Belt pitch length
-Speed ratio
-Output speed
-Wrap angle, small pulley
-All results
| Belt pitch length (exact) | - |
| Belt pitch length (common approximation) | - |
| Belt angle phi to the center line | - |
| Wrap angle, large pulley | - |
| Belt speed | - |
| Output speed with 2 % slip (V-belt estimate) | - |
| Center distance for the standard belt length | - |
| Center distance as a multiple of (d + D) | - |
Method and formulas
Plane geometry of an open belt around two circles, so every figure can be checked by hand. Use pitch diameters for V-belts and timing belts, and the belt thickness allowance for flat belts.
| Quantity | Formula |
|---|---|
| Belt angle | sin φ = (D − d) / (2 C) |
| Belt length, exact | L = 2 C cos φ + π (D + d) / 2 + φ (D − d) |
| Belt length, approximation | L ≈ 2 C + π (D + d) / 2 + (D − d)² / (4 C) |
| Wrap angles | θsmall = 180° − 2 φ, θlarge = 180° + 2 φ |
| Speed ratio | i = D / d, n2 = n1 / i |
| Belt speed | v = π d n1 / 60 (d in m, v in m/s) |
| Center distance from length | Solve the exact length equation for C numerically (length rises steadily with C) |
The pulleys are the same plane and parallel, the belt is open (not crossed) and has no stretch. The small pulley is the driver. Real belts need a take-up allowance on the center distance, see your belt maker.
Worked example: 100 mm and 250 mm pulleys, 500 mm center distance, 1450 rpm
sin φ = (250 − 100) / (2 × 500) = 0.15, φ = 8.63°. L = 2 × 500 × cos φ + π × 350 / 2 + φ (rad) × 150 = 1561.0 mm. Wrap angle on the small pulley = 180 − 2 × 8.63 = 162.7°. Speed ratio = 250 / 100 = 2.5; output speed = 1450 / 2.5 = 580 rpm. Belt speed = π × 0.1 × 1450 / 60 = 7.59 m/s.
Questions
How do you calculate belt length for two pulleys?
For an open belt on two pulleys of diameters d (small) and D (large) at center distance C, the exact pitch length is L = 2 C cos(phi) + pi (D + d) / 2 + phi (D - d), where sin(phi) = (D - d) / (2 C) and phi is in radians. The common approximation L = 2 C + pi (D + d) / 2 + (D - d)^2 / (4 C) is within a fraction of a percent for normal drives. Use the pitch diameters of the pulleys, not the outside diameters, for V-belts and timing belts.
How do I find the center distance for a standard belt length?
Belt length rises steadily with center distance, so you solve the length equation for C. Enter the standard pitch length of the belt you want to use and the calculator finds the matching center distance. Then check that your machine can adjust the center distance by at least the take-up and installation allowance given by the belt maker.
What is the wrap angle and why does it matter?
The wrap angle (arc of contact) on the small pulley is 180 degrees minus 2 phi, in degrees. It sets how much of the pulley the belt grips. Friction-driven belts lose power capacity quickly as it falls: most V-belt catalogues apply a correction factor below 1 under 180 degrees, and many recommend staying above about 120 degrees on the small pulley. Larger ratios and shorter center distances reduce it.
How do I calculate the speed ratio and output rpm?
The speed ratio is the large pitch diameter divided by the small one, i = D / d. The driven pulley speed is n2 = n1 d1 / d2 for the pulley diameters of the driver and driven. Belt slip lowers the real output speed by roughly 1 to 3 percent on V-belts. Synchronous (timing) belts do not slip, so the tooth counts give the ratio exactly.
What belt speed is acceptable?
Belt speed is v = pi d n / 60. Common limits are about 30 m/s (6 000 ft/min) for classical and narrow V-belts, with the most efficient range often quoted around 10 to 25 m/s. Higher speeds raise centrifugal tension and shorten life. Timing belts and flat belts have their own limits from the manufacturer's data.
What center distance should I choose?
A common design rule of thumb is a center distance between about 0.7 times and 2 times the sum of the two pulley diameters. Too short gives a small wrap angle and many belt flexes per second; too long lets the slack side flap. This is guidance only, follow your belt supplier's recommendations.
What does this calculator not cover?
It covers pulley geometry, ratio, speed and wrap angle for an open belt drive. It does not select a belt section, count belts for a given power, calculate belt tension, shaft loads, service factors, crowned pulleys, idlers, or crossed and serpentine layouts. Use the belt maker's selection method for power rating. Verify critical designs.
Engineering aid for guidance. It covers open two-pulley belt geometry, speed and wrap angle and does not cover belt selection, power rating, tension, shaft loads, service factors, idlers or crossed layouts. Verify critical designs.