Required roller chain link count

What does this quantity describe?

Roller chains are supplied only in whole links, usually an even count, to avoid needing an offset connecting link. From tooth counts, chain pitch and a desired center distance a₀, a theoretical, generally non-integer link count X₀ follows first; this is rounded up to the next even integer X. From X, the actually achievable center distance a follows, which deviates slightly from the original target a₀.

Picture shortening a pipe to a stock length: you want a specific length, but must round up to the next available stock size and adjust the rest of the design — here, the center distance — to that available length.

Formula and variables

X₀ = 2a₀/p + (N₁+N₂)/2 + (N₂−N₁)²p/(4π²a₀); X = 2·⌈X₀/2⌉

• X₀ = 2a₀/p + (N₁+N₂)/2 + (N₂−N₁)²p/(4π²a₀)
• X = 2·⌈X₀/2⌉
• a = p/4·[(X−(N₁+N₂)/2) + √((X−(N₁+N₂)/2)² − 2((N₂−N₁)/π)²)]

Choose the inputs correctly

Tooth counts N₁ and N₂, chain pitch p and the desired center distance a₀ set the theoretical link count X₀. The practical link count X is rounded up from that to the next even number; the actual center distance a can be computed either from the rounded X or independently for an already-known X, for example an existing chain.

How to use the calculator

Choose X as the target to find the link count needed for a desired center distance. If an existing chain with a known link count is given instead and the resulting center distance is wanted, choose a as the target and enter the existing link count as X.

Worked example

17/51 teeth, a 19.05 mm pitch (ANSI 60) and a 500 mm target center distance give X₀ ≈ 87.61 links. Rounded up to the next even number, X = 88 links, giving an actual center distance of about 503.8 mm — about 3.8 mm more than originally intended.

Understand the result and units

The 3.8 mm gap between target and actual center distance is typical and is compensated in practice with adjustable motor plates or a tensioner. The closer X₀ already sits to an even number, the smaller this gap becomes.

Center distance and chain pitch are given in mm; link count is dimensionless and an integer.

Why an even link count is preferred

A roller chain alternates between outer and inner links. With an even link count, the loop closes exactly with one connecting link between an outer and an inner link. With an odd link count, an offset connecting link would instead be needed, which has lower allowable tension than a regular link because of its asymmetric shape. In practice, the link count is therefore almost always rounded to an even number, even though this shifts the center distance slightly from the target value.

Typical applications

The calculation is used when designing a new chain drive to get from a desired center distance to an orderable chain length, and during maintenance to find the resulting center distance — and therefore the required tensioner travel — for a given chain length.

Assumptions, limits and common mistakes

Rounding to an even link count avoids an offset connecting link, which has lower load capacity than a regular link. For unfavorable tooth-count/pitch combinations the gap between a₀ and a can be larger than in the example; an adjustable shaft mounting or idler sprocket is therefore recommended for most chain drives.

Common mistake: A common mistake is ordering the unrounded theoretical link count X₀ or rounding to an odd number, forcing the use of an offset link with reduced load capacity. It is also easy to overlook that the resulting center distance a differs from a₀ after rounding, so the design must allow for adjustment.