Generator

The energy of a crossing against how flat it is

The energy of a crossing against how flat it is

The energy of a crossing against how flat it is. A sheeting at 0.20 N per crossing, with both sections taken to the same aspect ratio and the cloth's thread lengths held where the loom left them. The bending energy rises with flattening — Peirce's arc has radius D/2 and squashing the threads shrinks D, so the curvature rises faster than the angle falls. The compression energy rises as the square of the log aspect. The load's work falls as the cloth thins. Their sum is least at an aspect ratio of 1.44. At no load the same curve is least at exactly 1, and its slope there is positive, which is the whole reason a relaxed cloth keeps its threads round. What the plot cannot show is that only the compression term has a material constant in it, and that constant has no lower bound.

A sheeting at 0.20 N per crossing, with both sections taken to the same aspect ratio and the cloth's thread lengths held where the loom left them. The bending energy rises with flattening — Peirce's arc has radius D/2 and squashing the threads shrinks D, so the curvature rises faster than the angle falls. The compression energy rises as the square of the log aspect. The load's work falls as the cloth thins. Their sum is least at an aspect ratio of 1.44. At no load the same curve is least at exactly 1, and its slope there is positive, which is the whole reason a relaxed cloth keeps its threads round. What the plot cannot show is that only the compression term has a material constant in it, and that constant has no lower bound.

11 essays call press-curve. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the weave its essay argues about.

Where it is called

Every figure listed here is drawn from this one rule.

A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.40 N per crossing the sections flatten to aspect ratios of 1.75 and 1.84, the cloth thins to 0.263 mm, and the warp runs flat for 0.109 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

Mechanics and drape

A flattened thread is a record of a force

A yarn in cloth is not round, and this site has modelled the flattening for as long as this collection has run with the amount of it left as a number somebody chose. Give the section a stiffness and ask what the cloth prefers, and the answer is a circle — at every stiffness, for every balanced cloth in the table. Flattening does not happen by itself; it happens because something pressed.

A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.42 N per crossing the sections flatten to aspect ratios of 1.79 and 1.88, the cloth thins to 0.260 mm, and the warp runs flat for 0.113 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

Setting and geometry

Peirce and Kemp are one cloth at two moments

This site has run two thread sections side by side since the setting field was built — a circle and a flattened racetrack — and said honestly that they disagree and that the disagreement is the point. They are not rival descriptions of the same fabric. They are descriptions of the same fabric before and after something pressed it, and the difference between them is a pressure that can now be named.

How much too thick a round section is, and what reconciles it. For each cloth in the standard cloth table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

Compound and figured cloths

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.

The bracket that closes and the bracket that does not. A 25 tex cotton yarn at a packing factor of 0.6. Its bending rigidity lies between 0.00117 N·mm² — the sum of its fibres', with them free to slide — and 0.478, a solid rod of its own diameter: a factor of 408, and both ends are derivations. Its resistance to being squashed out of round has an upper bound of the same kind, 369 N/mm² for a solid section, and no lower bound at all, because fibres free to slide resist a change of shape with nothing. That is why the aspect ratio of a flattened yarn has been a free parameter here since the setting field was built: a quantity bounded below by zero cannot be estimated from its bounds, and has to be measured.

Mechanics and drape

The stiffness with no lower bound

A yarn's bending rigidity lies between two derivable ends and the ratio is the fibre count — wide, but closed. Its resistance to being squashed out of round has an upper bound of the same kind and a lower bound of exactly nothing, because fibres free to slide resist a change of shape with nothing at all. That is why nobody could ever compute the aspect ratio of a flattened thread.

From a calender's line load to a force at one crossing. A sheeting through a nip loaded at 30 N per millimetre of bowl width, with the cloth in contact over 5.0 mm. The pressure is the first divided by the second, 6.00 N/mm², and the force at one crossing is that pressure times the area a crossing owns — the product of the two thread spacings, 0.1374 mm². So the crossing carries 0.824 N, the sections flatten to 2.42 and 2.63, and the cloth thins from 0.388 mm to 0.218 mm. What the drawing cannot show is that two cloths through the same nip are not given the same treatment: the area a crossing owns varies fivefold across the standard cloth table, and it is a factor in the force.

A calender spends the compression for good

Calendering was described here as moving a cloth from one thread-section model to another, which was right and had no number in it because the amount of the move was a free parameter. It is a pressure now — and the same nip setting turns out to give two cloths quite different treatments, because the force at a crossing is the pressure times the area a crossing owns.

The crimp ratios a dense shirting can have. Poplins in 15 tex warp and 20 tex weft at 22 picks, from 32 ends per centimetre to 52. Each bar is the interval of warp-to-weft crimp ratios the construction admits at all: the warp must supply at least the thickness the weft cannot reach, and at most what it can reach itself. The rule at one is this site's standing default. It sits inside the interval up to 44.18 ends per centimetre and outside it beyond — so for a dense shirting an equal division of the crimp is not merely the wrong assumption but a geometric impossibility. The 44-end poplin the standard cloth table called impossible for a long time sits a fifth of an end below that limit, which is why the solver failed on it: its feasible interval was real and narrow, and a bisection on the whole range walked away from it.

Setting and geometry

The cloth that was called impossible

This site's own table of fabrics carries a note saying a real 44-end poplin has no solution in its geometry at all, and that the poplin row was therefore set at 32 ends. The cloth solves. What had no solution was the search — a bisection that treated a state it could not reach as evidence of having gone too far, and walked away from the answer every time.

How much too thick a round section is, and what reconciles it. For each cloth in the standard cloth table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

Mechanics and drape

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

An operation multiplies a spread by its own log-slope. A transformation does not leave a population's spread alone: if y goes as the kth power of x then a small spread in x becomes k times that spread in y, exactly in the limit and nearly so at the CVs a yarn has. So an operation with an exponent below one narrows the population it acts on — a thickness that goes as the square root of a load comes out at half the spread it went in with — and one with an exponent of four widens it fourfold. This is the same derivative that decided every bias in this ladder, read for its magnitude rather than for its curvature, and it is why a finish can be a variance-reducing operation without anyone having chosen it for that. The straight line through the origin is the whole of the rule; the departure from it at the right-hand end is the second-order term arriving, which is where the linearisation stops being one.

A finish spends a spread before it spends a mean

Every operation on a cloth multiplies the variation it inherits by its own log-slope, so an operation with an exponent below one makes the cloth more even and one above it makes the cloth less even. Calendering, which is bought for evenness, has an exponent of 1.4.

Where a calender works, and where recovery has been measured. Each fibre's measured elastic-recovery span drawn against the shape strains a calender imposes, on one axis. The settings run from 11.2 per cent shape strain at a flattening of 1.25 to 54.9 per cent at 3. cotton is measured to 5 per cent; wool is measured to 20 per cent; silk is measured to 5 per cent; flax is measured to 2 per cent; viscose is measured to 5 per cent; nylon is measured to 8 per cent; polyester is measured to 8 per cent. Only wool reaches any setting at all, and only the lightest.

A calender works where nothing has been measured

This account has twice recorded that the missing piece is plasticity — what fraction of a flattening survives the nip. The piece is missing for a sharper reason than nobody having written it down. A calender's flattening is a shape strain, and at the lightest setting in this account's own series that is 11 per cent while cotton's elastic recovery is measured from 2 to 5. Six fibres of seven have no data at any setting the machine has, and wool reaches only the lightest. The law cannot be had from the measurements; what can be had is the bracket, and it is a factor of twenty-four.

How much plateau a cloth of stated cover can be given. The flat run a calender can put on a 200 micrometre yarn, against the cover the cloth already has. A thread may widen until it meets its neighbours, so the flattening is capped where the section's width equals the pitch; at a cover of 0.3 that is a flattening of 13.9 and 619 micrometres of plateau, and at 0.9 it is 1.32 and 53 micrometres. At full cover both are nothing.

A covered cloth cannot be calendered

A calender widens a section at conserved area, and a thread may widen until it meets its neighbours. So the flattening is capped by the cover the cloth already has — ×30 on a scrim, ×1.32 at a cover of nine tenths, and exactly one at full cover. The plateau a calender buys falls to nothing along with it, which makes the lustre a nip can add and the cloth's opacity the same constraint read twice: a cloth that cannot be seen through is a cloth a calender cannot help.

A calender's best cloth is the one its nip fills

A calender's lustre is a length of crown line times a width of plateau. A closer cloth has more crown line and less room to widen each thread into, and the product was expected to peak at some interior cover that would move with the weave. It does not peak at all on the cap alone — per unit of crown line it falls all the way from the most open cloth. The peak is made by the nip: a nip that flattens by f has one best cloth, the one it exactly fills, at a cover of d over the flattened width, and there the lustre is (f − 1)/f. No yarn, and no weave, moves it.

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