Chain Length in Pitches

A chain drive is a belt drive that has given up on friction. Instead of gripping, it meshes, which means it does not slip, does not need initial tension, and does not lose capacity at low speed. What it gains in certainty it pays for in quantisation: a chain is made of whole links, so its length is not a continuous variable.

The length in pitches:

\[ L = \frac{2C}{p} + \frac{N_1 + N_2}{2} + \left(\frac{N_2 - N_1}{2\pi}\right)^2 \frac{p}{C} \]

The structure mirrors the belt equation exactly, and for the same reasons. The first term is the two straight spans measured in pitches. The second is the wrap — a chain lying on a sprocket occupies one pitch per tooth, and half the teeth of each sprocket are engaged, so the two half-wraps sum to $(N_1 + N_2)/2$ pitches. The third is the small correction for the fact that the wraps are not exactly half each when the sprockets differ in size.

Round up, and round to even

The answer will not be a whole number and it must be. Round up, because a chain that is slightly long can be taken up by moving the centres or fitting a tensioner, and one that is short cannot be stretched. Then round up again to an even count if you can, because an even number of pitches closes with a plain connecting link. An odd count needs an offset (cranked) link, whose plates are bent and therefore weaker than the rest of the chain — it becomes the designed-in weak point, and a chain that fails at its offset link almost always failed because it was an odd length.

Chain drives want slack

This is the opposite instinct from belts and it catches people. A tight chain has nothing to gain and a great deal to lose: it loads the bearings, it prevents the chain from articulating freely onto the sprocket teeth, and it accelerates joint wear. Aim for about 2% of the centre distance as sag in the slack span, and arrange the drive so the slack span is on the bottom, where gravity holds the sag rather than fighting it.

Chordal action, and the number 17 again

A chain does not wrap a sprocket in a circle — it wraps it in a polygon. As each link seats, the effective radius changes between the pitch radius and the radius at the flat of the chord, so the chain speed pulses at tooth frequency. The variation is $1 - \cos(\pi/N)$, which is 4.9% on an 11-tooth sprocket, 2.1% on 17, and under 1% at 25. This chordal action is the source of chain drive vibration and noise, and it is why the practical minimum is around 17 teeth on the driver for anything running at speed. It falls away quickly with tooth count, which is the whole argument for a larger driver sprocket.

Sizing the pitch

Pitch is not free — it comes in a standard series. The ANSI number gives the pitch in eighths of an inch, so number 40 is ½ in (12.7 mm), 50 is ⅝ in, 60 is ¾ in and 80 is 1 in; ISO 606 lists the same family in metric designations. A larger pitch carries more load per strand but worsens the chordal action and the noise, which is why a heavily loaded high-speed drive often uses multi-strand chain in a smaller pitch rather than a single strand in a larger one. And a chain "stretches" only in the sense that its pin and bushing joints wear — the plates do not elongate — so a chain replaced for being long is really being replaced for being worn.

The window for centre distance

Thirty to fifty pitches is the usual target. Shorter than about thirty and the wrap on the small sprocket suffers while each link articulates more often per hour, which is what actually wears a chain out. Longer than about eighty and the slack span needs supporting or it whips.