What a pulley ratio actually is
A belt drive has one belt, and that belt has one speed. Whatever length of belt leaves the driver sheave in a minute must arrive at the driven sheave in the same minute. That single fact is the whole of pulley mathematics.
The length of belt a sheave pays out per minute is its circumference times its speed: π·D·N. Set the two equal and π cancels, which leaves the identity every millwright carries in their head:
D₁ × N₁ = D₂ × N₂
Diameter and speed trade off exactly. Double the driven sheave and you halve its speed. Halve it and you double the speed. Nothing about belt width, belt section, number of grooves, motor power or shaft size enters this relation — those decide whether the drive can carry the load, not how fast it turns.
The same product also fixes torque. Power is torque times angular speed, so if the belt delivers power P to the driven shaft, the torque there is P divided by that shaft's speed. Slowing a shaft by a factor of three multiplies its torque by three, less whatever the belt loses to bending and slip. This is why a 1 hp motor on a 5:1 belt reduction can drive a machine no 1 hp direct-drive spindle could turn: you are buying torque with speed, at roughly a 5 percent commission.
The word sheave (pronounced "shiv") is the correct term for a grooved V-belt pulley; pulley strictly means a flat- or round-belt wheel. Shops use the words interchangeably and so does this page.
The formula explained, term by term
Rearranged for each unknown, the identity gives you four working formulas:
- Driven speed: N₂ = N₁ · D₁ / D₂ — the everyday case. You have the motor and both sheaves and you want to know what the machine will do.
- Driven sheave size: D₂ = N₁ · D₁ / N₂ — the shopping case. You know the speed you need and you are buying a sheave to get it.
- Driver sheave size: D₁ = N₂ · D₂ / N₁ — the retrofit case. The big sheave is already on the machine and you can only change the motor sheave.
- Driver speed: N₁ = N₂ · D₂ / D₁ — the diagnostic case. You measured the machine shaft with a tachometer and want to know what the motor is really doing.
Pitch diameter is the diameter that counts. A V-belt does not ride on the bottom of the groove or on the rim. It sits somewhere in between, on the pitch line, and that is the effective diameter the belt wraps. On a classical sheave the pitch diameter runs roughly a quarter to a half inch under the outside diameter depending on the section, and the manufacturer stamps or catalogues it. Using outside diameters on two similar sheaves gives an almost-right ratio because the two errors partly cancel; using them on a 2 in driver and a 12 in driven does not, because a 0.35 in error is 17 percent of the small sheave and 3 percent of the large one. Flat belts, round belts and toothed timing belts are different again — for a timing belt, count teeth and use the tooth counts directly in place of diameters, exactly as you would for a gear ratio or a chain and sprocket drive.
Belt surface speed is π·D·N, converted to feet per minute by dividing by 12 when D is in inches. It matters for two reasons. A V-belt's power rating rises with belt speed up to a peak and then falls away as centrifugal force starts lifting the belt out of the groove, so a slow drive needs more belts than a fast one for the same horsepower. And there is a practical ceiling: standard cast-iron sheaves with classical belts are normally applied to about 6,500 ft/min, above which the sheaves need dynamic balancing and the belts need a high-speed construction.
Worked example: slowing a 1,750 rpm motor to 500 rpm
You are building a shop machine that must turn at about 500 rpm. You have a 1 hp single-phase motor whose nameplate says 1,750 rpm, and a 3 in pitch diameter sheave already bored to fit its shaft. What sheave goes on the machine?
- Write the identity. D₁·N₁ = D₂·N₂, so 3 × 1750 = D₂ × 500.
- Work out the belt travel. 3 × 1750 = 5,250 inch-rpm. This number stays constant everywhere in the drive.
- Divide by the speed you want. D₂ = 5250 ÷ 500 = 10.5 in pitch diameter. A 10.5 in sheave is a stock size in the A and B sections, so you are in luck.
- Check the ratio. 10.5 ÷ 3 = 3.5:1 — a comfortable single-stage reduction, well inside the roughly 8:1 practical ceiling for one belt drive.
- Check the belt speed. V = π × 3 × 1750 ÷ 12 = 16,493.4 ÷ 12 = 1,374 ft/min, or 1374 × 0.00508 = 6.98 m/s. That is a slow but perfectly serviceable drive, and far below the 6,500 ft/min ceiling.
- Check the torque. The driver shaft carries 5252.11 × 1 ÷ 1750 = 3.00 lb-ft. Allowing 95 percent drive efficiency, the machine shaft receives 5252.11 × 1 × 0.95 ÷ 500 = 9.98 lb-ft. That is the 3.00 lb-ft multiplied by 3.5 and then by 0.95: 3.00 × 3.5 × 0.95 = 9.98 lb-ft, which confirms the two routes agree.
If a 10.5 in sheave will not clear the frame, work the problem the other way. Keep a 12 in sheave on the machine and solve for the driver: D₁ = 500 × 12 ÷ 1750 = 3.43 in. The nearest stock size, 3.5 in, gives 1750 × 3.5 ÷ 12 = 510 rpm — within 2 percent of target, and closer than most machines care about.
How to read the result
The ratio tells you the trade you are making. A ratio above 1 means the driven shaft turns slower and carries proportionally more torque; below 1 it turns faster and carries proportionally less. There is no free lunch and no efficiency gain hiding in a sheave change — only a redistribution.
Belt speed decides how much belt you need. Between roughly 1,000 and 5,000 ft/min a classical V-belt is in its comfortable range. Under 1,000 ft/min each belt carries very little power, so a slow drive that must move real horsepower needs either a bigger driver sheave or more grooves. Above about 6,500 ft/min you are into special-order territory.
Wrap angle limits how much ratio you can take in one stage. The belt grips by wedging into the groove over the arc it touches. As the ratio climbs, the arc on the small sheave shrinks, and once it drops much below 120 degrees the belt's rated capacity falls off sharply. That, and the physical size of the large sheave, is why single-stage V-belt reductions above about 8:1 are rare — a countershaft or a gearbox takes over. The V-belt length calculator reports the wrap angle on both sheaves for a given centre distance, which is the number to check before committing to a big ratio.
Nameplate speed is not synchronous speed. A four-pole induction motor on 60 Hz has a synchronous speed of 1,800 rpm but a full-load speed of about 1,725 to 1,760 rpm because of slip. Use the nameplate figure. If the motor runs lightly loaded it will sit closer to 1,800 and every driven speed will come out a percent or two high.
Slip and creep make the real speed slightly low. A correctly tensioned V-belt loses on the order of 1 to 2 percent of the ideal speed to elastic creep in the belt as it enters and leaves the groove. That is inside the accuracy of most sheave selections, but it is why a tachometer reading rarely matches the arithmetic exactly, and why a toothed timing belt is the right answer when the ratio must be exact.
Driven speed from a 1,750 rpm motor for common sheave pairs
| Driver sheave (in) | Driven sheave (in) | Ratio | Driven speed (rpm) | Belt speed (ft/min) |
|---|---|---|---|---|
| 2 | 4 | 2.00:1 | 875 | 916 |
| 2 | 6 | 3.00:1 | 583 | 916 |
| 2.5 | 5 | 2.00:1 | 875 | 1,145 |
| 3 | 6 | 2.00:1 | 875 | 1,374 |
| 3 | 9 | 3.00:1 | 583 | 1,374 |
| 3 | 10.5 | 3.50:1 | 500 | 1,374 |
| 3 | 12 | 4.00:1 | 438 | 1,374 |
| 4 | 8 | 2.00:1 | 875 | 1,833 |
| 4 | 12 | 3.00:1 | 583 | 1,833 |
| 6 | 6 | 1.00:1 | 1,750 | 2,749 |
| 6 | 3 | 0.50:1 | 3,500 | 2,749 |
Ratios of 2:1 appear three times at three different belt speeds — a reminder that the ratio fixes the speed and the driver sheave fixes the belt speed, independently of each other.
Mistakes that put a drive on the wrong speed
- Measuring outside diameter instead of pitch diameter. The single most common error. On a small sheave the difference is a large fraction of the diameter, and it biases the calculated ratio the same way every time.
- Using synchronous speed instead of nameplate speed. 1,800 rpm instead of 1,750 rpm puts every result about 3 percent high before you have made any other mistake.
- Forgetting which sheave the belt speed depends on. Belt speed is set by the driver sheave and its speed alone. Changing the driven sheave changes the ratio and the machine speed but leaves belt speed untouched.
- Assuming a step pulley gives evenly spaced speeds. A four-step cone gives four ratios, and because the ratios are quotients of two changing diameters they bunch up at one end. Compute all four rather than trusting the spacing to look even.
- Ignoring the driven machine's rating. A fan on a step-up drive can be spun past its maximum safe speed with one sheave change, and centrifugal fan power rises with the cube of speed, so a 20 percent speed increase asks the motor for about 73 percent more power.
- Changing the sheave without re-checking centre distance and belt length. A new sheave diameter changes the belt length required; check it before ordering the belt.
- Treating the ratio as exact under load. V-belts creep. When the speed must be exact, use a toothed synchronous belt or a gear train.
When a belt is the wrong answer
Use a toothed synchronous belt when the ratio must be exact and repeatable — indexing, timing, positioning. Tooth counts replace diameters in the identity, and there is no slip to correct for.
Use a roller chain when the drive is slow, heavily loaded, or exposed to oil that would destroy a rubber belt. The arithmetic is identical with tooth counts, and the chain length calculator handles the wrap geometry.
Use gears when the shafts are close together, the ratio is large, or the drive must handle shock and reversing loads. See the gear module and pitch diameter calculator for the equivalent diameter arithmetic.
Use a VFD when the required speed changes during the job. A variable-frequency drive on a three-phase motor changes speed electronically, but remember that it changes torque capability too: below about 25 percent of base speed a standard TEFC motor cannot shed its own heat, and constant-torque operation needs a motor rated for inverter duty. A belt reduction plus a VFD covers a far wider useful range than either alone.
Machine tools are a special case. On a lathe or mill the spindle speed you want is derived from the cutting speed of the material and the diameter being cut, not chosen for its own sake. Work that out first with the cutting speed calculator or the spindle RPM calculator, then come back here to find the sheave pair that gets you closest.
Key terms
- Sheave
- A grooved pulley designed for V-belts. The groove angle wedges the belt in under load, which multiplies the effective friction well beyond that of a flat belt on the same wrap.
- Pitch diameter
- The diameter of the circle on which the belt's tension member actually travels. It is the diameter that governs speed ratio, and it is smaller than the sheave's outside diameter.
- Drive ratio
- Driven pitch diameter divided by driver pitch diameter. Above 1 it is a reduction; below 1 it is a step-up. Speed divides by it and torque multiplies by it.
- Belt surface speed
- How fast the belt itself travels, π·D·N/12 in feet per minute with D in inches. It governs the power each belt can carry and the maximum safe sheave speed.
- Slip and creep
- Slip is gross loss of grip under overload. Creep is the small, unavoidable elastic stretching and relaxing of the belt as it moves between the tight and slack sides, and costs about 1 to 2 percent of ideal speed.