Belt Length — Open Drive

A belt on two pulleys is two straight tangent spans and two arcs. If both pulleys were the same size, the arcs would each be exactly half a circle and the length would be $2C + \pi D$ with nothing else to say. They are not the same size, so the belt leans, one arc grows and the other shrinks, and the straight spans are slightly longer than the centre distance. That is what the third term corrects for:

\[ L = 2C + \frac{\pi}{2}(D + d) + \frac{(D-d)^2}{4C} \]

The first two terms are the exact answer for equal pulleys; the last is a series approximation for the lean. It is accurate to a fraction of a percent over any sensible drive geometry, and it is the form printed in every belt manufacturer's engineering guide.

Pitch length, not outside length

The length a catalogue lists is the pitch length, measured along the belt's own neutral axis — the layer of cords that neither stretches nor compresses as the belt bends. It is not the outside circumference and it is not the inside. Likewise the diameters in the equation are pitch diameters of the sheaves, measured where that same cord line rides in the groove, which is below the rim of a V-groove. Mixing an outside sheave diameter with a pitch length is a small error on a big sheave and a significant one on a small.

The design order runs backwards

Belts come in standard lengths. So the real sequence is: choose the sheaves for the ratio, estimate a centre distance, compute $L$, pick the nearest stock belt, and then solve this equation for $C$ to find where the motor base actually has to sit. That solved centre distance is a target, not a fixed dimension — a belt drive needs adjustment range in both directions: enough shortening to fit the belt on without prying it over the rim (which breaks cords and is the single worst thing you can do to a new belt), and enough lengthening to take up the stretch that happens in the first few hours of running.

Wrap angle is the real constraint

The geometry also sets how much of the small pulley the belt touches:

\[ \theta_{small} = \pi - 2\arcsin\!\left(\frac{D-d}{2C}\right) \]

and it is this, not the belt's tensile strength, that usually limits the drive. Below about 120° of wrap the tension ratio a belt can hold collapses — see the capstan equation — and it slips long before it breaks. The cures are a longer centre distance, a smaller ratio, or an idler pressed into the slack span from the outside to force more wrap. Note that an outside idler bends the belt backwards, which shortens its fatigue life; an inside idler on the slack side is gentler but reduces wrap on one pulley while adding to none.

The crossed belt

Cross the belt and the driven shaft reverses. The length becomes $L = 2C + \frac{\pi}{2}(D+d) + \frac{(D+d)^2}{4C}$ — the sign inside the squared term flips — and the wrap on both pulleys increases, which is genuinely useful. It is nearly extinct because the two spans rub where they cross, which wears the belt from the outside in, and because a reversing gearbox or simply reversing the motor is easier.