A chain is not a strand

Everything else in this field is a continuum. A belt, a rope, a tendon and a winch line are all treated as a line of no thickness that can be any length and can lie on any curve, and the whole of the geometry — tangency, wrap, turning number, the shortest path — depends on that.

A chain has a smallest piece. Two things follow that the continuous model cannot express, and both of them are things a designer has to deal with.

The pins are on a polygon #

A chain’s rollers seat in the sprocket’s teeth, so its pins sit at the vertices of a regular polygon inscribed in the pitch circle rather than on the circle. The chord between two of them is the pitch; the pitch radius is therefore $p / (2\sin(\pi/z))$, which is 22.539 mm for an eleven-tooth sprocket at 12.7 mm pitch and 107.190 mm for a fifty-three.

The polygon’s perimeter is $zp$, and the circle’s is $2\pi R$. They differ:

The consequence for the drive is that the effective radius — the perpendicular distance from the axis to the taut strand — is not the pitch radius. It swings between $R\cos(\pi/z)$ and $R$ within every tooth, as a pin comes into engagement and rides round, so it varies by $1 - \cos(\pi/z)$:

That is chordal action, which this site has already measured from the drive’s side — where the interesting result is that the mean comes out at exactly the tooth ratio, and how much of the fluctuation reaches the output is decided by the fractional number of pitches in the taut span. What belongs here is only the shape of the dependence: it falls as $1/z^2$ for large $z$, so doubling the smallest sprocket quarters the problem, and there is no other lever on it at all.

A strand has no such number, because a strand has no pitch. Everything in this field’s continuous geometry is the $z \to \infty$ limit of a chain.

The loop is an integer #

The second consequence has no continuous analogue whatever.

A chain closes on itself, so its loop contains a whole number of pitches. The run’s length is therefore not free: it is $Np$ for integer $N$, and since the length is a function of the centre distance, the centre distance that closes the loop is quantised.

Take 53 teeth driving 11 at 12.7 mm pitch, at a nominal 415 mm between the axles. The taut loop there is 98.8097 pitches, which is not a number of links. The admissible centre distances near it are:

The step is 6.4834 mm, and the reason it is about half a pitch is immediate: the loop crosses the gap twice, so lengthening it by one pitch moves the axles apart by half of one. The reason it is not exactly half a pitch — 6.4834 against 6.3500, which is 2.1% more — is that moving the axles also changes the wrap angles, taking a little arc off the large sprocket and adding it to the small one, and the straight spans have to make that up as well.

The remainder has to go somewhere #

A designer who wants 415 mm between axles cannot have it. The options are the ones every chain drive uses:

Move the axle. Take 416.2342 mm and use ninety-nine links. This is what a bicycle’s horizontal dropouts are for, and the adjustment needed is at most half the step — 3.24 mm — from wherever was wanted.

Take up the remainder with a tensioner. Fit ninety-eight links at 415 mm and the loop is 0.81 of a pitch too short; fit ninety-nine and it is 0.19 too long. An idler set into the slack span absorbs the difference, and how much it can absorb is the take-up range of the tensioner — which for a 100 mm arm swinging through 37° on a similar drive was 17.6 mm, comfortably more than one step.

Use an offset link. A chain with an odd number of links needs a half-link to close, which is a part with a different geometry and a reputation for being the weakest piece in the drive. Avoiding it makes the quantum a whole link rather than a half, so the centre-distance step becomes 12.9626 mm — and the adjustment needed can then be up to 6.5 mm.

The last of those is the one worth stating as a rule: an even-link chain doubles the centre-distance quantum. It is a constraint that comes entirely from the part count and has nothing to do with any length in the drawing.

Why the step is not exactly half a pitch #

The 2.1% by which the centre-distance step exceeds half a pitch is worth taking apart, because it is the only place in this arithmetic where the wrap does any work.

Going from ninety-eight links to ninety-nine — 409.7464 mm to 416.2342 — the run’s parts move like this:

The total goes up by exactly one pitch, which is the condition being solved. But the straight spans go up by 13.257 — more than a pitch — and the arcs come down by 0.557, because pulling the axles apart takes 0.377° of wrap off the large sprocket and gives it to the small one, and the large sprocket’s arc is worth four and three-quarter times as much per degree.

So the two spans have to make up the pitch and the arc’s loss together: $2 \times 6.4834 = 12.9668$ mm of centre distance buys 13.2571 mm of span, of which 12.7001 survives as extra loop. Half a pitch would be the answer for two sprockets of equal size, where moving the axles transfers no wrap at all.

The link that meets one tooth for ever #

There is a second integer condition on a chain drive, it has nothing to do with lengths, and it is the one that decides how the drive wears.

The chain has $N$ links and the sprocket has $z$ teeth. After one lap of the chain the sprocket has turned through $N/z$ of a revolution, so a given link comes back to the sprocket at a tooth offset by $N \bmod z$. The number of distinct teeth that link will ever meet is therefore

$$\frac{z}{\gcd(N, z)}.$$

Which is a table worth reading:

A ninety-nine-link chain on an eleven-tooth sprocket puts every link on the same tooth, for ever. Whatever variation exists between the links — and there is always some — is delivered to one tooth on every lap, and whatever variation exists between the teeth is delivered to one link. Change to ninety-eight or a hundred and each link visits all eleven teeth in turn.

Nothing in the geometry reports this. The drive with a common factor is exactly as long, exactly as taut and exactly as smooth as the ones either side of it, and it draws identically. It is arithmetic on two integers that appear nowhere in the drawing.

It also collides with the even-link rule from two sections up, which is the useful part. An odd link count avoids the offset link’s weakness only by needing one; an even count avoids the offset link; and either may or may not share a factor with the tooth count. On this drive the three candidates are ninety-eight (even, no common factor, 5.75 mm from the wanted centre distance), ninety-nine (odd, common factor of eleven, 1.23 mm away) and a hundred (even, no common factor, 7.72 mm away). The nearest is the worst on both other counts, which is a fair summary of how chain drives are actually chosen.

What a chain’s wrap means #

The wrap angle is still computed the same way, and it now has to be read differently.

On this drive the wraps are 203.539° on the large sprocket and 156.461° on the small one. Divided by each sprocket’s own tooth pitch — 360/53 and 360/11 degrees — those are 29.966 teeth and 4.781 teeth in engagement.

Neither is a whole number, and neither should be: the chain’s pins arrive where the geometry puts them and the last tooth at each end of the wrap is partly engaged. But only whole teeth carry, so what a designer reads off the wrap is the floor: twenty-nine and four. The fractional part is the phase relationship between the two sprockets, and it changes as the chain runs — which is precisely the quantity the chordal-action analysis finds decides how much of the four per cent reaches the output.

Five teeth of engagement on the small sprocket is the standard minimum in chain practice, and the geometric reason for it is here: below about 17 teeth the wrap starts falling towards a half turn as the drive ratio rises, and each tooth of engagement is worth $360/z$ degrees, which is more on a small sprocket. The two effects work against each other and the sum is what has to be checked.

The strand is still the right model for the spans #

None of this means the continuous geometry is wrong for a chain. It means it is right about the spans and wrong about the wraps.

The straight parts of a chain are genuinely straight lines tangent to the two pitch circles, to whatever accuracy the pins are made to, and the tangency solve gives them exactly. It is only on the sprockets that the strand and the chain part company — the strand lies on the circle and the chain sits on the polygon inside it — and the difference is the 1.4% of a small sprocket’s circumference computed above.

So the useful division is: compute the run as a strand, then correct the wraps for the polygon. The length comes out within a fraction of a per cent, which is enough to pick a link count, and the link count is an integer so it is either right or wrong rather than approximately right.

Where else a smallest piece appears #

A chain is the clearest case and not the only one. A toothed belt has the same integer constraint: its length is a whole number of belt pitches, and its wrap has to contain enough teeth to carry the load. A cable in a groove with a fixed lay has a periodicity too, though a much weaker one. And a rope on a drum has a smallest piece in the other direction — its thickness, which is what makes the layer arithmetic a staircase rather than a smooth spiral.

What all of them share is that the discreteness enters as an integer condition on a length, and the length was a continuous function of the geometry. So the design problem changes shape: instead of solving an equation for a configuration, a designer solves it for a length, rounds the length to the nearest admissible value, and then has to find somewhere to put the difference.

That is a recognisable pattern from elsewhere on this site — a reverted gear train’s coaxial condition is a second Diophantine equation on top of a ratio — and it fails in the same way. Rounding is not a small perturbation when the thing being rounded is the count: the nearest admissible answer can be a long way from the one that was wanted, and no amount of care in the continuous part of the design changes where the admissible values are.

The polygon is where the two models actually differ #

One last comparison, to say precisely how much of this field’s continuous machinery survives.

The straight spans are exact. A chain’s pins on the two tangency lines sit where the strand’s tangency solve puts them, and the only error is that the pins are at discrete stations along the line rather than everywhere on it — which changes nothing about the line.

The arcs are where the two models part. The strand lies on the pitch circle and the chain sits on the polygon inside it, so the chain’s path round a sprocket is shorter than the strand’s by the difference between $zp$ and $2\pi R$ — 1.35% on eleven teeth and 0.06% on fifty-three. On the loop above, the arcs are 442.79 mm and the polygon’s version of them is about 6 mm less.

That is the whole of the model error, and its size explains the practice. Six millimetres on a 1,245 mm loop is half a pitch, so a link count picked from the strand model can be one link out — which is why chain length is chosen from a formula, checked, and then adjusted by the tensioner or the dropout in any case.

The one thing the chain shares with the belt #

For all the differences, the chain’s design problem has the same shape as the belt’s, and the shape is the field’s.

Both are one strand with one length. Both close on one scalar equation. Both need a tensioner when the length available and the length required do not match, and in both cases the tensioner’s authority is its own wrap. The chain’s extra ingredient is that its length comes in lumps, so the mismatch is guaranteed rather than accidental — a belt can be made to whatever length is wanted and a chain cannot.

The two integer conditions disagree, and this drive shows it #

The link-count parity and the tooth-meeting condition are named above as colliding, and the collision is worth working out on this essay’s own drive, because it changes which of the three remedies is right.

The parity rule wants an even number of links, so the chain closes without an offset link. The wear rule wants $\gcd(N, z)$ to be one, so a given link does not meet the same tooth for ever. Both are conditions on $N$, both are cheap to check, and neither is in the length arithmetic.

Run them on the two candidates. Ninety-eight links: even, so no offset link; and $\gcd(98, 11) = 1$, so every link meets every tooth. Ninety-nine links: odd, so an offset link is needed; and $\gcd(99, 11) = 11$, which is the worst possible case — every link meets exactly one tooth, for the life of the chain.

So the candidate that is closer to the wanted 415 mm is the one that fails both integer conditions. 416.2342 mm with ninety-nine links is 1.23 mm from the target and is a drive with an offset link and a fixed link-to-tooth pairing; 409.7464 mm with ninety-eight is 5.25 mm away and is clean on both counts.

That inverts the obvious ranking, and it is the useful form of the finding. The nearest closing centre distance is not the best one, because two of the three conditions on a chain drive have nothing to do with length. A designer choosing by proximity alone will pick the wrong link count about half the time, and neither failure announces itself — an offset link is a component that works, and a fixed link-to-tooth pairing is a wear pattern nobody sees for a year.

The practical procedure that follows is short and it is not the one a length calculation suggests. Enumerate the link counts that close near the wanted centre distance, discard the odd ones, discard the ones sharing a factor with either tooth count, and then take the nearest survivor — moving the axle to suit, since the axle is the adjustable thing and the chain is not. On this drive that means ninety-eight links and a 5 mm move, which is what a chain-drive designer would do by habit and now has three reasons rather than one.

What is not modelled #

Nothing here models the chain’s articulation: a real link has a roller, a bush and a pin with clearance between them, and the drive’s actual pitch is a function of wear — a “stretched” chain is one whose pins have worn, so its pitch has grown by a fraction of a per cent and its rollers sit further out on the sprocket teeth. That wear is the reason chain drives are replaced, and it is a contact-mechanics argument this field has nothing to say about. The tooth form is absent entirely: the sprocket here is a circle through the pin centres, with no flanks, no clearance and no seating geometry. And the chain is treated as taut throughout, where a real one has a slack span that hangs — a catenary, decided by weight, which is not a quantity this field has.