Six things a strand is not

Every field on this site collects the things that are said confidently and are not true. This one has a particular crop, because its object is so familiar: everybody has used a rope over a pulley, and the intuitions that come with that are mostly right, and the places they fail are places where the geometry is doing something a hand cannot feel.

Six claims, each with the measurement that settles it. Every number below comes from the same routine — a list of bodies with wrap senses, handed to a function that returns the tangent runs between them and the arcs on them — so none of them is a correction to a formula. They are what the mechanism does.

One: a four-part tackle gives four to one #

It gives four to one when its blocks are infinitely far apart.

The rule comes from counting the parts of line supporting the moving block, and its derivation assumes those parts are parallel to the lift. They are not: the sheaves in each block sit side by side, so each part leaves at an angle, and a part at angle $\beta$ shortens by only $\cos\beta$ of the movement.

The measurement. Differentiating the strand’s own length gives 3.927 with the blocks 220 mm apart, 3.774 at 120 mm, and 3.617 at 90 mm. At twenty metres it is 3.99999.

The second route. Measure each part’s angle off the drawn run and add the cosines: 3.92728 at 220 mm against the derivative’s 3.92728, and 3.77392 at 120 against 3.77392. Two computations that share no arithmetic.

Where the intuition goes wrong. The lean is small — 12.76° on the worst part at 220 mm — and a small angle feels like a small error. Its cosine is 0.975, and there are four parts.

And a trap next door. The strand alternates between the blocks, so an even tackle must have its dead end on the standing block and an odd one on the moving block. Reeve a three-part tackle with the dead end above and it measures 1.973: the last span runs from a block back to itself, taking its share of the line and none of the load. Nothing about the drawing looks wrong.

Two: a belt wraps 180° of each pulley #

Only in the limit, and the error runs the wrong way.

The measurement. On 40 and 24 mm pulleys 160 mm apart, the wraps are 191.478° and 168.522°. Pull the centres in to 70 mm and they become 206.426° and 153.574°; push them out to 400 and they narrow to 184.585° and 175.415°.

The small pulley always gets less than half a turn, and the small pulley is usually the driven one — the one whose grip matters. The gap widens exactly as the drive is made more compact.

The exact statement. The two wraps add to precisely 360°, always, because a closed strand’s signed wraps are its turning number and an open belt’s is one. So one pulley’s excess is the other’s deficit, and both are $\pi \pm 2\gamma$ with $\sin\gamma = (r_1 - r_2)/C$.

The case that reverses the intuition. A crossed belt gives both pulleys exactly the same wrap — 227.156357° each on the pair above — for any two radii at any centre distance. Crossing a belt is described as a way to reverse the output; it is also the only way to make the wrap equal.

Three: a tensioner takes up slack in proportion to how far it moves #

It takes up slack in proportion to the sine of half its own wrap, which is a different thing entirely at the end of the travel that matters.

The law. Moving an idler’s centre by $d\mathbf{c}$ changes the run’s length by $2\sin(\theta/2)$ per millimetre, along the bisector of its two spans. Measured against the run’s own numerically differentiated gradient across a whole contact range: agreement to $2 \times 10^{-7}$ mm per mm.

The measurement that matters. At the edge of contact, on the serpentine drive this field uses, the idler’s wrap is 0.0141° and it takes up 0.000245 mm of belt per millimetre of its own travel. Move it a whole millimetre and the belt length changes by a quarter of a micron.

Why it is exactly zero at the boundary. The extra strand a body costs when it intrudes into a span goes as the square of the intrusion — exponent measured at 2.0000, 2.0002 and 2.0014 over successive decades — so the length is stationary at first contact.

The consequence. A tensioner has a contact range and a useful range, and the second is strictly inside the first with nothing marking the boundary. An idler drawn just kissing the belt passes every check a drawing can be given and is a decoration.

Four: a winch’s line speed is a property of the winch #

It is a property of how much line is already on the drum.

Line speed per turn is $2\pi r$, and $r$ is the radius of the layer being wound rather than anything about the machine.

The measurement. On a drum with a 30 mm core, 6 mm line and twelve turns to a layer: 207.35 mm per turn on the first layer and 395.84 on the sixth. A factor of 1.909, with nothing changed.

What the mean costs. The honest average over a full drum is 301.59 mm per turn, and it is exactly right over the full drum and wrong everywhere inside it. Twelve turns from empty pays out 2,488.1 mm; the mean predicts 3,619.1 — 45.5% high. Twelve turns from five layers on pays out 4,750.1 and the mean is 23.8% low.

The shape of it. Length is quadratic in turns, so turns from length is a square root: one metre takes 9.646 turns from an empty drum, 7.074 with two layers on, and 5.585 with four.

Five: enough cables will hold it still #

No number of strands holds anything still, and the reason is a two-line geometric argument rather than a matter of degree.

A strand constrains one way: it holds a point at no more than a distance. So the set of positions permitted by a collection of strands is an intersection of discs, which is convex. Unless that set has degenerated to a single point, it has an interior, and a point anywhere on its boundary — including at a corner, with two strands taut together — can always move inward, at which every strand it was touching goes slack and none of them objects.

The measurement. For anchors at three corners and strands of 130, 130 and 120 mm the region has an area of 2,721.0 mm² and three corners. Shorten the third strand and the region shrinks smoothly: 1,425 mm² at 100 mm, 474.8 at 80, 14.4 at 60, and 0.67 at 57.

The degenerate case, and what it costs. The region is a single point when the third strand is exactly 56.191685 mm — the distance from the third anchor to the intersection of the other two circles. That is the only configuration in which three strands determine a position, and it takes a length specified to six figures.

What the claim should be. Something must push outward — gravity on a suspended load, a spring, a body in compression — and the moment that something is named the argument has left geometry. The rig plus the load can be rigid; the rig cannot.

Six: a shaped pulley can be given any ratio at all #

The variation available falls away as the square of how quickly it is asked for, and the demand that sounds simplest is not a shaped pulley at all.

A strand leaving a body feels only the perpendicular distance from the axis to the tangent — the support function $h(\psi)$ — and the strand paid out per radian is that distance. So a demanded rate is a demanded support function, and the shape comes back from it in closed form.

The ceiling. The shape’s radius of curvature is $h + h''$, so a demand $h_0(1 + a\cos n\psi)$ has a body behind it only while $a < 1/(n^2-1)$. Bisecting on whether the construction refuses:

And the first harmonic is an eccentric. Ask for an arm varying once per turn and the answer is a circle of the mean radius with its centre moved off the axis — the profile departs from that circle by $1.4 \times 10^{-14}$ mm, on an offset of 15.3 mm. So the demand that looks easiest to draw is answered by a round pulley bolted on off-centre, and the shaped part of a shaped pulley starts at the second harmonic, which is where the ceiling starts too.

Read backwards, this is the useful half. A pulley mounted 1 mm off-centre on a 34 mm radius is not a pulley with a run-out; it is a variable-ratio drive with a 2.9% swing once per revolution, indistinguishable from one that was designed.

A seventh, which is a modelling mistake rather than a claim #

One more is worth recording, because it is what a computation of a strand gets wrong rather than what a person does, and it caught this field’s own machinery.

The claim: if the geometry returns an answer, there is a strand there.

There is not. Given two bodies and two wrap senses, exactly one tangent line exists, and the formula returns it whether or not a strand could ever lie on it. Build a run out of such tangents with one body the strand does not actually reach and the result is a smooth, closed, plausible path of a reasonable length — which wraps that body by 348° instead of 6°, cuts through everything between, and crosses itself once.

It draws perfectly. Every wrap in it is a legitimate arc of a real circle; every span is a genuine tangent; the length is a number a designer would accept.

What catches it. Three tests, and the cheap one fires first. No span may pass inside a body; no two spans may cross; and the signed wraps must add to the turning number the route says. The third is a sum that has already been computed, and it comes out at two instead of one.

That is the shape of most of the errors this field’s library has made. The closed forms here have no residual — nothing converges, so nothing can fail to converge — and a construction that is wrong produces a picture that is wrong in a way no drawing check can see. Every result above is therefore checked against a second computation of a different kind: an integer, a hull’s perimeter, a sum of cosines, a numerically differentiated length, or a classical formula from a book.

What these six have in common #

Four of them are the same mistake: a ratio quoted as a number when it is a function. The tackle’s four, the belt’s half turn, the winch’s line speed and the shaped pulley’s demand are all quantities that vary across the mechanism’s own working range, and in every case the quoted value is a limit or a mean rather than a value.

That is this site’s oldest theme and the strand’s version of it has a particular flavour: the varying quantity is always the arm — the perpendicular distance from an axis to the strand — and it varies because the body is not round, or is not on its own centre, or has line wound on it, or because the strand is not perpendicular to the thing it moves.

The other two are about the one-sidedness: that a tensioner’s authority vanishes where its contact does, and that strands never fix anything. Those are the claims a familiarity with rope makes hardest to see, because a rope in the hand is always taut and always being pulled against something.

Two more that are nearly true #

Neither of these is wrong enough to be one of the six, and both are worth the correction.

“A chain drive’s ratio is its tooth ratio.” It is, exactly, as a mean over a whole turn — measured at 4.8182 against a tooth ratio of 4.81818 by integrating a velocity law that is never told the answer. Within a turn it is not: the pins sit on a polygon, so the effective radius swings by $1 - \cos(\pi/z)$, which is 4.05% on an eleven-tooth sprocket and 0.18% on a fifty-three. The claim is right about the quantity people usually mean and wrong about the one that makes the noise.

“A belt drive’s centre distance is a free length.” For a belt, yes. For a chain it is quantised: the loop holds a whole number of pitches, so the admissible centre distances come in steps of 6.4834 mm on a 12.7 mm pitch, and the drive that was wanted at 415 mm can have 409.746 or 416.234. Insisting on an even link count doubles the step to 12.9626 mm.

What each varying number varies with #

Four of the six are a ratio quoted as a number when it is a function, and lumping them together loses the most useful thing about them: they are functions of three different kinds of argument, and the kind decides what can be done about each.

Functions of the configuration. A tackle’s velocity ratio depends on how far apart the blocks are, which is where the mechanism is. Measure the separation and the ratio is known; the quantity is recoverable at any instant from the state of the machine, and a controller with a position sensor can compensate for it exactly.

Functions of the design. A belt’s wrap angles depend on the two pulley diameters and the centre distance, which are decided once and then do not change. Such a number is not a variable at all in service — it is a constant that was quoted from the wrong formula, and the remedy is arithmetic rather than instrumentation.

Functions of the history. A winch’s line speed per turn depends on how much line is already wound, which is not a property of where any part is. Two states with every part in the same angular position differ in it, so no measurement at the shaft recovers it and something must count. That is the only one of the three that cannot be sensed away.

Sorted like that, the four failures have three different remedies rather than one. The tackle needs a sensor. The wrap needs a better formula. The winch needs a counter, and needs it structurally — an absolute encoder of any resolution whatever is the wrong instrument. And the shaped pulley is a fourth kind again: its ceiling is not a varying quantity at all but a bound on what can be demanded, which is a design-time refusal rather than a runtime variation.

That sorting is worth more than the list, because it answers the question a reader of six refutations actually has. Knowing that a number varies is the beginning; knowing what it varies with says whether the variation can be measured, computed away, or only counted. Three of those are cheap and one of them changes what instrument the machine has to carry.

It also predicts where the next member of the list will come from. Any strand quantity that depends on accumulated turns rather than on positions is in the third category, so a capstan’s several wraps and a multi-layer hoist are the places to look — and both are mechanisms this field already flags as sitting awkwardly with the rest of its arithmetic.

What this essay does not claim #

None of the six is a claim that the usual rules are useless. A tackle’s count, a belt’s half turn and a winch’s nameplate are all right to within a few per cent over most of their range, and each is the correct limit of the exact answer. What is claimed is that the departures are geometric rather than incidental, that they are largest exactly where the mechanism is being used hardest, and that every one of them can be computed before anything is built.

And none of it is about friction, which is the largest term in a real tackle and the reason a real capstan holds. The sixty per cent a six-part tackle actually delivers is mostly bearing loss; the two per cent computed here is the part that survives with every force unknown.