Belt Speed

The belt travels at the rim speed of the pulley it is wrapped around, and rim speed is circumference times rotational speed:

\[ v = \pi d n \]

with $n$ in revolutions per unit time. In North American practice this is nearly always quoted in feet per minute, and the arithmetic collapses to a tidy shortcut: with $d$ in inches and $n$ in rpm, $v\,[\text{ft/min}] = \pi d n / 12$, so a 6 in sheave at 1160 rpm runs 1,822 ft/min.

Why anyone cares

Because power is the net pull times this speed. A belt of a given cross-section can only hold so much pull before it slips or fatigues, so the only lever left for carrying more power is running it faster. Doubling the speed roughly doubles the power a belt can carry — up to a point.

That point is centrifugal force. The belt is a mass going round a corner, and at speed it wants to fly off the sheave. The centrifugal tension $F_c = \rho_L v^2$, with $\rho_L$ the belt's mass per unit length, is subtracted from the tension available for gripping, and because it scales with $v^2$ it eventually eats the gains. V-belt drives are usually happiest around 20–25 m/s, or roughly 4,000–5,000 ft/min, and past about 30 m/s an ordinary drive is out of its comfortable range: balance quality, sheave material and belt construction all stop being afterthoughts.

At the other end, a slow belt carries very little power for its physical size. Below about 5 m/s a belt drive is usually the wrong choice, and that is exactly the territory where chain earns its keep — a chain meshes and does not depend on friction at all, so its capacity does not fall away with speed.

Pitch diameter again

A V-belt rides down inside its groove, and its neutral axis sits below the rim of the sheave. Use the outside diameter and the speed comes out a few percent high — small on a large sheave, and larger than you would like on a small one. Sheave catalogues list both the outside and the pitch (or datum) diameter for exactly this reason.

Belt speed and the ratio

One quiet consequence worth naming: since the belt is a single continuous loop, its speed is the same on both pulleys. Set $\pi d_1 n_1 = \pi d_2 n_2$ and the speed ratio of a belt drive falls straight out as $n_1/n_2 = d_2/d_1$ — the inverse ratio of the diameters, exactly parallel to the tooth-count ratio of a gear pair. Belt drives creep slightly (a percent or two under load, as the belt stretches on the tight side and relaxes on the slack), so the real ratio is never quite the nominal one. Where it must be exact, that is what a toothed synchronous belt is for.