Chain length and sprocket ratio: the formulas, a worked example and how many links to cut

A chain drive is two sprockets and a centre distance, and three numbers come out of them: the ratio, which sets the output speed; the chain length, in links; and the real centre distance once the chain is rounded to a whole number of links. This guide gives the formulas, a worked example for an ANSI 40 drive and the rules of thumb from the chain makers' catalogues.

The sprocket generator draws both sprockets once the tooth counts are chosen; the pitch and roller of each chain are in the roller chain size chart.

The formulas

Ratio and output speed

i = z2 / z1 ; n2 = n1 × z1 / z2 z1 is the driving sprocket, z2 the driven one and n the speed in rpm.

Chain length, in pitches

Lp = 2 C / p + (z1 + z2) / 2 + p × ((z2 − z1) / 2π)² / C C is the centre distance and p the chain pitch. Round Lp up to an even number: an odd count needs an offset link, the weakest link in the chain.

Real centre distance for a whole chain

C = (p / 4) × [A + √(A² − 8 × ((z2 − z1) / 2π)²)], with A = Lp − (z1 + z2) / 2 With the rounded Lp, the centre distance the chain actually gives, to set the tensioner or the slots of the motor base.

Wrap on the small sprocket

wrap = 180° − 2 × asin((d2 − d1) / 2C) d1 and d2 are the pitch diameters, p / sin(180° / z). Below about 120° the chain can jump the teeth.

Worked example: an ANSI 40 drive, 17 to 42 teeth

An electric motor at 1450 rpm, a 17-tooth driving sprocket, a 42-tooth driven one, and about 500 mm between centres.

The chain comes to 108.64 pitches, so it is cut at 110 links -- 55 inner and 55 outer -- which moves the centres to 508.7 mm. Slot the motor base for that, plus room to take up the stretch as the chain wears.

Sizing a drive, step by step

1. Pick the chain size from the chain maker's power ratings for the speed of the small sprocket.
2. Choose the teeth of the small sprocket, then the large one's from the ratio: z2 = z1 × n1 / n2, rounded to a whole tooth.
3. Choose the centre distance, 30 to 50 pitches, and work out Lp. Round it up to an even number of links.
4. Work out the real centre distance for that chain and check the wrap on the small sprocket: at least 120°.
5. Draw both sprockets in the generator with the chain's pitch and roller, and cut the chain at the link count.

Rules of thumb

• Centre distance between 30 and 50 pitches -- 380 to 635 mm on a half-inch chain. A shorter drive wears the chain faster, because each link flexes more often; with a pulsating load, though, keep it under about 20 pitches.
• More than 15 teeth on the small sprocket, and 17 or more on a drive that runs fast; no more than 120 on the large one.
• Up to about 7:1 in one stage. Beyond that, two stages run better than one huge sprocket.
• An even number of links, and some slack: a sag of about 4 % of the span on a normal drive, and 2 % on a vertical one, one over a metre between centres, or one that starts, stops or reverses often.
• At least 120° of wrap on the small sprocket.

Common mistakes

• Counting link pairs: the length is in pitches, and each inner or outer link is one. 110 links is 55 inner and 55 outer.
• An odd number of links on a fixed centre distance: it needs an offset link, which is weaker and wears first.
• Cutting the chain to the centre distance wanted: a chain is a whole number of links, so the shafts move to the real centre distance.
• Swapping z1 and z2: the output speed is the input speed times z1 / z2, with z1 the driving sprocket.

Frequently asked questions

How do I calculate chain length?

In pitches: Lp = 2C/p + (z1 + z2)/2 + p × ((z2 − z1)/2π)²/C, then round up to an even number. A 40 chain on 17 and 42 teeth at 500 mm centres: 108.64, so 110 links.

How do I work out a sprocket ratio?

Divide the driven sprocket's teeth by the driving sprocket's: 42 / 17 = 2.47. The output turns at the input speed divided by the ratio: 1450 rpm becomes 587 rpm.

How do I find the centre distance for a given chain?

With A = Lp − (z1 + z2)/2, C = (p/4) × [A + √(A² − 8 × ((z2 − z1)/2π)²)]. A 110-link 40 chain on 17 and 42 teeth gives 508.7 mm.

Why should a chain have an even number of links?

Because the links alternate inner and outer. An odd count needs an offset link, which is the weakest link of the chain and the first to wear.