Hong Life CalculatorsUnderstanding the belt length and pulley speed calculator — concept and principle
For an open belt drive between two pulleys, you need the belt length to order the right belt, the driven pulley speed to check the machine, and the wrap angle on the small pulley to judge grip. This calculator gives all four from the pulley diameters, the centre distance and the driver speed.
How it is calculated
- Belt length: L ≈ 2C + π(D₁ + D₂)/2 + (D₂ − D₁)²/(4C), all in mm.
- Driven speed: n₂ = n₁ × D₁ / D₂.
- Belt speed: v = π × D₁/1000 × n₁/60, in m/s.
- Wrap angle on the small pulley: θ = 180° − 2·asin[(D₂ − D₁)/(2C)].
For V-belts, use the pitch (datum) diameters, not the outside diameters.
The principle
The belt is made of two arcs around the pulleys plus two straight spans. If the pulleys were equal, the length would be exactly 2C + πD. With unequal pulleys the straight spans tilt and the belt wraps more of the large pulley and less of the small one; the third term is a series approximation of that geometry, very accurate when C is large compared with the diameter difference. Because the belt does not stretch or slip (ideally), its surface speed is the same on both pulleys, so πD₁n₁ = πD₂n₂, giving the speed ratio. The tilt angle of the spans, asin[(D₂ − D₁)/2C], is what reduces the wrap on the small pulley.
Worked example
- Inputs: D₁ = 100 mm, D₂ = 250 mm, C = 500 mm, n₁ = 1,450 rpm.
- Length = 1,000 + π × 350/2 + 150²/2,000 = 1,000 + 549.779 + 11.25 = 1,561.03 mm.
- Driven speed = 1,450 × 100/250 = 580 rpm.
- Belt speed = π × 0.1 × 1,450/60 = 7.592 m/s.
- Wrap angle = 180 − 2 × asin(0.15) = 180 − 2 × 8.627 = 162.75°.
| Centre distance (mm) | Belt length (mm) |
|---|---|
| 125 | 844.7787144 |
| 250 | 1,072.278714 |
| 375 | 1,314.778714 |
| 500 | 1,561.028714 |
| 625 | 1,808.778714 |
| 750 | 2,057.278714 |
| 1,000 | 2,555.403714 |
How to read the result
Belts come in standard lengths, so pick the nearest stock size and adjust the centre distance; drives need some take-up for installation and tensioning anyway. Wrap angles well above 120° give good grip; small wraps need more tension or a larger centre distance. The real driven speed is usually 1–2% lower than calculated because belts creep under load. Higher belt speed carries more power for the same tension but raises centrifugal loads, so manufacturers give speed limits for each section.
Common mistakes
- Using outside diameters for V-belts, which overstates length and speed.
- Treating the approximate formula as exact for very short centre distances.
- Applying the open-belt formula to a crossed belt, where the last term changes to (D₁ + D₂)²/(4C).
- Ignoring pulley material speed limits.
Frequently asked questions
How do I find the centre distance for a stock belt length?
Rearrange the formula into a quadratic in C, or iterate in this calculator until the length matches the stock size.
What speed ratio should a single belt stage have?
V-belt drives are usually kept at or below about 6–7:1; beyond that the small wrap angle limits grip.
Why does my measured rpm differ?
Belt creep, slip and actual pitch diameters all shift the real speed by a few percent.
Sources: Budynas & Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill · Machinery's Handbook, Industrial Press