A patterned warp knit needs a stroke as long as its stripe

Two bars cannot share a beam found that the difference between two guide bars’ appetites for yarn needs no model of a loop. Each bar makes one overlap a course round its own needle, and the overlap is the same object on every bar — same needle, same loop, same yarn — so it cancels from any difference. What is left is the underlap, the straight run from one needle to the needle a shog away on the next course, whose length is a hypotenuse:

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

with $w$ the wale spacing and $c$ the course spacing. A cord bar wants 111 metres more yarn per wale than a tricot bar over a hundred-metre piece, and a shared beam between them is a course spacing out within one course.

That essay’s bars were uniform, the same shog at every course. It ended on the fabrics anybody buys, which are not: a patterning bar changes its shog from course to course, so what it takes from its beam is a sequence rather than a constant. A beam delivers at one rate. So the question is what has to give between them, and how much.

The overlap cancels a second time #

Course $k$ costs the bar one overlap $o$ and one underlap $u(s_k)$. A beam set to the repeat’s mean delivers $o + \bar u$ every course. After $k$ courses the bar has taken $\sum (o + u(s_i))$ and the beam has given $k(o + \bar u)$, so the bar is short by

$$E_k = \sum_{i \le k} \bigl(u(s_i) - \bar u\bigr).$$

The overlap has cancelled again, this time between a course and the average course rather than between two bars, and for the same reason: every course throws the same loop. So the whole of what a patterning bar asks of its beam is exact and needs nothing but the shogs, the gauge and the course density — the same three things the uniform bars needed.

$E_k$ returns to nought at the end of every repeat, because the beam was set to the repeat’s mean. The fabric does not drift; the question a figured warp needs a beam for every share of its figure asked of a woven figure — whether one beam can serve at all — has the answer yes for any patterning bar whose beam is set right. What does not return to nought within the repeat is the shortfall itself, which rises and falls through each repeat. Its range, the top of $E$ less the bottom, is the stroke: the length something between the beam and the needles has to take up and give back every repeat.

The stroke of a stripe is a triangle’s height #

The simplest patterned lapping is a stripe: $k$ courses at one shog, then $k$ at another, repeated. At 28 needles an inch and 14 courses a centimetre a tricot underlap is 1.155 millimetres and a satin underlap, three needle spaces, 2.814. The mean of the two is 1.984, so every tricot course leaves the bar 0.830 millimetres ahead of the beam and every satin course puts it 0.830 behind.

So the block stripe’s shortfall runs down in a straight line through the tricot courses and back up through the satin courses: a triangle whose height is $k\,\Delta u/2$, where $\Delta u$ is the difference between the two underlaps. Six courses of each need a stroke of 4.98 millimetres a thread; twelve of each need 9.95; twenty-four need 19.9. The stroke grows with the height of the stripe and has nothing to do with how many stripes a repeat contains.

The same twelve courses alternating need 0.83 millimetres — a single course’s excess, taken and given back at once. Nothing about the fabric’s total consumption differs between the two: over a repeat both take exactly the same yarn, and a run-in measured over a rack of courses would not tell them apart. What differs is how the demand is arranged in time, and that is worth a factor of six in the stroke.

An atlas needs none, however far it wanders #

An atlas lapping walks several needle spaces in one direction, one space a course, and then walks back. It is the lapping warp knitting is a different thing entirely drew to show a thread that ranges far and still makes cloth. Its thread ranges across four wales, more than any lapping here, and it looks like the patterned case par excellence.

It needs no stroke at all. Every one of its shogs is one space, so every underlap is a tricot underlap, the demand is the same every course, and the running shortfall is nought throughout. The shog is the anisotropy found the atlas’s underlaps all identical when it asked what they do to the fabric’s give; here the same fact decides what they ask of the beam. What a beam has to absorb is a change of shog, not a range of travel, and a lapping that wanders a long way in equal steps is, to its beam, a uniform bar.

A stripe needs a stroke about its own height #

The stroke of a stripe is $k\,\Delta u/2$, and $k$ is the stripe’s height in courses. Measured against the stripe’s height in the cloth — $k$ times the course spacing — the stroke is a nearly fixed multiple of it.

For tricot beside satin the multiple is 1.16 millimetres of stroke for every millimetre of stripe. A satin stripe a centimetre deep in the fabric asks the beam’s slack for 11.6 millimetres of travel; one two centimetres deep, for 23. Chain beside satin, the largest difference among the four common shogs, asks for 1.47 millimetres of stroke a millimetre of stripe; tricot beside cord for 0.56; chain beside tricot for 0.31.

The multiple has a closed form in the limit of long underlaps. When $s w$ is much larger than $c$, an underlap is nearly its sideways run, $u(s) \approx s w$, so $\Delta u \approx \Delta s\, w$, and the stroke over the stripe’s height $k c$ is

$$\frac{k\,\Delta u / 2}{k\,c} \approx \frac{w}{c}\cdot\frac{\Delta s}{2}:$$

the fabric’s own aspect, wale spacing over course spacing, times half the shog difference. At 28 gauge and 14 courses a centimetre $w/c$ is 1.27, and the exact stroke is 1.16, lower by the part of each underlap that is course spacing rather than sideways run.

The machine’s fineness does not enter #

That form has a consequence a machine builder might not expect. A finer gauge has a smaller wale spacing, so each course’s underlap excess is smaller, and it would be natural to expect a finer machine to need a smaller compensator. It needs the same one for the same stripe, because a finer machine also knits more courses to a millimetre, and the two cancel in $w/c$.

Across five machines from 18 gauge at ten courses a centimetre to 40 gauge at twenty-four, the stroke per millimetre of stripe runs from 1.16 to 1.43, and it tracks the aspect $w/c$ of each machine’s fabric — 1.27 to 1.52 — at nine tenths of it every time. The stroke is set by the shape of the fabric and the height of the stripe, not by the fineness of the machine. It is the same shape of result as the one two bars cannot share a beam found for the per-piece difference between bars: a shorter underlap per course and more courses per metre very nearly cancel across the ordinary range.

Of 924 orderings, one needs the least #

The factor of six between blocks and alternation raises the obvious question: how much stroke do the orderings in between need? A twelve-course repeat with six courses of tricot and six of satin can be ordered in $\binom{12}{6} = 924$ ways. Every one of them can be lapped — give the tricot courses alternate directions and the satin courses alternate directions and the walk closes on itself whatever the order — so every one is a real pattern.

Every stroke is a whole number of $\Delta u/2 = 0.830$ millimetres, from one to six. That is because each course moves the running shortfall by exactly that amount, up for a satin course and down for a tricot course, so the stroke is the height range of a walk of six steps up and six down, times the step. The census is then a census of such walks:

• 2 orderings need the least stroke, 0.83 millimetres, and they are one pattern started at two places: strict alternation.
• 124 need two steps, 390 three, 300 four and 96 five.
• 12 orderings need the most, 4.98 millimetres, and they too are one pattern: the two solid blocks.

Counted up to where the repeat starts, which does not change the stroke, the 924 orderings are 80 patterns: 1, 12, 33, 25, 8 and 1 at the six strokes. The median pattern needs three times the least stroke and half the most, and a designer choosing a pattern by its look alone chooses among these with no idea which column it falls in.

The cheap ordering is a different look #

The census invites a conclusion it does not support: that a designer should pick the alternating order and save five sixths of the stroke. The two orderings contain the same courses and the same yarn, but they are not the same fabric to look at.

Two solid blocks make a stripe: six courses of short underlaps, then six of long ones, each band 4.3 millimetres deep at 14 courses a centimetre — a visible horizontal bar of different density and lustre on the back of the fabric, where the underlaps lie. Alternating courses make a fine mixture of the two, too small for an eye to resolve into bands at all. That is the same distinction a stripe is a partition of the warp drew for a woven cloth, where a band of one weave beside a band of another is a design and the two weaves mixed end by end is a different weave altogether.

So the stroke is not a price a designer can shop around once the look is fixed. The look fixes the stripe’s height, and the height fixes the stroke, at about a millimetre of travel for every millimetre of band. What the census does show is that among orderings that all read as stripes, the difference between the most and least demanding is set by the tallest band in the repeat. A pattern of narrow bands with one wide one costs what the wide one costs, and why a knit shows a thick place found the same thing from the other side: in a knit, the machine’s own repeat decides what the eye sees as a stripe.

Why a warp knit cannot pay the way a woven figure does #

A figured warp pays in tension before it needs a beam found that a woven figure whose regions take up warp at different rates can often share one beam anyway. The ends that consume more pull harder, straighten and give up crimp until every end consumes alike. The price was a few hundredths of a newton an end for the closest pairings, and the ground’s crimp budget capped it.

A patterned warp knit has no equivalent budget. The only length a warp-knit course can give back under tension is in its loop, the overlap this arithmetic has cancelled twice. Suppose the loops absorbed the whole shortfall with nothing else to take it. Then every satin course would knit its loop 0.83 millimetres shorter than the mean and every tricot course 0.83 longer, so the loops of the two stripes would differ by 1.66 millimetres. Two bars cannot share a beam took an overlap of about two millimetres for the sake of a figure; against that, the two stripes’ loops would differ by four fifths of a loop.

That is not a fabric with a faint tension bar in it. It is two fabrics knitted at wildly different stitch lengths. The woven figure could regulate itself because crimp is a few per cent of a thread and a figure’s demand differed by less. A warp knit’s patterning changes the demand by most of an underlap, and the loop is too small a reservoir to regulate it. Something outside the fabric must take up the stroke, and the arithmetic says how big it must be.

What takes up the stroke #

The beam itself cannot, for the reason two layers need two beams found for a double cloth: a beam turns at one rate for every thread on it. A beam delivering at one rate to a bar whose demand swings through a stroke has two ways to cope. The warp between them can stretch and relax: over a free length $L$ a stroke $S$ is a strain of $S/L$, and at a length of the order of a metre a 20-millimetre stroke is a two-per-cent swing, far past any strain a warp should be asked to cycle through every repeat. Or a compliant element can take up the length — a tension bar or compensating rail that moves as the demand swings, which is on a warp-knitting machine what an easer gives back the kink the crossed shed puts in is on a leno loom.

The comparison with the leno easer is close. A leno easer should be a light weight found the crossing end needing some fifteen millimetres of give at every crossed shed, and a weighted bar the right way to give it. A patterning bar’s stroke is of the same order, and it has the same answer: a compliant element with little spring rate, since any spring in it turns the stroke into a tension that ramps through the stripe — the tension at the last course of a stripe different from the first by the spring rate times the stroke — and a tension ramp is a loop-length ramp.

The other answer is to stop delivering at one rate. A let-off whose rate follows the pattern, faster through the satin courses and slower through the tricot, removes the shortfall at its source. The stroke is then set by how quickly the beam can change speed rather than by the stripe. The arithmetic here says how much work that saves: the whole stroke, for a let-off that can follow each stripe.

How the numbers were produced #

The underlap is the straight line from one needle to the needle a shog away on the next course, as in the essays on two bars and on the shog, at 28 needles an inch and 14 courses a centimetre unless another gauge is named. The overlap is taken as the same on every course and never given a value.

The stroke is the range of the running sum of each course’s underlap less the repeat’s mean, computed course by course. The ordering census enumerated every twelve-bit word with six ones — 924 of them — and reduced them under rotation of the repeat to 80 patterns. The gauge table took a stripe twenty millimetres deep on each machine at the course density named with it.

What the arithmetic is required to do. A uniform bar of any shog must need no stroke, and so must an atlas. A block stripe of $k$ courses must need exactly $k\,\Delta u/2$ and the same courses alternating exactly $\Delta u/2$, for $k$ from one to twenty-four. The census must find 924 orderings, with its least and most exactly those two. And one thing that could have come out otherwise: the stroke per millimetre of stripe must lie within a fifth of $(w/c)\,\Delta s/2$ below it at every gauge in the table, which it does, at nine tenths.

What the arithmetic cannot say #

It assumes the loop is held. The overlap cancels only if every course throws the same loop, which is exactly what a compensator exists to ensure. A machine without one does not obey this arithmetic; it knits a different fabric, as the section on paying through the loops describes.

The underlap is a straight line, which is its shortest length, as the uniform essays also assumed. A real underlap lies over the loops of its own course and bows, more for a long shog than a short one, so every excess here is a floor and every stroke is a floor with it.

And the machine’s own compliance is left out. A warp sheet between a beam and a guide bar has a length, a stiffness and a friction over its guide rods, and some of the stroke will be taken up there whatever else is fitted. The stroke is what must be taken up somewhere; where the machine takes it is a matter of its design.

Who found which part #

Run-in, tension bars and pattern let-off are ordinary warp-knitting practice, and patterned lappings have been knitted on bars driven by pattern chains since the nineteenth century. Run-in tables list totals per rack of courses, and a total per rack is exactly the quantity that cannot see a stroke: the block stripe and the alternating stripe have the same run-in.

What is added here is the stroke as an exact quantity with no loop in it, the overlap cancelling between a course and the mean course as it cancelled between bars. A stripe’s stroke is $k\,\Delta u/2$, about the stripe’s own height in the cloth at $(w/c)\,\Delta s/2$, and independent of the machine’s fineness. The census of 924 orderings has every stroke a whole number of half-excesses, and one pattern at each end. An atlas needs nothing, and a warp knit cannot regulate a pattern through its loops the way a woven figure regulates through its crimp.

Still open: two patterning bars in opposition #

A fabric with two patterning bars has two strokes, one on each beam, and each is fixed by its own bar’s shogs alone. But the fabric sees both: a course in which the front bar laps satin and the back bar tricot is a course in which the two threads’ underlaps lie across each other at different lengths. If the two patterns are arranged so that each bar’s long underlaps fall where the other’s are short, the fabric’s total underlap per course is constant even though neither bar’s is.

Whether that arrangement does anything useful — whether a fabric whose two bars’ demands are complementary knits more evenly, or relaxes more evenly off the machine, than one whose bars’ long courses coincide — is not a question about beams at all, since each beam still needs its own stroke. It is a question about the fabric’s own course-by-course length, which is where the shog is the anisotropy located everything that decides how a warp knit gives. The arithmetic of the two strokes is here; how the fabric responds to their relative timing has not been worked out.