Two bars cannot share a beam

What a second guide bar is for answered the structural question: one bar leaves the wales in as many independent fabrics as its shog shares factors with the width, two bars leave the greatest common divisor of both, and a pair that each fail alone can succeed together. That is an argument about connectivity and it has no lengths in it.

The machine has lengths in it. A guide bar is fed from a beam, a beam delivers yarn at one rate, and if two bars want different rates then one of them is short. Whether they do is not a question about the fabric’s structure; it is a question about how much thread each lapping actually uses, and the answer turns out to need almost no modelling at all.

The overlap is the same on every bar, so it cancels #

A warp-knit bar does two things per course. It throws an overlap — the loop round its needle, drawn through the loop below — and it makes an underlap, the sideways run to whichever needle it goes to next.

The overlap is the same object on every bar. The same needle, the same loop shape, the same yarn, at the same course. A bar with a shog of one and a bar with a shog of three each lap one needle and each make one loop; the loops do not know what the guide did before or after.

So in the difference between two bars’ consumption, the overlap cancels exactly. That is the whole reason this arithmetic has no free parameter in it, and it is worth stating, because the loop is precisely the object this account has spent a field failing to compute from first principles: a knit’s dimensions come from its loop and the loop’s own length is measured rather than derived.

Here it is not needed.

An underlap is a hypotenuse #

An underlap runs from a needle on one course to a needle s spaces away on the next, and the shortest path between them is a straight line. So

$$u(s) = \sqrt{(s\,w)^2 + c^2}$$

with w the wale spacing and c the course spacing — both of which a warp-knitting machine sets directly, as a gauge and a course density — unlike a loom, where the warp sett is a reed and the pick density a gear and the two are chosen at quite different granularities.

At 28 needles an inch and 14 courses a centimetre, the wale spacing is 0.907 millimetres and the course spacing 0.714:

A chain lapping’s underlap is the course spacing alone, because it goes nowhere sideways — and that is consistent with it making a set of independent cords rather than a fabric, which the essay before it the last one established by a completely different argument.

What that comes to over a piece #

A hundred metres of fabric at 14 courses a centimetre is 140,000 courses, and every one of them costs each bar one underlap.

A cord bar wants a hundred and eleven metres more thread than a tricot bar, per wale, over a hundred metres of cloth. More than the piece is long. A satin bar wants more than twice the piece.

That is a larger number than it sounds, because it is per wale and a machine has thousands of them. It is also the number that decides the question, and it decides it in one direction only.

The failure is one course, not four millimetres #

Two layers need two beams is this account’s own version of the same argument on the woven side, and it is worth setting the two beside each other because the margins are not comparable.

In a double cloth, a plain face over a five-end satin back differs by twelve percentage points of crimp — twelve metres of warp over a hundred-metre piece, and one pick spacing of slack after four millimetres of weaving. That is fast enough to be fatal and slow enough to be surprising.

In a warp knit, the difference between a tricot and a cord bar is 0.795 millimetres a course against a course spacing of 0.714. A shared beam is one course spacing of slack out after 0.43 courses — before the first course is finished.

So the two cases are not the same result at different scales. A woven double cloth’s two layers differ by a crimp, which is a few per cent of a thread length; two warp-knit bars differ by most of an underlap, which is most of what distinguishes their lappings in the first place. The double cloth’s shared beam is a bad idea and the warp knit’s is not a thing that can be attempted.

Which is why every warp-knitting machine has a beam for each bar, and why a machine’s bars are described by their run-in rather than by their yarn. The beam is not an economy; it is a consequence.

The difference does not go away on a finer machine #

A finer gauge is a smaller wale spacing, so the obvious expectation is that the difference shrinks with the machine. It does per course and it does not per piece, because a finer machine also knits more courses in the same length.

The per-course difference falls by more than half from an 18 gauge to a 40 and the per-piece difference rises. The two effects — a shorter underlap and more courses in a metre — very nearly cancel across the ordinary range and then the second wins.

And the number that matters least to the machine builder moves least of all: a shared beam is a course spacing of slack out inside one course at every gauge in the table. There is no machine fine enough or coarse enough to make the shared beam work, which is a stronger statement than any single row of it.

How much more, as a ratio — and why that one is a bracket #

The difference is exact. The ratio — the number a machine is actually set by, the run-in — needs the overlap after all, because a ratio does not let it cancel.

$$\frac{\text{run-in of bar }s}{\text{run-in of the tricot}} = \frac{o + u(s)}{o + u(1)}$$

and o, the overlap, is the length this arithmetic declines to model.

Over overlaps from 1 to 5 millimetres — which brackets anything a loop at this gauge could be — a cord bar’s run-in is 1.15 to 1.30 times a tricot bar’s, and a satin bar’s is 1.32 to 1.63.

That is a useful answer even though it is a band, and it is the honest shape of it. The uncertainty is a tenth in the ratio and nothing at all in the difference, so a mill setting two beams by their lengths has an exact number and a mill setting them by their rates has a bracket. The first is what a beam is wound to and the second is what a machine’s gearing sets, which is an unfortunate division.

What a beam has to hold, which is a specification a mill can check #

The difference is per wale and a beam serves every wale, so the beam’s own size follows immediately and is the form a mill would use.

A tricot machine 130 inches wide at 28 gauge carries 3,640 needles, so a beam feeds 3,640 threads. Over a hundred-metre piece each tricot thread runs 140,000 × (o + 1.155) millimetres and each cord thread 140,000 × (o + 1.950).

Taking the overlap at 2 millimetres for the sake of a figure, that is 442 metres a thread on the front bar and 553 on the back — 1,609 kilometres of yarn on the front beam and 2,014 on the back, for one piece.

The two beams therefore differ in content by a quarter and they are the same physical size, which means the back beam runs out first and a mill changes beams at different times on the two bars. That is an operational consequence with a number on it, and it follows from nothing but the two shogs.

It is also the check a mill can make without stopping the machine. The run-in difference is the one quantity here that is exact, so a mill that measures the two beams’ consumption over a known length of fabric and finds a ratio other than the one its lappings imply has a fault rather than a preference — a slipping brake, a wrong gear, or a lapping that is not what the pattern chain says it is.

Which bar goes in front #

The arithmetic settles a practical question the connectivity argument could not touch.

A two-bar machine has a front bar and a back bar, and the front bar’s loops sit on the face of the fabric. The back bar should carry the longer shog, and there are now two independent reasons.

The structural one is the connectivity argument. Two shogs join the wales into $\gcd(s_1, s_2, \text{width})$ groups, so the pair only has to be coprime between them — which does not say which is which.

The yarn one is here. A longer shog is a longer underlap, and an underlap lies behind the overlaps of its own course. So the bar with the longer shog is laying more thread on the back of the fabric, and putting it on the front bar would put that thread on the face where it would show and snag.

And the beam one is here too. The back bar’s beam is larger, because it holds 25 to 60 per cent more yarn for the same piece, and a machine with the larger beam behind is a machine that can be threaded.

Three reasons and they agree, which is the sort of thing that makes a convention look like common sense rather than like arithmetic. It is arithmetic.

The bar that consumes least is the bar that makes nothing #

The underlap curve starts at the chain lapping, and the chain is the cheapest bar there is — its underlap is the course spacing alone, 0.714 millimetres, which is 38 per cent less thread than a tricot and 63 per cent less than a cord.

It is also the bar that makes no fabric. A thread that never leaves its wale makes a chain of loops and nothing else, so a cloth of chains is a set of independent cords — and the connectivity criterion this account is built on refuses it, by the same coprimality condition that refuses a six-end satin.

So the two questions a warp knit asks of a lapping run in opposite directions, and it is worth naming the trade because it is exact rather than approximate.

Connectivity wants a shog that shares no factor with the width, which at any ordinary width means a shog of one or more and rules out only the chain among the small ones.

Yarn wants the shortest shog there is, which is the chain.

So the cheapest lapping that makes cloth is the tricot, at one needle space, and every longer shog is buying something else — cover, weight, stability, a longer underlap lying across the back — at a cost this arithmetic now prices. A cord’s extra 111 metres a wale is not waste; it is the underlap the fabric is bought for, and the number says what it costs.

That reframes the essay before it’s result. Two bars found that a pair of shogs that each fail alone can succeed together, which is a statement about what is possible. This one says what each possibility costs, and the two together are what a designer actually chooses between.

What was counted, and how #

The underlap is a straight line between two needles on successive courses, which is the shortest path and therefore a lower bound on the thread. A real underlap is not straight — it lies over the loops of its own course and is pushed out by them — so every difference here is a floor, and the true difference is larger.

The gauge and the course density are the machine’s, 28 needles an inch and 14 courses a centimetre, which is an ordinary tricot. Both appear in the answer only through the wale and course spacings, and both scale it: a coarser machine has a larger wale spacing and a proportionally larger difference.

The overlap is not modelled and is not needed for the difference. That is the part of this essay worth carrying: an argument that needs a loop length is an argument this account cannot make from first principles, and the difference between two bars is an argument that does not.

The monotonicity is required — a longer shog is a longer underlap — because it is a hypotenuse and a sign error in one of the two terms would produce a plausible-looking table that ran the wrong way.

And the two headline claims are checks rather than observations: that the back bar wants more extra yarn than the piece is long, and that a shared beam is a course spacing out within two courses. Either failing would mean the arithmetic had changed rather than that the sentence needed softening.

What the arithmetic cannot say #

Nothing here is about tension. Two bars at the correct run-ins are still two threads at whatever tension their beams are braked to, and a warp knit’s fabric quality is notoriously a matter of the two tensions rather than the two lengths. The lengths decide whether the fabric can be made at all; the tensions decide what it is like.

The underlap length assumes both bars lap the same way. A machine can run its bars in opposition — one lapping left as the other laps right — which is how a balanced fabric is made, and the underlap lengths are the same but the two threads’ paths cross rather than lie parallel. That changes nothing in this arithmetic and changes a great deal about the fabric.

And the piece is taken as knitted at a fixed course density. A warp knit relaxes off the machine like any knitted fabric — the same move a woven cloth makes coming off the loom — so the hundred metres on the beam is not the hundred metres in the roll; the course density used here is the machine’s, which is the right one for a beam calculation and the wrong one for a specification.

Who found it, and when #

Run-in is the central setting of a warp-knitting machine and every practical account of the trade gives run-in values for named lappings — in millimetres of yarn per rack of 480 courses, which is the unit the industry uses. Those are measured figures, tabulated by machine and by lapping, and the tables have been in the manuals since the 1950s.

What the tables do not have is the reason the numbers differ, and the reason is one line: the overlap cancels. A published run-in table is a list of totals, so the loop is in every entry and nothing in the table suggests that the differences between entries are computable without it.

The comparison with the double cloth is this account’s, and it is the point of putting the two constructions in one place. Both are two thread systems fed at different rates from a machine that would prefer one; the woven case fails in four millimetres and the knitted one in less than a course; and the ratio between those two margins is the ratio between a crimp and an underlap, which is the difference between the two constructions stated in one number.

Still open: what a patterning bar does to the arithmetic #

Every lapping here is uniform — the same shog at every course — and the fabrics anybody buys are not. A patterned warp knit changes a bar’s shog from course to course, which makes its consumption a sequence rather than a constant, and a beam delivering a constant rate to a varying demand is the same problem this essay solved in the steady state.

The quantity wanted is the running difference between what the beam has delivered and what the bar has taken, over a pattern repeat. If the repeat’s mean matches the beam’s rate the fabric is stable; the excursion within the repeat is what has to be absorbed, and it is absorbed by exactly the compliant element a leno’s easer is on a loom.

So a patterning bar needs a tension compensator and a uniform one does not, and how large it has to be is the sum of the underlap differences over the repeat — which is this arithmetic run over a sequence rather than over a constant, and is a few lines rather than a new model.