Where a strand stops touching

Every mechanism in this field has a boundary that rigid mechanisms do not: the configuration at which one of its bodies stops being part of the run.

Nothing about it looks like a limit. The mechanism does not lock, nothing loses rank, no equation runs out of solutions and no assembly becomes impossible. A pulley simply stops touching the strand, and everything carries on — except that from then on the pulley’s position has no effect on anything at all.

This essay is about how that boundary is approached, and the answer is a single exponent.

The quadratic #

Take a straight span of strand — a tangent run — and push a body into it. Let $d$ be the depth of the intrusion — how far the body’s surface has passed the line the strand used to run along — and ask how much extra strand the detour costs.

Measured on a 200 mm span with the body pressed into the middle of it:

Fitting the exponent between successive decades gives 2.0000, 2.0002 and 2.0014. The cost is quadratic in the intrusion, and stays quadratic over four decades before the higher-order terms are visible at all.

The closed form is the sagitta relation. A detour of depth $d$ into a span whose two parts are $a$ and $b$ long costs

$$\Delta L \approx \frac{d^2}{2}\left(\frac{1}{a} + \frac{1}{b}\right),$$

which for $a = b = 100$ mm is $0.01\,d^2$ — against the 1.0008 × 10⁻² measured at $d = 1$ and 1.0063 at $d = 10$, where the approximation is starting to earn its wiggle.

Why that makes the boundary stationary #

A quantity that grows as the square of a displacement has zero derivative at zero. So at the exact configuration where a body begins to touch a run, the run’s length is stationary with respect to that body’s position: moving the body does not change the strand’s length to first order.

The three quantities involved fit together exactly, and measuring all three at once is what makes the picture close.

The wrap grows linearly with the intrusion, at $1/a + 1/b$ radians per millimetre — 0.02 for the symmetric 200 mm span, measured at 0.020000. The take-up rate is the wrap, since $2\sin(\theta/2) \to \theta$ for small angles. And the length is the integral of the rate, which is $\tfrac12 \theta d$: quadratic, at half of $1/a + 1/b$, measured at 0.010000.

So the three statements — the wrap is linear, the authority is the wrap, the length is quadratic — are one statement differentiated twice, and each of them was measured separately off the run’s own geometry.

The numbers at the boundary #

On the serpentine drive, the idler’s contact interval runs from 4.1224 to 4.7740 radians of arm angle. At the lower end:

• wrap: 0.0141°
• take-up: 0.000245 mm of belt per millimetre of the idler’s own travel
• length: 1,013.373 mm, against the bare four-pulley loop’s 1,013.373 mm

That last line is the point. The run with the idler just touching is the same length, to the digits printed, as the run with no idler at all — because a body at the boundary of contact contributes an arc of no length and a detour of no depth.

So a tensioner has two ranges and they are not the same. Its contact range is where it touches — 37.3° of arm travel here. Its useful range is where it takes up an appreciable amount of belt, and it is strictly inside the contact range, with no marker at the boundary between them. An idler drawn just kissing the belt satisfies every drawing check anybody would apply and does nothing.

What the intrusion is measured against #

One subtlety decides whether the quadratic above is a fact about strands or an artefact of how the intrusion was defined, and it is worth settling.

The depth $d$ is measured from the strand’s own straight span as it would run with the body absent — not from the body’s centre, not from the line of centres, and not from where the strand ends up. For a body moving perpendicular to that span, the centre’s displacement and the surface’s intrusion are the same number, so the distinction does not bite here; for a body on an arm, swinging rather than translating, they differ by the cosine between the arm’s tangent and the span’s normal, and it is the intrusion that carries the law.

That is why the serpentine’s take-up per radian of arm does not vanish quite as cleanly as its take-up per millimetre of idler: the arm’s geometry multiplies the law by a lever that is itself changing. The take-up per millimetre at the contact boundary is 0.000245; the take-up per radian is 0.019. Both are as good as zero against their mid-travel values of 0.4387 and 25.90, and only the first of them is the geometry.

A run’s length is a hull, plus what the idlers add #

The quadratic has a global consequence worth stating, and the four-pulley loop above is the proof of it.

A taut loop over a set of bodies wrapped the same way is the boundary of their convex hull, and there is exactly one of it. Every additional body — an idler pressed in from outside, with a negative wrap — makes the run longer: the strand has to detour to reach it, and the detour costs the quadratic.

So the hull perimeter is a floor. No routing, no idler and no arrangement produces a strand shorter than the hull of the bodies that must be wrapped, and every tensioner in the world is a device for spending a controlled amount above that floor.

That also settles a question the field’s arithmetic raises and does not otherwise answer: why an idler’s wrap is negative. It is negative because the idler is outside the hull, and a body outside the hull is reached by bending the strand the other way — which costs length, adds to the unsigned wrap sum, and subtracts from the signed one.

The other version: joining the hull #

The same boundary appears in a closed loop as a membership question, and it can be bisected.

Take a taut loop over four pegs and bring a fifth in from inside the hull. While it is inside, the strand does not touch it and its position is irrelevant. Move it outward, and at a definite place it joins the hull and the strand starts to lie on it.

Bisecting on the number of bodies the loop touches puts that threshold at y = 107.888889 mm for the arrangement drawn here. Half a millimetre past it, the peg’s wrap is 0.8111° and the loop’s length has gone from 591.1005 mm to 591.1039 — three and a half microns for half a millimetre of intrusion, which is the same quadratic seen in a loop.

The two descriptions of the transition are worth keeping apart. The length passes through the boundary smoothly, with a continuous first derivative — that is the quadratic. The combinatorics do not: the run has four stations on one side of the threshold and five on the other, the senses list gains an entry, and every routine in this field takes a different argument. A quantity that is smooth and a description that jumps, at the same place.

What this is, in the vocabulary of the rest of the site #

A linkage’s characteristic boundary is a singularity: a configuration at which its constraint Jacobian loses rank, so that some output velocity becomes unobtainable or some input loses control of the mechanism. Toggles, dead centres and the two kinds of parallel-platform singularity are all of that kind, and this site’s linkage and platform fields are built around them.

The boundary here has the same flavour and a different mechanism. Nothing loses rank; the strand’s constraint is a single scalar equation and it has a perfectly good derivative everywhere. What vanishes is the gradient of the length with respect to one body’s position — so that body loses its ability to influence the constraint, while every other body keeps its own.

The practical form is the same as a singularity’s: a control input with no authority. The difference is that a mechanism’s singularity is a property of its configuration and this one is a property of a pair — the body and the run — so a mechanism can have several bodies at their own contact boundaries at once and be perfectly well behaved with respect to all the rest.

The same boundary in the other mechanisms #

Every mechanism in this field has one, and they are worth listing because the quantity that vanishes is different each time.

A tensioner loses its take-up. The wrap goes to zero and the belt stops noticing the arm.

A tendon over a joint loses its arm. Each idler’s wrap is its own joint’s angle plus a constant, so the wrap reaches zero at a definite joint angle — 202.8° of travel on the arm measured in the tendon essay — and past it the strand runs straight by and the drive is gone.

A peg between two anchors stops obstructing. At an offset equal to its own radius the taut path is the straight line, exactly, and the peg’s class stops being a wrap.

A tackle never reaches one, and the reason is instructive: every sheave in a tackle has a wrap near a half turn, and the geometry that would take one to zero — pulling the blocks so far apart that the parts of line become parallel and a sheave stops deflecting anything — is the limit in which the tackle’s ratio becomes its integer. So the tackle’s contact boundary is at infinity, and its whole design space is the interior.

A drum has no such boundary at all. Nothing is tangent to anything, the wrap is many turns, and the line is on the drum or it is off it. That is the sense in which the drum sits outside this field’s geometry while still being one of its mechanisms.

What the boundary does to a derivative #

The quadratic settles more than the take-up: it says exactly how smooth the length is at the boundary, and that has consequences for anything that differentiates it.

Approach from outside and the body is not touched, so the run’s length is whatever it is with the body absent — constant in the intrusion, and its derivative is zero. Approach from inside and the length grows as the square, so at the boundary the length is continuous and its first derivative is zero on both sides. The function is therefore once differentiable across the boundary and no more: the second derivative jumps from nought to the constant the sagitta relation supplies.

The wrap behaves one order worse and in the same place. It is zero outside and grows linearly inside, so it is continuous and its first derivative jumps. That is the kink the take-up figures are reporting, and it is why an idler at its contact edge has an authority of 0.000245 mm per millimetre rather than a smaller number of the same kind — the authority is not small because the mechanism is nearly at a limit, it is small because it is on the wrong side of a corner.

The practical consequence is about instruments rather than about belts. A gradient-based search over a strand mechanism’s geometry — placing an idler to achieve a wanted take-up, say — sees a perfectly well-behaved objective at the boundary, because the gradient exists and is continuous there. What it does not see is that the curvature is discontinuous, so every method that estimates a Hessian or fits a local quadratic is fitting across a kink and will produce a step based on a model that is wrong on one side. The search does not fail loudly; it converges slowly and to somewhere slightly wrong, which is the failure mode hardest to attribute.

The same shape of trap catches a finite-difference check. A central difference of the length taken across the boundary averages a quadratic with a constant and returns a number that is neither, and the departure shrinks with the step, so refining the step makes the reading look better while never converging to anything meaningful. The site’s habit of halving the step and watching the ratio is what catches it: at an ordinary point the second difference converges, at this boundary it approaches the one-sided value at half the rate, and the two are told apart by the same procedure that classifies a mutilated gear’s missing acceleration.

So the design rule set a tensioner well inside its contact range has a second justification alongside the one about authority. Inside the range every quantity of interest is smooth, the closed forms describe the mechanism, and both the optimiser and the check behave. At the edge the mechanism is still well defined and every instrument pointed at it degrades — which is a good deal more troubling than a small take-up, because a small take-up announces itself and a mis-fitted curvature does not.

Where the design rules come from #

Three, and all three fall out of the quadratic.

Set a tensioner well inside its contact range. Not because performance degrades near the edge, but because it is zero there and rises as the square root of the intrusion. The wrap angle is the honest reading: an idler with less than about ten degrees of wrap has less than a tenth of the authority of one with a hundred.

Do not design a pulley to be just touching. A body at its contact boundary is one manufacturing tolerance from not being in the mechanism. There is no warning in any drawing, and the failure — the strand runs past — is silent, since the drive continues to work at a slightly shorter belt length.

An idler that intrudes deeply is doing something expensive. The take-up exceeds one millimetre per millimetre past 60° of wrap, which sounds efficient and means the strand is being bent through 60° twice per lap for it. That is the unsigned wrap sum, and it is the cost half of the same decision.

Why this rung is where the field stops #

The contact boundary is the deepest thing this field has to say about its own object, and it is worth saying why.

The two sentences the field started from were that a strand has no shape of its own and that it constrains one way. The first gave the tangent construction, the wrap arithmetic, the turning number and the hull; the second gave slackness, the reachable region and the mobility that changes underfoot. This boundary is where the two meet: a body leaves a run because the strand’s shortest path stopped needing it, which is the first sentence, and the length constraint stops depending on it, which is the second.

Everything above it in the ladder was about a run that was assumed to exist. This is the arithmetic of the run’s membership, and it is the last question in the field that has a closed-form answer — the next ones are combinatorial. Which bodies a run should touch, given a set of them and a length, is a search; which routing is best is a search; and both go to a subject that owns search.

What is not modelled #

The strand has no thickness and no stiffness, so it turns through its wrap as a corner would; a real belt has a bending radius, so a body intruding by less than that radius is not deflecting the strand at all and the quadratic above is a statement about a line rather than about a belt. Nothing here models what happens at the boundary dynamically — a body oscillating about its contact threshold makes and breaks contact, which is an impact problem and needs masses. And the intrusion depth is measured geometrically against the strand’s own straight span; on a real drive the span is not straight, because a belt under load carries a catenary sag decided by its weight and tension, neither of which this field has.