Pitch length
Length measured along the pitch line.
Mechanics · Transmission and rotation
Belt length and center distance between two pulleys
Calculate the pitch length of an open belt or recover the exact center distance from the two pulley pitch diameters.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Center distance
Distance between both pulley axes.
Small-pulley wrap
Contact angle α₁.
Large-pulley wrap
Contact angle α₂.
Relation and numerical substitution
L = 2√(C²−e²) + D₁(π−2β)/2 + D₂(π+2β)/2—
Open belt, parallel shafts and pitch diameters. Commercial inside, outside or datum length then depends on the manufacturer.
01Formulas and symbols
Exact open-belt geometry
e
(D₂ − D₁) / 2Pitch-radius differenceβ
asin(e / C)Tangent angleα₁
π − 2βSmall-pulley wrapα₂
π + 2βLarge-pulley wrapL
2√(C²−e²) + D₁α₁/2 + D₂α₂/2Exact pitch lengthC
numerical solution of L(C)Monotonic bisection02Assumptions and limits
- Two parallel-axis pulleys connected by an open belt.
- D₁, D₂ and L use the same pitch line.
- The calculation does not size belt section, tension or transmitted power.
- Idlers, elasticity, misalignment and tolerances are excluded.
03Validation example
- D₁ = 100 mm, D₂ = 250 mm and C = 500 mm.
- e = 75 mm and β = 0.150568 rad.
- L = 1,561.05 mm, α₁ = 162.75° and α₂ = 197.25°.
- The inverse calculation returns C = 500 mm.
04References
- Shigley’s Mechanical Engineering Design — belt drives.
- Geometry of common external tangents to two circles.
- BIPM — SI Brochure.