Belt, pulley and gear ratio calculator

A belt or gear drive does one thing: it trades speed for turning effort, in exact proportion and in both directions. Put a 3 inch pulley on the motor and a 10.8 inch pulley on the machine and the machine turns at 3.6 times less than the motor and pulls 3.6 times harder. That single ratio is the whole calculation, and it is the same number whether you measure pulley diameters, count gear teeth or compare the two shaft speeds — which is exactly what Bando says in its V-belt design manual: "Speed Ratio = Faster Sheave's rpm / Slower Sheave's rpm" or "Large Sheave's Outer Diameter / Small Sheave's Outer Diameter". What catches people is everything hanging off it: the belt you have to buy is not the distance round the pulleys, the belt speed decides whether the drive is safe, and the torque figure everyone quotes has a motor rating hidden inside it. This page works all of them from figures printed in a published design manual rather than remembered.

How it is calculated

ratio = driven ÷ driver · L = 2C + 1.57(D + d) + (D − d)² ÷ 4C · HP = torque × rpm ÷ 63,025

The ratio is the driven size over the driver size, in whatever unit you like as long as both are in the same one — inches, millimetres or teeth. Output speed is the input divided by it; output torque is the input multiplied by it, because an ideal drive passes the power straight through and power is torque times angular speed. The belt length is Bando's formula for pitch length from the centre distance C and the large and small pitch diameters D and d, and the 1.57 in it is half of pi: the belt wraps roughly half of each pulley, and the last term is the correction for the fact that it does not wrap exactly half when the pulleys differ. The power line takes torque in pound-inches, so this page converts your newton metres first using the NIST factor of 0.1129848 newton metres per pound-inch, then converts horsepower to kilowatts at 745.6999 watts each.

Source: Bando USA, V-Belt Design Manual BU-143/05-06 — Useful Formulas — "Belt Pitch Length: L = 2C + 1.57 (D + d) + (D - d)² / 4C", where L is belt pitch length (inches), C centre distance (inches), D large sheave pitch diameter (inches), d small sheave pitch diameter (inches)

Questions people ask

Which pulley is the driver?

The one on the thing with the power in it — the motor, the engine, the treadle. The driven pulley is on the thing being turned: the spindle, the drum, the saw arbor. Getting them the wrong way round inverts the ratio, and the giveaway is the output speed: if you expected a slow, strong spindle and this page shows a fast one, swap the two figures. A ratio above one means the drive gears down, which is what almost every workshop machine wants; below one it gears up, which is what a lathe overdrive or a centrifugal blower wants.

Why is the belt length not the distance round the pulleys?

Because a belt sits down in the groove, not on the rim, and its working length is measured on the pitch line inside it rather than on either surface. Bando quotes the length formula in terms of pitch diameters and then works its own example with outer diameters, which tells you how much slack there is in practice: the answer lands you on a stock belt, and you take up the rest by moving the motor. That is also why the result here is called a pitch length. If you buy by a catalogue number such as A48 or 5V850, the number is a length in that maker's own convention, and the two are not interchangeable between sections.

How fast is too fast for a belt?

Bando draws the line at 6,500, in a sentence with a typo worth knowing about: "Belt speeds in excess of 6500 rpm require special sheaves. Consult the sheave manufacturer." Belt speed is not measured in rpm and the manual's own belt speed formula produces feet per minute, so that is the unit the limit is in. It sneaks up on you because it depends on the pulley diameter as much as the motor speed — a 12 inch pulley at 1,750 rpm is already almost 5,500 feet per minute. That is why the speed sits next to the ratio here rather than buried: a drive can be right on ratio and wrong on surface speed at the same time.

Does the small pulley need enough belt wrapped round it?

Yes, and it is the thing that makes an otherwise correct drive slip. The wrap on the smaller pulley shrinks as the two pulleys become more unequal or the shafts move closer together, and Bando's correction table puts a price on it: a belt wrapping 180° carries its full rating, 157° derates it to 0.94, 133° to 0.87 and 127° to 0.85. Losing an eighth of the drive is usually cheaper to fix by moving the motor further away than by adding another belt. One wrinkle worth knowing: the manual also prints a shortcut on its formula page, "Arc of Contact: q = 180° - 57.3 (D - d) / C", and that shortcut does not agree with the manual's own table once the pulleys are very unequal — at (D − d) / C of 0.8 it gives 134° against the table's 133°, and at 0.9 it gives 128° against 127°. This page follows the table, not the shortcut.

Does the drive really multiply torque for nothing?

Not quite. The proportion is exact in an ideal drive, because power is conserved and power is torque times speed, so dividing the speed by the ratio multiplies the torque by it. Real drives lose a few percent to slip, belt flex and bearing friction, all of which comes out as heat. The figure on this page is the ideal one, which is the right number to design with and the wrong number to promise. It also assumes your input torque is real: a motor nameplate gives power, not torque, so if that is all you have, work the torque back from the power and the nameplate speed rather than guessing.

Can I chain two stages together?

Yes, and the ratios multiply. A 3:1 first stage into a 4:1 second stage is 12:1 overall, and torque goes up by the same twelve. That is how a drill press gets from a 1,400 rpm motor down to 250 rpm at the chuck without a pulley the size of the table. Run this page twice, taking the driven speed and driven torque from the first pass as the driver figures for the second. Belt length has to be worked out separately for each stage, since each has its own pair of pulleys and its own centre distance.

Where do the numbers come from?

The belt length, belt speed, speed ratio and power formulas are transcribed from the Bando USA V-Belt Design Manual, BU-143/05-06, which is linked below; the worked example on its page 4 — a 3 inch and a 10.6 inch sheave 32 inches apart giving an 85.8 inch belt — is one of the checks this page has to pass before it ships. The unit conversions are NIST Special Publication 811 Appendix B.8. Gear tooth ratio and the torque trade are not quoted from anywhere: they follow from what meshing teeth and conservation of power mean, and the page says so rather than dressing them up in a citation.

Sources

The documents this page reads its numbers out of, linked so you can check them yourself.

1. Bando USA, V-Belt Design Manual BU-143/05-06 — Useful Formulas — "Belt Pitch Length: L = 2C + 1.57 (D + d) + (D - d)² / 4C", where L is belt pitch length (inches), C centre distance (inches), D large sheave pitch diameter (inches), d small sheave pitch diameter (inches)
2. NIST SP 811 (2008), Appendix B.8 — inch (in) to meter (m) = 2.54 E−02

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