Calculation
L = 2·C + (N₁+N₂)·p/2 + (N₂−N₁)²·p²/(4·π²·C)
For an open roller chain drive between two sprockets; the calculated length is rounded up in practice to a whole, usually even, number of pitches.
L = 2C + (N₁+N₂)p/2 + (N₂−N₁)²p²/(4π²C)
Chain length and center distance for a roller chain drive
For an open roller chain drive between two sprockets; the calculated length is rounded up in practice to a whole, usually even, number of pitches.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Chain length L — Effective roller chain length between sprocket centers.
- Center distance C — Distance between driving and driven sprocket centers.
- Chain pitch p — Distance between adjacent pin centers of the roller chain, e.g. 19.05 mm for ANSI 60.
- Small sprocket teeth N₁ — Tooth count of the smaller (usually driving) sprocket.
- Large sprocket teeth N₂ — Tooth count of the larger (usually driven) sprocket.
Example
19.05 mm pitch (ANSI 60), 17/51 teeth and a 500 mm center distance give about 1,668.95 mm, or 87.6 pitches, of chain.
Assumptions and limits
Open chain drive without an idler; the real chain needs a whole, usually even, number of links, so the center distance is adjusted slightly in practice.
Understand Chain length and center distance for a roller chain drive
This calculator finds the missing quantity among chain length L, center distance C, chain pitch p and the two sprocket tooth counts N₁ and N₂ from the other four — for an open roller chain drive between two sprockets. It answers two common practical questions at once: what chain length is needed for a planned center distance, and what center distance suits an already available chain?
What does this quantity describe?
For an open chain drive, chain length follows from the straight run between the sprockets, the average tooth count and a correction term for the difference in tooth counts: L = 2C + (N₁+N₂)·p/2 + (N₂−N₁)²·p²/(4π²C). This design equation follows from the geometry of an ANSI B29.1 chain drive and is used identically in the engineering catalogs of Martin Sprocket & Gear and Tsubaki.
Like a belt, a roller chain must match its center distance and tooth counts exactly — too short a chain won't fit over the sprockets, too long a one sags and runs unevenly. Unlike a belt, chain length must additionally correspond to a whole number of links, and preferably an even one, so that no special offset link is needed. Both the Martin Sprocket engineering catalog and Tsubaki's chain drive selection literature independently confirm the same formula.
Formula and variables
L = 2·C + (N₁+N₂)·p/2 + (N₂−N₁)²·p²/(4·π²·C)
L = 2C + (N₁+N₂)p/2 + (N₂−N₁)²p²/(4π²C)C = p/4 · [(L/p − (N₁+N₂)/2) + √((L/p − (N₁+N₂)/2)² − 2(N₂−N₁)²/π²)]
| Symbol / input | Meaning |
|---|---|
| Chain length L | Effective roller chain length between sprocket centers. |
| Center distance C | Distance between driving and driven sprocket centers. |
| Chain pitch p | Distance between adjacent pin centers of the roller chain, e.g. 19.05 mm for ANSI 60. |
| Small sprocket teeth N₁ | Tooth count of the smaller (usually driving) sprocket. |
| Large sprocket teeth N₂ | Tooth count of the larger (usually driven) sprocket. |
Choose the inputs correctly
Enter four of the five quantities to find the fifth. For N₁ and N₂, use the actual tooth counts of the two sprockets; which sprocket drives and which is driven does not matter for the length calculation, since only the difference between the two tooth counts enters the formula. Use the standard pitch of the roller chain for p (e.g. 12.7 mm for ANSI 40, 15.875 mm for ANSI 50, 19.05 mm for ANSI 60), not a measured approximation.
How to use the calculator
Select the target quantity and enter the four known values with units. In practice, round a calculated chain length up to the nearest whole, preferably even, number of links, and then recalculate the actual center distance to use with that rounded length.
Worked example
Given an ANSI 60 chain with p = 19.05 mm, a driving sprocket with N₁ = 17 teeth, a driven sprocket with N₂ = 51 teeth and a center distance of C = 500 mm, substitution gives L = 2·500 + (17+51)·19.05/2 + (51−17)²·19.05²/(4π²·500) ≈ 1,668.95 mm, or about 87.6 links.
Understand the result and units
A chain length of 87.6 pitches is initially just a theoretical value; since a real chain has a whole number of links and an even number avoids the special offset link, it is rounded up to 88 links in practice. That rounded length is then used to back-calculate the exact center distance to set, which comes out slightly larger than the originally planned 500 mm.
Chain length L, center distance C and pitch p are given in the same length unit, usually millimetres. N₁ and N₂ are dimensionless tooth counts.
Typical applications
The formula is used for new chain-drive designs, changing tooth counts to adjust the drive ratio, and retrofitting conveyors, agricultural machinery and drives with different sprockets.
Assumptions, limits and common mistakes
The formula applies to an open chain drive between two sprockets without an idler or guide rail. As a rough design guideline, a center distance of 30 to 50 pitches is recommended; for center distances above about 80 pitches, idlers or guide rails should be used to support the chain. Chain sag, wear elongation and the polygon (chordal) effect are not included.
Common mistake: Do not substitute a plain outside measurement or a roughly measured pitch for the standard chain pitch. Also, when calculating center distance, remember that a real chain is only available in whole links — the calculated value must be rounded up to the nearest whole, preferably even, number of links before assembly, and the center distance adjusted accordingly.
Frequently asked questions
Why should the number of chain links be even?
An even number of links closes using the two matching link types (inner and outer link) with no extra piece; an odd number needs a special offset link, which has lower load capacity than a normal link and should be avoided where possible.
Which tooth count is N₁ and which is N₂?
It doesn't matter which sprocket drives for the length calculation — only the difference between the two tooth counts enters the formula. N₁ is conventionally the smaller sprocket, N₂ the larger one.
How large should the center distance be at minimum?
Center distance must always exceed half the sum of the two sprockets' outside diameters so the teeth don't touch; as a guideline, 30 to 50 pitches is recommended, and at ratios above about 3:1, at least the sum of both sprocket diameters.