What holds a tuft in, in newtons

How a tuft is held set out the two classical ways of putting a pile end into a ground cloth. A V fastening dips between two ground picks and comes back up: two legs, one turn, one half-turn of wrap. A W fastening goes under one pick, over the next and under the third: three turns, three half-turns of wrap.

It computed the difference as a capstan ratio, exp(μθ), and it said what that was worth in as many words: the anchorage ratio is the capstan on the wrap angle, and the answer is a ratio at a stated μ. That was the honest limit at the time. A capstan ratio multiplies a tension, and this site had no tension anywhere.

The contact-force ladder supplies one, and the pile ladder’s oldest unanswered question can be asked properly.

The claim #

A W-fastened tuft in a woven ground is held by a few newtons of friction, and a carpet specification asks for tens. The mechanical anchorage falls short of a domestic carpet’s requirement by about a factor of three and of a contract carpet’s by about six.

That is not a failure of the model. It is the model saying something true about carpets: the interlacing is not what is holding the tuft in. The rest comes from somewhere else, and the somewhere else is the back-coating.

The capstan is a series, not a factor #

The arithmetic has one subtlety and it is worth taking slowly, because the obvious version of it is wrong by more than half.

The normal force pressing the pile yarn against a ground pick is the same one every crossing on this site now carries: 2·T·sin θ, with T the tension in the ground pick and θ its weave angle. The ground is doing the gripping, so the ground’s tension is what matters and the pile’s does not appear.

Friction at one wrap resists sliding with μ times that. The question is what the capstan does with it.

A capstan ratio exp(μθ) is a statement about tension on both sides of a wrap. A tuft’s far end is free — nothing is pulling it from the other side — so there is no tension to amplify across the whole thing. What there is instead is this: each wrap’s own friction has to be dragged around every wrap between it and the end being pulled. Numbering the wraps from the pulled end,

F = μ·N · (1 + exp(μπ) + exp(μ2π) + …)

which is one term for a V and three for a W.

The first version of this computation multiplied the summed friction by the capstan ratio of the whole wrap angle, which treats the near wraps as though they were at the far end. It overstates a W fastening by about sixty per cent and was caught by asking what the ratio between the two fastenings ought to be.

What the two fastenings are worth #

At an ordinary friction of 0.3, with a heavy ground held at a newton per pick:

The ratio is 10.2, and it runs from 5.2 to 16.9 across the reported friction range. The previous rung’s pure capstan ratio was 3.5 to 12 — the same shape and a somewhat larger figure here, because a W fastening has three contacts to a V’s one as well as a larger wrap angle, and the previous rung counted only the second.

Pile is a third thread system, and that is what makes the question askable at all: the pile end is not part of the ground’s interlacement and is held to it only by wrapping. W is worth about an order of magnitude and it costs a third of the pile density, since each W tuft consumes three ground picks where a V consumes two. That trade is the one the pile ladder computed and it is unchanged; what is new is that both halves of it can now be quoted in the units a specification uses.

Where the rest of the anchorage comes from #

A tufted carpet is specified by a tuft withdrawal force in newtons — twenty or so for domestic use, forty for contract — and the numbers above do not reach it.

The gap is not a mystery and the trade closes it deliberately. A tufted carpet is back-coated: a latex or polymer compound is applied to the reverse, penetrates between the ground yarns, and sets around the tuft’s legs. What holds the tuft in after that is adhesion to a film, not friction against a thread.

So the model’s shortfall is the measurement of something real. Friction contributes a few newtons and the coating contributes the rest, and the ratio between them is roughly three to one for a domestic specification and six to one for a contract one.

Two consequences follow that a purely mechanical account would miss.

A carpet’s anchorage is a coating property, so it degrades the way coatings degrade — with heat, with cleaning solvents, with age and with flexing — rather than the way friction degrades. A carpet that has been steam-cleaned repeatedly has lost something the interlacing never had.

And a woven carpet is a different object from a tufted one. Velvet is cut apart from a double cloth and corduroy’s pile is a cut float; both hold their pile by weaving rather than by adhesion. A Wilton or an Axminster holds its pile by weaving it in, at wrap angles and ground tensions much larger than anything above, and needs no coating to meet a specification. The distinction between woven and tufted carpet is usually described in terms of how they are made; this rung says it is also a distinction between two entirely different mechanisms of anchorage.

What the shortfall says about the two carpet trades #

The gap between a few newtons and a specification’s tens is a measurement of the coating, and reading it that way says something about the two ways a carpet is made that neither trade puts in those terms.

A tufted carpet’s anchorage is three quarters coating and a woven carpet’s is none. So the two products have their anchorage in completely different places, and the properties that follow from anchorage follow differently.

A woven carpet’s tuft is held by a mechanism that does not age, does not dissolve and does not soften. It is held harder when the carpet is compressed underfoot, because compression raises the ground’s own contact forces, and it is held by a quantity that a manufacturer can compute from the construction before weaving.

A tufted carpet’s tuft is held by a film whose bond is a chemical property, whose behaviour under heat and solvent is a formulation question, and whose contribution is measured rather than designed. That is not a criticism — the coating reaches numbers the interlacing cannot, at a fraction of the cost — and it does mean the specification is being met by the part of the product furthest from the fibre.

The practical consequence is about what a withdrawal test is testing. On a woven carpet it measures a construction, so it is repeatable, predictable from the draft, and stable over the product’s life. On a tufted one it measures an adhesive bond, so it varies with cure, with age, with what the carpet has been cleaned with, and with how well the coating penetrated on the day. The same number, in the same units, is a geometric measurement on one product and a chemical one on the other.

That is worth knowing before comparing two carpets on it. A tufted carpet at forty newtons and a woven one at forty newtons are not equally likely to be at forty newtons in five years, and nothing in the figure says which is which.

The arithmetic gives one more thing a specification could carry and does not: the fraction of the anchorage that is mechanical, which is computable from the construction and is between a sixth and a third for the cases above. A carpet whose friction share is a third has a floor it cannot fall below however the coating ages; one whose share is a tenth has almost none. That is a durability statement available from a draft and a friction range, and it needs no test at all.

It is also the number a purchaser would most want and the one least likely to be offered, because it is a statement about how much of a product’s performance depends on the part of it that ages.

Why the ratio survives and the values do not #

The two fastenings’ ratio is worth more than either force, and the reason is structural rather than a matter of confidence.

The ratio is a sum of exponentials in μ and has no tension in it at all. So it is the same whatever the ground is held at, which the machinery asserts by recomputing the whole table at two and a half times the tension and requiring agreement to twelve figures. A tension that had leaked into the ratio would be a sign that the contact force had been applied to the wrong thread — the pile’s rather than the ground’s — and that is a mistake that would produce entirely plausible numbers.

The forces themselves are proportional to the ground pick’s tension, which is stated rather than derived. A newton per pick is a plausible figure for a heavy ground cloth and is not a measurement of any particular carpet. Everything in the table scales with it exactly, so a reader who prefers another figure can rescale the page by one multiplication — and the shortfall against the specification is a statement about that stated tension as much as about the construction.

What does not rescale is the conclusion. Reaching a domestic specification by friction alone would need about three newtons in every ground pick, and a contract specification about six. Those are large tensions for a ground cloth, and they would have to be maintained in the finished carpet rather than on the loom — which is the point at which the mechanical account stops being plausible and the coating becomes the only available explanation.

What was counted, and how #

The ground state is Peirce’s solution at a heavy canvas’s quoted construction, verified against its own equations. The weave angle comes out of that; the normal force is twice the stated pick tension times its sine.

The wrap counts are the constructions’: one half-turn for V and three for W, which are geometric facts about the two fastenings and are the same numbers the pile ladder has used since it was built.

The capstan series is computed term by term rather than as a closed form, so that the number of terms is visibly the number of wraps.

Three assertions guard it. W must anchor harder than V at every friction, which would fail on a sign error in the series. The ratio must be independent of the ground tension to twelve figures, as above. And no fastening may reach a carpet specification by friction alone — which is an assertion in the direction that could embarrass the model, because a computation that quietly produced a comfortable twenty newtons would be much easier to believe and much harder to check.

The friction range is quoted at 0.2 to 0.4 for yarn on yarn, from the table this site has used since the pile ladder was built, and it is a range because yarn-on-yarn friction is not a material constant.

Where the model stops #

The pile yarn’s own strength is not modelled, and it is the other way a tuft can leave a carpet: a tuft can break rather than pull out, and which happens is the same crossover question the contact ladder asks of a woven thread. At these grip lengths sliding always wins, so it does not change the answer here — but a longer-anchored pile, a woven carpet’s for instance, could be in the other regime.

The coating is not computed, only inferred. Saying that the gap must be closed by the backing is not the same as computing what a backing contributes, and doing that needs the film’s modulus and its bond to the yarn — which is exactly the shortfall the coating ladder records and which this rung now records from a second direction.

The ground tension is stated. As above, and it is the least defensible number on the page.

And nothing here is dynamic. A tuft is pulled out by a testing machine slowly and is loosened in service by repeated small tugs from feet, vacuum cleaners and furniture. Friction under repeated small displacements is not the same quantity as friction under one steady pull, and the difference is the same one the felting ratchet runs on.

The generalisation #

The lesson is about what a ratio is worth and when.

A ratio is the right answer when the quantity it multiplies is unknown, and it stops being enough the moment there is an external standard to meet. For as long as it stood alone, the pile ladder’s anchorage ratio was genuinely the best available statement, and it supported a real conclusion: W buys a large multiple and costs a third of the density. What it could never do is answer is this enough, because enough is a number.

The second lesson is about what a shortfall means. A model that falls an order of magnitude short of a specification is not necessarily wrong; it may be measuring the part of the mechanism it was given. Here the mechanical part is a few newtons, the specification is tens, and the correct conclusion is not that the arithmetic is broken but that something else is carrying most of the load. A model that had been tuned until it reached the specification would have hidden exactly the thing worth finding.

That is the general form of a useful negative result: the gap between a computed mechanism and an observed performance is itself a measurement of the mechanism that was left out.

Who found it, and when #

The capstan equation is Euler’s, from 1762, and textile mechanics knows it as the reason a knot holds and a package of yarn does not collapse.

V and W fastenings are trade constructions of long standing, and the rule that W is specified for heavy traffic and V gives more pile for the money is in every carpet manual. Tuft withdrawal as a specified quantity belongs to carpet standards work of the twentieth century, and the role of the back-coating in meeting it is not a secret — it is why the coating is applied.

What is this site’s is putting the three side by side: the construction’s own mechanical anchorage, computed; the specification, quoted; and the difference, read as a measurement of the coating’s contribution. Each of the three is available separately in its own literature and they are not usually in the same place, because the mechanical calculation is not usually done at all.

Where the ladder goes next #

The next thing this ladder owes is the coating’s own contribution, which needs the film’s modulus and its bond — the same two quantities the coating rung records as missing, now wanted from a second direction.

Sideways, the same series applied to a leno’s crossed warp should give a much larger number, because a leno’s wrap angles are far larger than a weave angle and it is the one woven construction that is genuinely in the capstan regime.

Further out is the dynamic problem: a tuft is loosened by many small displacements rather than one pull, and the ratchet that makes that different from static friction is a mechanism this site already has in another field entirely.