, in which the bracket cancels exactly and the comparison survives however badly the rigidity is known. Where a single figure is unavoidable it is said to be one, and the direction of its error is stated.

What the picture cannot show #

Not the force. A force has no extent, and the spines in the figures are curvature — the moment, in fact, once the rigidity is multiplied in.

The force is the slope of that field along the thread, and no static drawing carries a slope in a way a reader can measure. What the figures can show, and do, is that the field has no step in it, which is the property that makes the slope exist. Beyond that, the number has to be read rather than seen.

Two independent numbers, not one #

There is a second multiplier and it is not the contact force. The endpoint condition has two coordinates, so the solve returns a force along the fabric as well as across it, and the two are different quantities with different uses.

The transverse one is the contact force. The one along the fabric is the force the thread’s own bending exerts on the fabric, tending to change its spacings — and it turns out to be the more interesting of the two, because it does not vanish where it should. Both are returned everywhere and neither is derived from the other.

The number that is not a force #

One more caution about what the multipliers are not. Neither of them is a tension in the thread.

A thread in a relaxed fabric may be under a residual tension left over from how it was made, and that is a real quantity with its own consequences — it is what a cloth relaxing until its threads stop pushing is about. It is also not what is computed here. The internal force in an elastica is whatever the shape requires, and for a thread that is merely bent into place it is small, transverse and has nothing to do with how hard anybody pulled the yarn.

The two can be told apart by an experiment rather than by argument: cut the thread. A residual tension releases and the fabric moves; a bending force does not release, because the bending is still there in both halves.

What it costs #

Nothing measurable. A solved thread with its two forces takes under a millisecond, which is the reason this ladder can sweep a fabric’s whole state space instead of solving one configuration and arguing about the rest.

That is worth stating because the alternative shaped everything before it. A quantity that costs a minute to obtain gets quoted once and reasoned about; a quantity that costs a millisecond gets plotted, and a plot of a force against a fabric dimension says things no single value ever does.

Both threads at a crossing, not one #

A crossing has two threads in it and the arithmetic above treats one. The other has its own span, its own length, its own spacings and its own multipliers, and there is no reason at all for the two to deliver equal loads.

At a real contact they must balance, of course: the load one thread puts on the other is the load the other puts back. What that means is that the shape is not free — the crimp heights the two systems take are whatever makes the two answers agree, which is the equilibrium condition this collection has been solving on the woven side since it acquired a crimp energy at all. The forces here do not replace that condition. They make it a statement about forces rather than about a stationary energy, which is the same statement said in the way a reader can check.

What the balance says about how the crimp divides #

The section above says the two threads at a crossing must deliver equal loads and that this makes the shape a consequence rather than a choice. Written out with a beam’s own scaling, the consequence is an explicit rule, and the rule reproduces three things the trade has always known.

A thread spanning between crossings behaves, to first order, like a beam of rigidity B deflected by its crimp height h over a span equal to the other system’s spacing. Its transverse force is then proportional to Bh/p³. Setting the warp’s equal to the weft’s,

B₁h₁ ÷ p₂³ = B₂h₂ ÷ p₁³,

and with the closure condition fixing h₁ + h₂ to the combined diameter, that determines both. The ratio is

h₁ ÷ h₂ = (B₂ ÷ B₁) × (p₂ ÷ p₁)³.

Three readings, and each is a rule somebody already had.

A balanced cloth of one yarn divides its crimp equally. Both rigidities and both spacings are the same, so the ratio is one. That is the standard result and the arithmetic gives it for nothing.

The densely set system crimps more, as the cube of the sett ratio. A cloth with twice as many ends as picks has p₂/p₁ = 2, so the warp takes eight ninths of the cloth’s thickness and the weft one ninth. The crowded system does nearly all of the bending — which is why a warp-dense shirting has a nearly straight weft, and why the crimp figures for such cloths are so lopsided.

The finer yarn crimps more, as the fourth power of the diameter ratio. Bending rigidity goes as the fourth power of a diameter, so at equal setts h₁/h₂ = (d₂/d₁)⁴. A warp only twenty per cent coarser than its weft takes barely a third of the crimp, and the finer system does two thirds of it — an enormous response to a small difference in count, and exactly the rule a weaver states as the softer yarn does the bending.

Which is why crimp is so hard to specify #

The fourth power is the useful surprise here, and it says something about why crimp figures scatter so badly between nominally identical cloths.

Every other quantity in this collection responds to a diameter ratio linearly or as a square. Crimp division responds as the fourth, so a five per cent difference between two yarns’ effective diameters — well inside what a spun yarn’s own variation supplies — moves the crimp division by twenty per cent.

A crimp ratio is therefore the least reproducible quantity a cloth has, and it is the input every geometric model needs and none of them derives. The crimp ratio is not a measurement makes that complaint from the other side; this says why the underlying quantity is so sensitive.

It also gives the specification a maker should write instead. Since the division depends on the two rigidities and the two spacings and on nothing else, quoting the two counts and the two setts determines the crimp division without measuring it — which is a conversion nobody performs, and which would replace a destructive measurement with two numbers already on the ticket.

The caveat is the beam. The scaling above is the small-deflection form, and a real crimp is a large deflection where the exponents soften: the cube and the fourth power are upper bounds on the sensitivity rather than values, and the full solve this ladder uses would give something a little gentler. The three orderings are robust and the three exponents are not, which is the same division this collection draws everywhere between what a shape argument gives and what a solve gives.

Where the ladder goes next #

The force exists and has been checked twice. The next question is where it is worth having, and the answer divides sharply.

A woven thread turns out to have no room to be an elastica at all — its whole crimp is spent going round the thread it crosses. A knitted loop has half its length spare and is free over all of it. The next two rungs measure both, and the gap between them is larger than anything that follows would suggest.

What would falsify it #

The claim is narrow enough to be worth stating as something that could fail. It is that the transverse multiplier returned by the solve is the load one thread puts on another in a fabric under no external load.

Three things would break it. If the moment-balance fit and the multiplier disagreed by more than the basis error, the solve would be returning something that is not an equilibrium. If the differenced energy and the multiplier disagreed, the identity would have a sign or a factor wrong. If the force did not converge as the expansion grew while the energy did, the shape would be right and its slope would not be.

All three are checked every time the site is built, and the first two are checked against tolerances tight enough that a factor of two could not hide in them. That is the difference between a computed quantity and an asserted one, and it is worth the arithmetic.

Who found it, and when #

The variational identity is Lagrange’s, and older than any of the textile arithmetic here by a century and a half. The moment balance is the standard statics of a rod. Neither is new, and neither was hard to apply.

What was missing was the shape to apply them to, and that is worth remembering when reading the concessions this collection has made about relaxed fabrics. The obstacle was never the mechanics. It was that a path built by hand from arcs and straights has no derivative, and every route to a relaxed contact force went round the problem rather than through it.