Pulleys and Belts- by W. E. Johns
(http://www.gizmology.net)

This excellent article is in imperial measurements but it it is very accurate

Pitch Diameter and Drive Ratio

Pulleys and belts have two uses; to increase or reduce speed or torque, or to transfer power from one shaft to another. If the transfer of power is all you need, then two pulleys of the same diameter will do the trick. But most of the time you'll also want to take the opportunity to trade speed for torque, or vice versa. This is done by using pulleys of different pitch diameters.

The pitch diameter of a pulley is not the outside diameter. Or the inside diameter. In fact, the pitch diameter is very difficult to measure directly. If you cut a belt and look at the end, you'll see a row of fibers near the outside surface. This is the tension carrying part of the belt; the rest of the belt exists only to carry the forces from the pulley to and from these fibers. The pitch diameter of any pulley is measured at these fibers. If you think about this for a moment, you'll see that the pitch diameter of a pulley depends not just on the pulley itself, but on the width of the belt. If you put a B series belt on an A series pulley, it will ride higher than usual, increasing the effective pitch diameter.

The ratio of the pitch diameters is called the drive ratio, the ratio by which torque is increased and speed is decreased, or vice versa. Power is the product of speed and force, or in the case of things that spin, speed and torque. Pulleys do not effect power; when they increase torque, it is at the expense of speed, and vice versa.

V-belts are not 100% efficient, however. While they transfer torque effectively, they loose a bit of speed as the belt stretches under load.

Maximum Load and Initial Tension

How much load can be put on a belt before it slips depends on a lot of stuff, but most importantly the initial tension, the force squeezing the pulleys toward each other at rest. Everyone has seen the results of too little initial tension - a slipping alternator belt that eventually results in a dead battery. Too much initial tension isn't good either, as it unnecessarily stresses the belt and wears the bearings. Initial tension is the force on a single strand; the force on the bearings will be twice this, as there are two strands.

This calculator generates an approximation of a minimum, so you'll want to add some to provide a safety margin. It assumes a type A 40° v-belt; wedge belts will require a bit less tension, and heavier belts a bit more.

Pulley and Belt Calculator

Power		hp
R.P.M. Pulley 1		rpm
Pitch Diameter 1 (Motor Pulley)		inches
Pitch Diameter 2 (Drive Pulley)		inches
Center Distance		inches
Initial Tension		lbs
Bearing Force		lbs
Maximum Tension		lbs
Minimum Tension		lbs
Cyclic Variation		lbs
Belt Speed		fpm
Belt Length		inches
Wrap Angle 1		degrees
Wrap Angle 2		degrees
R.P.M. Drive Pulley		rpm
Torque Shaft 1		ft-lbs
Torque Shaft 2		ft-lbs
Drive Ratio

Example: a ¹/₃ horsepower motor turning a 5" pitch diameter pulley at 1750 rpm, driving a 2.5" pulley 12" away.

The required initial tension will be 7.6 pounds on each strand, 18.3 total on the bearings. Interestingly, the bearing load decreases when running: under load, the belt will be under a maximum tension of 9.1 pounds on the tight side, and a minimum tension of only 4.3 pounds on the slack side, 13.4 pounds total. This may seem weird, but that's only because it is. What happens is that the belt stretches under load, becoming looser.

The cyclic variation is the difference between the maximum and minimum tensions, 4.8 pounds. It is the cyclic variation in tension, not the tension itself, that fatigues and eventually kills the belt.

Of course, you're usually stuck with a given rpm, rather than a given belt speed. If so, you face a trade-off between belt fatigue and bearing fatigue. If you use a bigger pulley, the belt will see less cyclic variation, but the bearings will see higher loads. A smaller pulley is the opposite. I usually figure that it is easier and cheaper to replace a belt than a bearing, so I use small pulleys.

Belt Length

The easy way to measure the circumference of a belt is to roll it along the wall, measuring the distance you've traveled when you get back to the same point on the belt. Subtract two inches to get the inside circumference.

If you don't have a belt, just the pulleys installed on the machine, you can run a string around the pulleys and measure that. If you don't have access to the machine, you can use a formula so royally obnoxious that I won't include it here, as the above calculator will do it for you. (If you insist, you can view the source code of this document and find it in the TensionCalc Javascript function).

It is important to remember when designing belt drives that belts come in discrete lengths, and pulleys come in discrete pitch diameters; you cannot just arbitrarily select dimensions hope to find such components.

If you're like me, you often scrounge up a belt and some pulleys, and then try to figure out the center distance. The easy way, of course, is to lay them out on the work bench and measure it. If you can't do that, just enter the pulley sizes into the above calculator, and reiteratively enter values for center distance until you manage to hit on the right belt length. It's a kludgy way to do it, but if you saw the formula, you'd know why I didn't want to solve it for center distance.