# Belt Length

First posted on April 27, 2020

I recently needed to calculate the relationship between pulley diameter, center to center spacing and the length of a belt going around those pulleys. The references I looked at all gave an approximation for the case where the difference in diameter between the pulleys is small. So I decided to do the geometry myself. We assume circular pulleys, and that belts lie tangent to pulleys. WLOG assume $$D1 > D2$$, where $$D1,D2$$ are the diameters of the pulleys, $$CC$$ is the center to center spacing.

$\beta = sin^{-1} \frac{D_1 - D_2}{2cc}$ $S = \sqrt{CC^2 - \left(\frac{D_1 - D_2}{2}\right)^2}$ $L = 2S + \pi\frac{D_1}{2} + \pi\frac{D_2}{2} + 2\beta\frac{D_1}{2} - 2\beta\frac{D_2}{2}$

That is: the total belt length is the sum of two the straight parts, the half circumferences of each pulley, and the two arcs of arclength $$\beta$$ on the larger pulley, minus the two arcs of arclength $$\beta$$ on the small pulley.

-- Haskell snippet
beltLength d1 d2 cc = 2 * s + 0.5 * pi * (d1 + d2) + beta * (d1 - d2)
where
beta = asin ((d1 - d2) / (2*cc))
s = sqrt(cc**2 - 0.5 * (d1 - d2))

In my particular application I need to know what center to center spacing will let me use a 610mm belt with pulley diameters of 40.74mm and 125.41mm. The small difference (small angle $$\beta$$) approximation gives a center to center spacing of 169.28mm, while the exact form gives 163.27mm. This difference is more than enough to make the exact formulation preferable.