The belt length formula
For two pulleys of diameters D and d with their centers a distance C apart, the length of an open belt, where both pulleys turn the same way, is:
L = 2C + π(D + d)/2 + (D − d)² / (4C)
For a crossed belt, which makes the driven pulley turn the opposite way, the last term becomes (D + d)² / (4C). The first term counts the two straight spans, the second the belt wrapped around the pulleys, and the last a small correction for the angle of the spans.
Use the pitch diameter of each pulley: for a V-belt, the diameter at the belt’s neutral line, slightly smaller than the outside diameter of the pulley. The result is then the belt’s pitch length, also called datum or effective length depending on the manufacturer.
Example
A 4-inch motor pulley drives an 8-inch pulley with the shafts 20 inches apart:
- 2 × 20 = 40 in
- π × (8 + 4) / 2 = 18.85 in
- (8 − 4)² / (4 × 20) = 0.2 in
L ≈ 59.05 in. With a motor at 1,750 rpm, the driven pulley turns at 1,750 × 4 / 8 = 875 rpm, and the belt moves at about 1,830 ft/min.
Wrap angle
The belt touches the small pulley over an arc called the wrap angle:
θ = 180° − 2 × arcsin((D − d) / (2C))
The smaller the angle, the less grip the belt has. Below about 120°, V-belts start to lose power. A longer center distance or an idler pulley increases the wrap.
Belt speed
v = π × d × rpm. Classic V-belts work best at about 1,000 to 6,000 ft/min (5 to 30 m/s). Too slow and the belt transmits little power; too fast and centrifugal force reduces the grip.
Choosing and fitting a belt
- Calculate the length, then pick the nearest standard size from the manufacturer’s list.
- Make sure the motor mount or tensioner can move enough to fit the belt and take up stretch.
- Align the pulleys with a straightedge: misalignment wears belts quickly.
- Tension so that the belt deflects about 1/64 inch per inch of span when pressed at its middle.
Always lock out the power before working on a belt drive.