Calculation
L = 2C + (z₁+z₂)p/2 + (z₂−z₁)²p²/(4π²C)
The same open two-pulley geometry used for flat belts applies to timing belts; pitch length must additionally be a whole multiple of the belt pitch.
L = 2C + (z₁+z₂)p/2 + (z₂−z₁)²p²/(4π²C)
Timing-belt pitch length and center distance
The same open two-pulley geometry used for flat belts applies to timing belts; pitch length must additionally be a whole multiple of the belt pitch.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Pitch length L — Inside length of the timing belt along its pitch line.
- Center distance C — Distance between driving and driven pulley centers.
- Belt pitch p — Tooth spacing of the timing belt along the pitch line.
- Small pulley teeth z₁ — Tooth count of the smaller timing-belt pulley.
- Large pulley teeth z₂ — Tooth count of the larger timing-belt pulley.
Example
16/48 teeth, an 8 mm pitch and a 200 mm center distance give a pitch length of about 664.3 mm, corresponding to 83.04 belt teeth.
Assumptions and limits
Open two-pulley timing-belt drive without an idler; pitch length is computed as an approximation identical to the flat-belt case and must be rounded to a whole tooth count for an orderable belt size.
Understand Timing-belt pitch length and center distance
This calculator finds the pitch length of a timing-belt drive from center distance, belt pitch and pulley tooth counts, or the center distance needed for an existing belt of known pitch length.
What does this quantity describe?
The geometry of an open two-pulley timing-belt drive exactly matches that of an open flat-belt drive: L = 2C + (z1+z2)p/2 + (z2-z1)²p²/(4π²C), where pitch diameters dw = z·p/π follow from tooth count and belt pitch. Unlike a flat belt, pitch length must additionally be a whole multiple of the tooth pitch so the belt closes without a tooth offset.
Picture two meshing racks wrapped around two circular disks: the straight sections between the pulleys plus the two wrap arcs add up to the total length — exactly the same geometry as an ordinary belt drive, except each "rung" here is a belt tooth that must fit a whole number of times into the total length.
Formula and variables
L = 2C + (z₁+z₂)p/2 + (z₂−z₁)²p²/(4π²C)
L = 2C + (z1+z2)p/2 + (z2-z1)²p²/(4π²C)dw = z·p/π
| Symbol / input | Meaning |
|---|---|
| Pitch length L | Inside length of the timing belt along its pitch line. |
| Center distance C | Distance between driving and driven pulley centers. |
| Belt pitch p | Tooth spacing of the timing belt along the pitch line. |
| Small pulley teeth z₁ | Tooth count of the smaller timing-belt pulley. |
| Large pulley teeth z₂ | Tooth count of the larger timing-belt pulley. |
Choose the inputs correctly
Tooth counts z1, z2, belt pitch p and center distance C set the pitch length L. For a fit-check question — what center distance matches an existing belt of known pitch length — C can be chosen as the target.
How to use the calculator
Choose L as the target to find the belt pitch length needed for a desired center distance. If an existing belt's pitch length is known instead and the matching center distance is wanted, choose C as the target.
Worked example
16/48 teeth, an 8 mm pitch and a 200 mm center distance give a pitch length of about 664.3 mm, corresponding to 664.3/8 ≈ 83.04 belt teeth. In practice, an available belt with a whole tooth count — say 84 teeth — and a slightly adjusted center distance would be used instead.
Understand the result and units
Since 83.04 is not a whole number, no standard belt can exist with exactly this pitch length; the center distance must be adjusted slightly until pitch length matches an available belt with a whole tooth count — in principle identical to the rounded chain link count.
Center distance, belt pitch and pitch length are given in mm; tooth counts are dimensionless.
Why pitch length must be a whole multiple of tooth pitch
A timing belt is an endless closed band with uniform tooth pitch along its entire length. For the belt to repeat exactly on each circuit without a tooth landing offset from the pulley teeth, pitch length L must be a whole multiple of tooth pitch p: the belt tooth count L/p must come out as a whole number. Manufacturers therefore produce timing belts only in stepped standard lengths with a whole tooth count, similar to the stepped link counts of roller chains.
Typical applications
The calculation is used when designing a new timing-belt drive to find the belt size needed for a desired center distance, and for replacement parts, to set the matching center distance for an already-available belt.
Assumptions, limits and common mistakes
As with flat belts, the formula is a very good approximation for small to moderate diameter differences and assumes an open two-pulley drive without an idler. Per Dubbel, center distance should roughly fall between 0.5 and 2 times the sum of both pitch diameters; markedly smaller or larger center distances need additional design measures such as idler pulleys. The computed pitch length must additionally be rounded to a whole multiple of the tooth pitch for an orderable belt size.
Common mistake: A common mistake is ordering the computed, usually non-integer belt tooth count L/p directly, even though timing belts are only manufactured with a whole tooth count. It is also easy to substitute the pulley's outside diameter for the pitch diameter dw = z·p/π, which distorts the computed length.
Frequently asked questions
Why is the formula identical to flat-belt length?
Because the open two-pulley geometry — straight sections plus wrap arcs — is the same for timing belts and flat belts; only the pitch diameters are derived from tooth count and pitch instead of measured directly.
What is pitch diameter and why not outside diameter?
Pitch diameter dw = z·p/π describes the circle of the belt's neutral bending layer, which actually determines length; the pulley's measurable outside diameter differs from it and would distort the computed length.
What if the computed tooth count is not a whole number?
Round to the next available whole belt tooth count and adjust center distance slightly to match, similar to the chain link count.
What center distance range is recommended?
As a rough guideline, the literature cites roughly 0.5 to 2 times the sum of both pitch diameters; outside that range, additional design measures such as idler pulleys become necessary.
Can this find the center distance for an already-purchased belt?
Yes, enter the existing belt's pitch length as L and choose C as the target.