Opposed bars even a warp knit by locking it

A patterned warp knit needs a stroke as long as its stripe followed one guide bar through a stripe of six tricot courses and six satin ones and found what its beam has to put up with: a bar whose demand swings from course to course, against a beam that delivers at one rate, runs up a shortfall and pays it back every repeat. The range of that swing, the stroke, came to 4.98 millimetres a thread at 28 gauge, and it was set by the stripe’s height and by nothing about the machine.

That essay ended on a second bar. Real patterned warp knits are rarely one bar; the commonest have two, and a fabric with two patterning bars can be arranged so that each bar’s long underlaps fall on the courses where the other’s are short. Then the fabric’s total underlap per course is constant even though neither bar’s is, and the obvious hope is that a cloth whose every course takes the same yarn knits and relaxes more evenly than one whose stripes coincide.

The hope is half right, and the half that is wrong is the interesting half. The opposed cloth does come out even, course for course, under any pull. But the constant total is not the reason, and the evenness is not free.

What each beam still has to absorb #

Start with the machine, because the beams are where the previous account left off and they are the part the arrangement does not touch.

Each bar is fed from its own beam. Two bars cannot share a beam found that a pair of uniform bars at different shogs want different amounts of yarn per course, so a shared beam runs a course spacing out of step within a course. Each patterning bar here makes six tricot courses and six satin courses in its repeat, in some order, and so each wants exactly the same yarn per repeat as the other. Its beam is set to that repeat’s mean, and its running shortfall is the sum of each course’s underlap less the mean. That sum depends on that bar’s own shogs and on nothing the other bar does. Moving the back bar’s repeat along the front’s changes when the back bar’s satin courses arrive; it does not change how many arrive in a row.

So the back bar’s stroke is 4.98 millimetres at every offset from coincident to opposed. In opposition its shortfall is exactly the negative of the front bar’s, course for course, and that symmetry buys the beams nothing, because two beams cannot lend each other length. A compensating rail across both sheets would have to slacken the front threads and tighten the back ones at the same moment, which a rail cannot do; each bar keeps its own compensator, and each compensator keeps the stroke the single-bar account gave it.

The only thing the opposition changes at the machine is the total. Two sheets whose demands are mirror images draw, between them, the same length every course — 3.97 millimetres of underlap a wale, against 2.31 where both bars lap tricot and 5.63 where both lap satin in a coincident cloth. No let-off sees only the sum, so at the machine this is a curiosity; it is real only in the fabric.

Where the course total does matter #

In the fabric the total is a weight. A course’s yarn, per wale, is two loops and two underlaps; the loops are the same on every course, and the underlaps are whatever the two bars lapped. So in a coincident cloth the satin band carries more yarn per course than the tricot band, by an amount that depends on how long a loop is taken to be. The single-bar essays never gave the loop a value, because it cancelled from every quantity they wanted. It does not cancel from a ratio, so here it is bracketed: with each loop between a millimetre and a half and three millimetres, a coincident satin course carries 1.4 to 1.6 times the yarn of a tricot course.

That is a visible band. A course with half as much yarn again in it is a denser, more opaque, heavier course, and six of them together make a stripe of weight across the back of the cloth — the same kind of band why a knit shows a thick place found a single course of thicker yarn making in a weft knit, multiplied by six. In opposition every course carries the same yarn, 1.20 to 1.31 times a tricot course on the same bracket, and the weight stripe is gone.

So the total does what the hope expected for the cloth’s weight and cover. It does not decide how the cloth gives when pulled, which is what “knits more evenly, or relaxes more evenly” was really asking.

Two circles, one course #

The shog is the anisotropy reduced a warp knit’s first give to one fact. An underlap is a straight run from a needle on one course to a needle some spaces along on the next; its length is fixed by the yarn, and the two spacings it spans are not. So the wale spacing $w$ and the course spacing $c$ that one underlap of shog $s$ allows lie on a circle,

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

with the as-knitted fabric at one point of it. Pull the fabric wider and the underlap turns towards the horizontal and pulls the courses together; narrow it and the courses can open. That essay also gave the rule for two uniform bars: a fabric with two loci can move only where both allow, so each bar stabilises the direction the other is loose in — the long shog stops the width, the short shog stops the length.

A patterned fabric has to apply that rule course by course, because its courses differ. Both bars’ underlaps on course $k$ run from course $k$ to course $k + 1$, so they share that course’s spacing $c_k$. Every course shares the fabric’s one wale spacing, because a wale runs the length of the cloth. So at a stated width each course can open only as far as the tighter of its two circles allows:

$$c_k = \min\Bigl(\sqrt{u(s_k)^2 - (s_k\,w)^2},\ \sqrt{u(t_k)^2 - (t_k\,w)^2}\Bigr),$$

with $s_k$ and $t_k$ the two bars’ shogs on that course. At the knitted width every circle passes through the knitted spacing, so every course starts in the same place. Away from it, the courses part company.

The two pure courses each follow their own circle. The mixed course follows whichever circle is lower, on each side. Narrowed, a tricot underlap lets its course open slowly and a satin underlap lets it open fast; the course can only open as far as its tricot underlap allows, so it opens like a tricot course. Widened, a satin underlap closes its course fast and a tricot underlap slowly; the satin closes it first, so it closes like a satin course. A course with one of each takes the stiffer response in both directions, which is the uniform-bar rule with the bars swapping roles at the knitted point.

Held at four fifths of its knitted width — roughly what a lengthwise pull narrows a fabric to — a course of two tricot underlaps can open to 1.26 times its knitted spacing and a course of two satins to 2.50; the mixed course, to 1.26. Held a fiftieth wider, a tricot course closes by 3 per cent, a satin course by 36, and the mixed course by 36. The satin underlap lies flat at 1.034 times the knitted width; at that width every satin course and every mixed course has closed to nothing, and the tricot courses still keep 94 per cent of their spacing.

What a pull does to coincident and opposed stripes #

That is enough to draw the cloth. In a coincident fabric the twelve courses of a repeat are six of one pure kind and six of the other. In an opposed fabric all twelve are mixed.

Coincident, the stripe comes back as a stripe of stretch. At four fifths of the width the tricot band has opened by a quarter and the satin band by a factor of two and a half, so a repeat that was knitted as two bands of equal height is now a short band and a long one, and the whole repeat is 88 per cent longer. A fiftieth wider, the tricot band has hardly moved and the satin band has lost over a third of its height; the repeat is a fifth shorter, nearly all of it taken from one band. Opposed, there is no band to move. Every course opens by a quarter when the cloth is narrowed and closes by over a third when it is widened, so the repeat keeps its proportions in both directions. That is the evenness the earlier account hoped for. But both of the opposed fabric’s numbers are the worse of the coincident fabric’s two bands: it opens like the tricot band and closes like the satin band. The opposed cloth is even because it is locked, by its tricot underlaps against lengthening and by its satin underlaps against widening, on every course at once.

Against the coincident cloth it gives up most of its length: 26 per cent at four fifths of the width against 88, and it loses 36 per cent of its length to a widening that costs the coincident cloth 20. These are first gives, before the loops begin to straighten — the shog is the anisotropy placed the loops’ contribution in series after the underlaps’, and nothing here moves it — but the first give is the part a wearer feels at small strains, and it is the part what stops a knit extending found opposed only by friction.

It is not the total #

The opposed cloth’s courses carry identical totals and move identically, and it would be easy to read the second as a consequence of the first. It is not, and a pair of courses shows why.

A course whose bars lap tricot and satin carries 1.155 and 2.814 millimetres of underlap, 3.968 in all. A course whose bars both lap cord, two needle spaces each, carries 1.950 and 1.950, 3.900 in all — 1.7 per cent less underlap, as near the same weight as two courses of different structure can be. Narrowed to four fifths, the first opens to 1.26 times its spacing and the second to 1.82. Widened a fiftieth, the first closes to 0.64 and the second to 0.86. A cloth alternating these two kinds of course would be even in weight to within two per cent and would pull into bands in both directions.

The reason is in the formula. A course’s spacing is the minimum of two circles, and a minimum does not care about a sum. What decides it narrowed is the course’s shortest shog, and what decides it widened is its longest. Two courses give alike only when they have the same shortest shog and the same longest, and for two bars that means the same pair. A course of chain and satin, which carries less yarn than either of the others, cannot lengthen at all — its chain underlap is the course spacing itself — and closes widened as a satin course does.

So equal totals are necessary but not sufficient: with two bars, the same pair on every course carries the same yarn, but the converse fails as soon as more than two shogs are in play.

Exactly one back ordering for every front #

The single-bar census counted every ordering of six tricot and six satin courses — 924 of them — by the stroke each asks of its beam. With two bars the question turns round: given a front bar’s ordering, how many of the 924 orderings of the back bar’s courses make every course give alike?

The answer is the same for every front ordering, and it is one. Every course has to pair a tricot underlap with a satin one, so wherever the front laps satin the back must lap tricot and the reverse: the back bar must be the front bar’s complement. An exhaustive check over all 924 front orderings, each paired with its complement, finds every one of them even, and the counts for one front ordering against all 924 back ones fall into seven classes by how many courses come out mixed — 1 ordering mixing none, 36 mixing two, 225 four, 400 six, 225 eight, 36 ten and 1 mixing all twelve.

The complement costs no stroke. A complement’s running shortfall is the front’s turned upside down, so its range is the same: the even cloth’s back beam needs exactly the compensator its front beam needs. A designer who has already chosen the front bar’s ordering for its stroke has, in the same choice, chosen the back’s.

Moving the back bar’s repeat along the front’s one course at a time shows how the evenness arrives. Each course moved converts two pure courses into mixed ones, so the mean course spacing falls in equal steps — at four fifths of the width, from 1.88 times the knitted spacing in coincidence to 1.26 in opposition, about a tenth of the knitted spacing for each course moved. The range, meanwhile, does not move at all until the last step. At every offset from nought to five the repeat still contains at least one pure tricot course and one pure satin course, and those two set the range: the cloth is as uneven at an offset of five as at nought, merely with fewer courses at the extremes. At six the last pure course goes and the range closes to nothing in one step.

That is a practical point for a pattern that is nearly, but not exactly, opposed. A cloth one course short of opposition has the full unevenness of a coincident cloth, concentrated in a pair of courses. Two odd courses in a repeat of twelve are a fine line across the cloth, the kind of fault a stripe is a partition of the warp found an eye picks out readily because it is narrow.

Which pairs it is worth doing for #

Tricot and satin is one pair. The same arithmetic runs for any two shogs, and what it says about each pair is how much give opposition costs.

Any pair with a chain in it locks lengthwise when opposed. A chain underlap runs straight down the wale, one course spacing long, so a course containing one cannot open by any amount however narrow the cloth is held. A coincident chain-and-satin stripe gains three quarters of its length at four fifths of the width, all of it from the satin band; opposed, it gains nothing. That is the two-bar construction the trade calls queenscord — a chain bar beside a long-shog bar — and its reputation is for a firm, stable cloth that does not grow. The circles say why in one line: every course of it is held against lengthening by a chain and against widening by the longer bar, which is the strongest lock any pair of shogs can make.

At the other end, a cord-and-satin pair gives up least: 116 per cent of length coincident against 82 opposed, because the shorter of its two shogs is already long. Tricot and cord sit between, giving up half of their lengthwise give to buy an even cloth, and tricot and satin seven tenths of theirs.

Widened, the cost runs the other way and is set by the longer shog. Opposition makes every course close like the longer bar, so a pair containing satin loses 36 per cent at a fiftieth wider whatever the other shog, against 18 to 25 coincident. The width at which the cloth locks solid does not move at all: width is one quantity for the whole fabric, and the first underlap anywhere to lie flat stops all of it, so a cloth containing satin locks at 1.034 times its knitted width in every arrangement of its bars.

Why the even cloth is not simply the better one #

The obvious conclusion is that a designer wanting an even patterned cloth should oppose the bars, and that is right as far as it goes. The circles add two things to it.

First, the evenness and the give are the same quantity, read twice. A coincident cloth is uneven because its tricot and satin bands give by different amounts, and its average give is high because the satin band gives a great deal. Opposition removes the satin band’s give and with it the unevenness. There is no arrangement of two bars in which every course gives alike and the cloth keeps the satin band’s lengthwise stretch, because every course must then carry a tricot underlap, and the tricot stops it.

Second, the choice is per direction. A garment pulled mostly along its length wants the coincident cloth’s stretch and will show its stripes opening, which may be acceptable on a stripe and not on a ground; one that must not grow wants the opposed cloth. A cloth has one budget for two directions made the same point about a woven cloth’s crimp: what one direction spends, the other cannot have, and the arrangement of the structure decides which direction pays.

So the trade’s habit of pairing a long underlap with a short one on every course — in locknit, in sharkskin and in queenscord, on uniform bars long before anyone patterned them — is not a habit about yarn. What a second guide bar is for found the structural half of it, that a second bar can join wales the first leaves apart; the mechanical half is that a course with one long and one short underlap is locked both ways. Opposing two patterned bars is that habit kept up course by course through a pattern, and the arithmetic says exactly what it costs.

How the numbers were produced #

Every underlap is a straight line from a needle on one course to the needle a shog away on the next, at 28 needles an inch and 14 courses a centimetre, as in the single-bar account. A course’s spacing at a stated width is the smaller of the two spacings its two underlaps allow, each underlap keeping its own length; the width is shared by every course. The overlaps are held at their knitted shape throughout and are given a length only for the weight comparison, where they are bracketed from a millimetre and a half to three.

The two states are stated, not loaded. “Four fifths of the width” and “a fiftieth wider” are widths imposed on the cloth, and each course is taken to its own limit at that width, which is where a pull along or across the cloth leaves it. No force enters, because the first give has none to speak of.

The census paired one front ordering with every back ordering of six tricot and six satin courses — 924 of them — and classed each by how many of its courses came out mixed. Separately, every one of the 924 front orderings was paired with its complement and checked to give alike on every course at both widths, with the complement’s stroke equal to the front’s.

What the arithmetic is required to do. Opposed block stripes must give alike on every course at both widths and carry the same total on every course; coincident ones must not. Each bar’s stroke must be the same at all seven offsets, and in opposition the two running shortfalls must be negatives of each other at every course. Every course at the knitted width must sit at the knitted spacing. The fabric’s widest must be the satin underlap’s own at every offset. And the case that could have come out otherwise: a tricot-and-satin course and a cord-and-cord course, within two per cent in total underlap, must differ by more than a quarter in give at both widths — they differ by 31 and 25 per cent of the larger.

What the circles leave out #

The loops. The first give ends where an underlap comes taut, and past that a warp knit gives through its loops straightening, which this account has no loaded model of. The opposed cloth’s 26 per cent is therefore a floor on how far it lengthens before anything stiff resists, not a value; a knit’s dimensions come from its loop is where the loop’s own shape is worked out, unloaded.

Friction, and time. A course moving along its circle strains no fibre; what opposes it is friction at the crossings, and a knit relaxes for as long as it is allowed found that kind of resistance giving way slowly. The arithmetic says where each course can go, not how fast it gets there or whether a coincident cloth’s bands relax at different rates off the machine.

Straight underlaps. A real underlap lies over the loops of its own course and bows, more for a long shog than a short one, so every spacing here is computed from the shortest an underlap can be.

And patterns of more than two shogs, which can be even in total and uneven in give; the cord-and-cord course above is the smallest example, and no census of them is taken here.

Who found which part #

Two-bar constructions pairing a long underlap with a short one — locknit, sharkskin, queenscord — are standard warp-knitting practice, and their stability is part of the trade’s ordinary description of them. Patterning with two bars whose lappings complement each other is also old practice.

What is added here is the course-by-course reading. A course’s first give is the minimum of its bars’ circles, so it is set by its shortest shog when the cloth is narrowed and its longest when it is widened; opposed stripes are even because every course carries the same pair, and an equal total is necessary but not sufficient. Each bar’s stroke is unaffected by the offset; the even cloth’s back bar is the front’s complement and costs the same stroke; one course short of opposition is as uneven as coincidence. And the evenness costs a measured share of the cloth’s give, from half for tricot and cord and seven tenths for tricot and satin to all of it for any pair containing a chain.

Still open: a third bar between the other two #

The minimum in the formula has a consequence for a three-bar fabric that the two-bar case cannot show. A course’s first give depends only on its shortest and its longest shog; a third underlap whose shog lies between them is never the one that binds, in either direction.

That makes a middle bar free, mechanically. A fabric with a chain on one bar and a satin on another has every course locked both ways, and a third bar lapping anything from one space to three on any course leaves that lock exactly where it was — so it could carry a pattern, of colour or of lustre, without moving the cloth’s give at all. Its own stroke would still be its beam’s problem, and its underlaps would still change the cloth’s weight course by course. Whether a middle bar’s pattern is truly invisible to the cloth’s first give, and how far its bowing over the other two bars’ underlaps disturbs that, is a three-bar calculation this account has not made.