The chain is a polygon

A gear tooth is an involute because that is the curve that keeps the velocity ratio constant while the contact point slides. The whole of the gears field is built on that: there is essentially one answer, and it is chosen precisely so that the ratio is a number.

A chain has no such curve. Its pins are pins, its links are straight and rigid, and what rolls on the sprocket is not a circle but a polygon through the pin centres.

That has a consequence everybody who has ridden a bicycle has felt at low cadence and almost nobody has been told the size of.

Where the four per cent comes from #

The pins sit on a circle of radius R = p / (2 sin(π/n)) for a chain of pitch p on a sprocket of n teeth. The strand leaves along a tangent, so the perpendicular distance from the sprocket’s centre to the strand — the radius that decides the chain’s speed — is R cos ψ, where ψ is the departing pin’s angular position measured from the tangent point.

Between one pin seating and the next, ψ runs from −π/n to +π/n. So the effective radius runs from $R\cos(\pi/n)$ up to $R$ and back, and the chain’s speed varies by $1 - \cos(\pi/n)$ for a sprocket of n teeth, whatever else is in the drive. It is 4.05% at eleven teeth, 1.70% at seventeen, 1.12% at twenty-one and 0.18% at fifty-three.

Every one of those is a closed form and every one is checked against it. The sprocket that dominates a drive is the small one, and on a bicycle the small one is very small.

The ratio, instant by instant #

Put two sprockets together and the ratio at any moment is the driver’s effective radius over the driven one — $\omega_2/\omega_1 = (R_1\cos\psi_1)/(R_2\cos\psi_2)$ — with each $\psi$ running through its own sprocket’s range and wrapping at its own pitch angle.

The corners are real. The identity of the pin the strand runs from changes discontinuously when the next pin reaches the tangent, and the derivative of the ratio jumps there. A chain drive is not a smoothly varying transmission; it is a sequence of smooth arcs joined at seatings.

The mean is exactly the tooth ratio, and that is the check #

The quoted ratio is not wrong. It is the mean, and the mean is exact.

That is not an assumption in the model — it is a test of it. The mean here is computed by integrating the instantaneous ratio over several turns of the driver, which uses only the velocity law and the phase relation. The tooth counts never enter the integration. The result is 4.81818 against 53/11 = 4.818181…, agreeing to three parts in ten million, which is the trapezoid rule’s own error at the sampling used.

If the cosine were wrong, or the wrap misplaced, or a factor of two lost, the mean would drift. It is the one property of the model that is known in advance, and it is therefore the one worth asserting.

The phase relation, which is exact and was nearly an integration #

There is a hidden variable in a chain drive and it is not obvious that it exists.

The two sprockets’ pin positions are not independent. They are connected by a taut strand of a whole number of links, so the phase of the arriving pin on the driven sprocket is set by the phase of the departing pin on the driver plus the fractional number of pitches in the strand.

The exact relation falls out of one observation. The chain’s travel while a pin swings from one seating to the next is R sin ψ, because the travel’s derivative is R cos ψ and that is exactly the effective radius. Over the full swing that gives 2R sin(π/n), which is the pitch — exactly. So the chain travel is a closed form in ψ on both sprockets, the two are the same travel offset by however much of a pitch the strand is out, and ψ₂ follows from ψ₁ with no differential equation at all.

The first version of this model did not notice, and integrated ω₂ with Runge–Kutta. It reported a fluctuation of 0.02% for two identical sprockets a whole number of pitches apart, which must be exactly zero: identical polygons in step deliver a constant ratio. Small enough to publish, and entirely the integrator’s.

What the chain’s length is worth #

Now the finding. Because the phase is a real variable, the same two sprockets can run smoothly or roughly depending on the chain’s length and where the wheel sits in the dropouts.

For two 21-tooth sprockets the phase decides everything: 0% to 2.25%, on the chain’s length alone, with no part changed. That is the strongest form of the result and it is the case a track cyclist can arrange, since a fixed drive has one chain and one pair of sprockets.

For sprockets that share no factor it is much weaker.

The general rule the two cases make between them: cancellation needs commensurate sprockets. When one tooth count divides the other, the two fluctuations have a common period and their relative phase is a fixed, adjustable thing. When they do not, the phase drifts through every value in a few turns and the best and worst are averaged out.

What the model is, and what it refuses to be #

It is worth being explicit about the model, because a chain drive can be modelled at several depths and this one is at the shallowest that is honest.

The chain is treated as pins on two polygons joined by a straight taut strand of a whole number of links. There is no mass, no tension, no sag, no tooth profile and no engagement impact. What the model contains is exactly the geometry: where the pins are, where the strand leaves and arrives, and the rigid-body consequence for the two angular velocities.

Two things follow from that, and they pull in opposite directions.

It is enough for the finding. Chordal action is a kinematic phenomenon — it happens with the chain massless and the drive turning arbitrarily slowly — so a kinematic model captures it completely, and the four per cent is not an approximation to something else.

It is not enough for anything a designer would do next. The reason chordal action matters in a real drive is that the ripple excites the chain’s own dynamics; the frequency at which it does so is n times the shaft speed, and whether it matters depends on where that falls relative to the strand’s natural frequency. None of that is here, and the site’s boundary is exactly at the point where mass enters.

The site has drawn that boundary in the same place three times now — in the cam field, where the acceleration is computed and the force it implies is not; in the practice field, where a clearance is a length and the impact when it closes is not; and here.

Wear moves a drive through the phase #

The phase relation is the finding of this essay and it is also the one quantity in the model that nothing in a workshop sets deliberately. It is worth following what happens to it over a chain’s life, because the answer is entirely inside the model and is not the effect anybody expects wear to have.

A worn chain has a longer pitch. That is what chain wear is — the pins and bushes lose material, each link grows by a fraction of a per cent, and the standard replacement criterion is elongation of about half to one per cent measured over a length of the chain.

Follow the pitch through the geometry and most of the results do not move at all. The pitch radius is $p / (2\sin(\pi/n))$, so both sprockets’ radii grow in exact proportion to the pitch, and the ratio of the two is unchanged. The fluctuation is $1 - \cos(\pi/n)$, which contains no length whatever, so it does not move either. The mean ratio is still the tooth ratio, exactly, and each sprocket still swings by four per cent or by whatever its tooth count dictates. On the two headline numbers of this essay, a worn chain and a new one are indistinguishable.

The strand is where it lands. The centre distance is set by the frame and does not change, so the strand’s length is fixed while the pitch it is measured in has grown — and the number of pitches in the taut strand, which is the hidden variable the phase depends on, falls. On a strand of forty pitches, a one per cent elongation removes four tenths of a pitch from the count. The fractional part, which is the whole of what the phase relation reads, has moved by nearly half a pitch: not a perturbation of the phase but a substantial traverse of it.

So a chain drive drifts through its own phase relation as it wears, with nothing else about it changing. For the commensurate case, where the phase decides everything between 0% and 2.25%, that is the difference between a drive that cancels its two fluctuations and one that adds them, arriving purely from elongation. A drive assembled at a fortunate strand length does not stay there, and one assembled at an unfortunate one will pass through the fortunate value on its way to somewhere else.

That is a genuinely useful consequence and it is worth being careful about how much it claims. It says that the smoothest chain drives are the ones whose smoothness does not depend on the phase — which is the ordering the design rules above already give, with tooth count first and strand length last. The third rule was called the interesting one and the least useful; this is the sharper reason. A design that relies on the phase term is relying on a quantity that its own wear destroys, and it will hold for a few thousand kilometres rather than for the life of the parts.

It also gives a kinematic account of something usually attributed to other causes. A drive that has grown rough with age is normally explained by hooked teeth, by a stiff link, or by the chain riding higher on worn sprockets — all real, and all outside this model. The geometry says there is a fourth mechanism that needs none of them: the same parts, undamaged, with the pitch a half per cent longer, sitting at a different point of the phase relation and adding two fluctuations that used to cancel.

Whether that is what anybody actually hears is not settled here and could not be, since the model has no masses in it and roughness is a dynamic response. What the model does establish is that the geometric input to that response changes over a chain’s life, by an amount comparable to the whole range of the effect, and from a cause that is measured in every workshop with a ruler.

Why nobody notices on a bicycle #

Four per cent of the back wheel’s speed, eleven times per crank turn, is a real oscillation. It is not felt for three reasons, and it is worth naming them because they are all outside the geometry.

The wheel and rider have inertia; the chain and the freehub have compliance; and the drive is being turned by a person, not a stiff machine. Between them they filter a small high-frequency ripple.

Where it is noticed is where those three are absent: in low-speed, high-torque chain drives with big loads and rigid mounts — conveyors, hoists, motorcycle final drives at low speed — where chordal action is a standard design consideration and the standard mitigation is the one this essay’s table supports: use more teeth on the small sprocket. Nothing else in the drive helps as much, because the variation is 1 − cos(π/n) and n is the only variable in it.

What a smooth chain drive would need #

Put the three results together and the design rules follow without a further computation.

More teeth on the small sprocket. The fluctuation is 1 − cos(π/n) and n is the only variable in it, so this is not one lever among several — it is the lever. Going from eleven teeth to seventeen cuts the variation by more than half; going to twenty-one cuts it by nearly four.

Commensurate sprockets, if the drive allows. Two sprockets whose counts share a factor have fluctuations with a common period, and their relative phase is then a fixed quantity that can be set once. A bicycle’s derailleur drive cannot use this — the sprockets change — but a fixed drive, a camshaft chain or a conveyor can.

Then the strand length. Only worth adjusting when the first two are settled, and worth a factor of two in the commensurate case and nine per cent otherwise.

The order matters: the third rule is the interesting one and the least useful, and presenting it first would be the sort of finding that is memorable and misleading.

A fourth rule is available and this essay cannot support it: run the drive fast enough that nothing responds. That is a dynamic argument — the ripple is at n times the shaft speed, and above the system’s natural frequencies it is filtered by inertia — and it is the reason a motorcycle’s chain is quiet at road speed and rough pulling away. It is named here because leaving it out would make the geometry look like the whole story, and it is not computed because a natural frequency needs a mass.

The tooth field, arriving at the opposite answer #

The gears field’s whole argument is that a tooth profile can be chosen so the ratio is constant, and it is: an involute flank delivers a constant velocity ratio because the common normal passes through a fixed point at every instant of the mesh.

A chain cannot do that, and the reason is structural rather than a failure of design. The chain’s links are rigid and straight, and a straight link between two pins on two circles cannot maintain a constant velocity ratio — there is no profile to choose, because there is no profile.

Which makes the chain the honest middle of the field’s ledger. Its number is not a quoted one — there is a quantity, the mean, and the mean is exactly what is written on the box. It is a number about a whole turn being read as a number about an instant.

What is not modelled #

The chain’s mass and tension, which is what actually makes chordal action audible — the strand’s transverse oscillation is a dynamic response to the geometric one, and this site computes only the geometry.

Wear, which lengthens a chain’s pitch and therefore changes both the pitch radius and the strand’s fractional length. A worn chain is a chain whose phase has drifted, which is a nice way to see why the same drive gets rougher with age even before the sprockets are damaged.

And the slack strand entirely. Everything here is about the taut side; the slack side’s geometry is a catenary problem with its own frequency, and it is what the tensioner on a derailleur drive is for.

One more thing is not here, and it is the one a reader may be waiting for: the tooth profile of the sprocket. A chain sprocket’s teeth are shaped to receive and release the rollers cleanly, and the shape is standardised — but it has no effect on any number in this essay, because the chain’s position is decided by where the pins seat and not by the flank they slide down on the way in. That is the exact opposite of the situation in a gear mesh, where the flank is the whole mechanism and the tooth’s position on the wheel is incidental.

What is here is the geometry, and the geometry says three things clearly: the fluctuation is 1 − cos(π/n) and nothing else, the mean is the tooth ratio exactly, and the phase is a design variable that no drawing records.

The next essay stays on the tooth ladder and takes the one drive in this field whose quoted number survives everything: a reduction that is a ratio of integers, exact at every position, and fragile in a completely different way.