A member with no length of its own

Every mechanism drawn on this site so far has been made of rigid bodies. A link holds two points a fixed distance apart; a pin removes two freedoms; the configuration is whatever satisfies all of it at once, which is why a position here is the output of a solve rather than a decision. That assumption is so deep in the machinery that it is easy to miss it being an assumption at all: it is in the loop-closure residual, in the constraint Jacobian, in the mobility count, and in the word length.

A belt is not that. Neither is a rope, a chain, a tape, a tendon or a winch line, and the difference is exactly two sentences long.

A strand has no shape of its own. Where it runs is decided entirely by the bodies it touches. Between two of them it is a straight line tangent to both; on one of them it lies along the surface. Nothing it carries decides anything, which is why a picture of a belt drawn freehand is almost always wrong in a way a picture of a four-bar is not — the illustrator has to invent a curve, and there is only one right one.

A strand constrains one way. A bar holds two points at a distance. A strand holds them at no more than a distance, and does nothing whatever until it is taut. The constraint is an inequality, and an inequality is not something the rest of this site’s machinery can take: a rank cannot be computed for it, a Jacobian has no row for it, and the mobility it removes depends on where the mechanism is standing.

One closed form, and the sign that is the whole of it #

The first sentence is what makes this field computable, and it collapses into three lines of arithmetic.

Take the direction of travel along a straight run to be $\mathbf{u}$, and let $\mathbf{n}$ be its left normal. Give each body a sense: $+1$ if the strand wraps it anticlockwise, $-1$ if clockwise. Then a body of radius $r$ and sense $s$ has its centre at $+s\,r\,\mathbf{n}$ from the strand — to the left if the sense is positive, to the right if it is negative. So the vector between two bodies’ centres decomposes as

$$\mathbf{c}_2 - \mathbf{c}_1 = L\,\mathbf{u} + (s_2 r_2 - s_1 r_1)\,\mathbf{n},$$

which is a right triangle with the centre distance as its hypotenuse. The run’s length is $\sqrt{D^2 - \Delta^2}$ with $\Delta$ the signed radius difference, and its direction is the direction between the centres turned by $\arctan(\Delta / L)$.

That is the whole of it, and the signed part is what makes it worth writing down. Two bodies wrapped the same way give $\Delta = r_2 - r_1$ and the outer tangent; wrapped opposite ways they give $\Delta = r_2 + r_1$ and the tangent that crosses between them. There is no case analysis, no branch and no search anywhere in this field’s geometry — a crossed belt is an open belt with one sign changed.

The refusal matters as much as the formula. When $D < |\Delta|$ there is no run: one body is inside the other for a same-sense pair, or the two are closer than the sum of their radii for a crossed one. The routine says so rather than returning a square root of a negative number and drawing something.

What a run returns, and which of it is a measurement #

Hand an ordered list of bodies and senses to one routine and three things come back.

The length is the constraint. It is the sum of the straight runs and the arcs, and it is the only quantity a strand imposes on the bodies it touches.

The wrap angles are what each body feels: how much of its surface the strand lies on, and — since a strand does not slip — how much of its rotation the strand can transmit.

The turning is the sum of the wraps, each signed by its sense, and it is a different kind of quantity from the other two. It is not a measurement of the geometry at all. Its value is fixed by the route the strand takes and is unchanged by moving any body anywhere the run still exists.

That is the cheapest check this field has and the one it leans on hardest. The wraps are computed one at a time, each from a pair of tangent lines that knows nothing about the others; adding them and finding exactly $2\pi$ — to $2 \times 10^{-15}$ radians across forty randomly proportioned belts — is arithmetic about a quantity nothing in the computation was ever given. It also fails loudly for the one mistake this geometry invites, which is taking a wrap the short way round when the strand goes the long way: that error adds a whole turn, and it is invisible in a drawing of a path that is nearly closed anyway.

A strand’s closure is one equation, not two #

The second thing worth saying about a run is what kind of constraint it is, and it separates this field from every other one here.

A linkage closes when a vector comes back to where it started. That is two scalar equations in the plane and six in space, which is why a four-bar with one free joint angle is determined and why the solver on this site is a Newton–Raphson iteration on a residual with components.

A strand closes when a number comes back: its length. One equation, however many bodies it runs over. So a run with one free body is determined and a run with two is not — a serpentine belt with two spring-loaded idlers has a one-parameter family of taut configurations, and choosing between them needs something this field does not have.

The consequence is not a technicality. It is why a belt drive can be designed by choosing a stock belt length and letting the idler find its own position, and why a chain drive cannot: a chain’s length comes in whole pitches, so the equation has no solution at all for most centre distances and the mechanism must be given somewhere to put the remainder.

Six systems, one routine #

The ledger at the top of this essay is the field’s argument in one table. Each row is a list of bodies with senses handed to the same function; the middle column never changes, and the right-hand column is what each row was checked against.

A belt on two pulleys — the object a variator solves for — is checked against the textbook length formula — the one with $\sin\gamma = (r_1 - r_2)/C$ in it — and agrees to $1 \times 10^{-13}$ mm over forty-two proportions, open and crossed.

A crossed belt is checked against an integer: its turning number is zero, because its two wrap angles are equal and opposite whatever the two radii are. Both wraps here come out at 227.156357°, which is not a coincidence and is proved in the essay on wraps.

A loop over five pegs is checked against the convex hull of the pegs. The strand’s straight runs are the hull’s edges pushed outward by one radius, and its arcs are the hull’s exterior angles, which add to one turn — so the whole strand is 503.136 mm of hull perimeter plus 87.965 mm of one circle, and the number of pegs does not enter it.

A block and tackle is checked against the integer it is sold as — and does not match it, which is the subject of its own essay.

A serpentine with a tensioner is checked by solving its length equation and reading the residual, which is $1 \times 10^{-12}$ mm on a run of a metre.

A tendon over two joints, which is how an arm is driven from its base, is checked against the radii of the idlers it runs over, which it reproduces to $2 \times 10^{-8}$ mm per radian — the floor of the finite difference that measures it, rather than a departure.

One sign, and the output turns the other way #

The signed radius difference is worth one more paragraph, because changing a single sense is a design decision rather than a notational one.

Cross the strand between two pulleys and both of them still turn — at the same ratio, since the arms are unchanged — but the driven one now turns the other way. Nothing else about the mechanism is different. There is no idler, no extra wheel, and no tooth count to rearrange; the reversal costs one crossing, and what it costs in the geometry is 24.358 mm of extra strand on this pair and a rise in both wrap angles from 191.478° and 168.522° to 227.156° and 227.156°.

That the two wraps come out equal is the first result in this field that could not have been guessed from a drawing. It says that a crossed belt shares its grip evenly between a large pulley and a small one, which an open belt emphatically does not: the open pair above gives the small pulley 168.5° and the large one 191.5°, and it is always the small pulley — the one with less surface and usually the driven one — that gets the smaller share.

A length is worth differentiating #

The last thing to say about a run before the ladder starts is what to do with the number it returns.

A length on its own is a constraint. Its derivative is a ratio. Move one body and ask how much strand that costs, and the answer is a velocity ratio; turn one body and ask the same, and the answer is an arm. Neither of those quantities is in the geometry as such — both are obtained by differentiating a length that was computed for other reasons — and between them they account for most of what this field measures.

The reason to measure them this way rather than to write them down is that the closed forms are all correct for the arrangement they were derived for and silently wrong for any other. A tackle is sold as four to one; the derivative says 3.927 with the blocks 220 mm apart. A tendon over a joint is quoted as a radius; the derivative says the radius exactly, and only while the idler is on the axis.

What the strand ignores #

A peg inside the convex hull of the others is not touched at all. Its wrap is not small; it is absent, and the strand’s length is completely indifferent to where that peg sits as long as it stays inside.

This is a failure mode with no analogue among rigid mechanisms, and it is the reason this field’s routines refuse as often as they compute. A tangent line between two bodies exists whether or not a strand could ever lie on it. Ask for a run over a body the strand does not reach and the formula still answers — it returns a path that wraps that body by nearly a whole turn and cuts straight through everything in between. It draws perfectly. The three tests that catch it are geometric and arithmetic: no straight run may pass inside a body, no two runs may cross, and the turning must be what the route says. The last of those is the cheapest, and it is the one that found the case: an idler set just outside the span it was meant to press into, with a wrap of 348° where 6° was intended.

Where the boundary is #

Three things that belong to a strand in the world are deliberately not in this field, and the line between them and what is here is sharp enough to state as a test: every number here survives with every force in the mechanism unknown.

Tension is not here. What a strand pulls with, how that tension varies round a wrap, and how much of it friction will hold — the capstan relation and everything downstream of it — is a force argument and belongs elsewhere. What is used here is only that a taut strand lies on the shortest path its route permits, which is a statement about geometry.

Stretch is not here. Every strand in this field is inextensible. A real belt lengthens under load and creeps round its pulleys, and both of those need a stiffness. The word this field uses is taut, and it means the length constraint is active, not that the material is doing anything.

The smallest piece is mostly not here. A rope is a continuum for these purposes and a chain is not; the difference gets an essay of its own, and everywhere else the strand is continuous.

What this changes about the rest of the site #

Two of the essays that came before this field already met a strand and treated it as a length. The variator’s ratio with no steps in it solves for a running radius by holding a belt length constant, which is this field’s one scalar equation seen from the transmission side. And the chain is a polygon measures the four per cent of chordal variation an eleven-tooth sprocket imposes, which is a statement about pitches rather than about strands.

Neither of them needed the object. What they needed was one number, and both got it. The reason to build the object anyway is that the number is the least interesting thing a strand has: what a strand really brings is a constraint that switches on and off, a path with more than one class, an arm that is not a radius, and a mobility that is a function of position.

The ladder from here #

The field is built in the order the geometry allows. Where a strand leaves a body, and why tangency is a consequence rather than a construction. What the wraps add up to, and why that is an integer. Then the two things a length can be differentiated into — a velocity ratio, and an arm — and the two things an inequality does: a region instead of a configuration, and a mobility that changes underfoot.

After that, the applications that are really about the object: a drum whose working radius is a function of how much line is on it, a chain whose loop is an integer, a tendon that is linear only when its idlers are on the joint axes, and the peg that gives a strand two shortest paths with no way between them.

What is not modelled #

No strand here has a thickness, except on the winding drum, where thickness is the entire subject. No strand has mass, so nothing sags; a slack strand in this field is a length inequality that is not active, and its shape is not computed because nothing here decides it. No pulley has a groove, no sheave has a bearing, and no chain has a roller: a sprocket is a circle through the pin centres and nothing else. Tension, friction, creep and wear are all absent, and every claim in the field is written so that they can be.

It is worth naming what makes this field’s members so different from the rest of the site’s in one sentence, because the two defining properties are really one. A strand has no shape of its own and constrains in one direction only, and both follow from its being a length with no rigidity: a member that cannot push cannot hold a shape, and a member that holds no shape takes the shape of whatever it touches. So the two sentences the field opens with are not two independent assumptions to be checked against a real rope; they are one idealisation stated twice. That matters for knowing what breaks it. Anything that gives the strand a bending stiffness gives it both a shape of its own and the ability to push a little, at the same time and in proportion — a wire rope departs from both idealisations together, and a chain departs from the first while keeping the second exactly. Which is why a chain is the honest middle of the field: it has a smallest piece and therefore a shape, and it still cannot push.