# Pulleys and Belts

Pulleys, at least the kind we're talking about here, are also called sheaves. Ask for a sheave at the hardware store and count all the blank looks!

### 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 inches Pitch Diameter 2 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. Pulley 2 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.

### How to read V-Belt numbers

Automotive belts start with either 4L (12.5mm wide) or 3L (9.5mm). The number following it is the outside length of the belt in tenths of inches. The inside length of the belt is typically 2" less for a 4L belt, and 1-1/2" less for a 3L belt. An example would be 4L460, which would be 46" long outside, 44" inside.

"Classic" v-belt numbers start with a letter identifying the cross section, A through E - see V-Belt Dimensions below. A series belts are the most common. The number following it is the inside length in inches. The outside length is typically 2 inches more. An example would be A44, 44" long on the inside, 46" outside; the equivalent of the 4L460 above.

4L and A series belts are interchangeable, even though automotive belts are technically 0.3mm narrower.

All v-belts have a 40° angle between the faces, except the V series (aka "Harvester" or "Wedge" belts), which have a 30° angle between the faces.

You will notice that the inside edge of the belt is wider than the base of the V in a v-belt pulley - the belt touches the pulley only on the sides.

### How to Make your Own Pulleys

First you need a to cut a circle out of wood, thick enough to hold the belt - in the case of an A/4L series belt, at least 5/8". Then you need to put the v-shaped notch in the edge.

There are a variety of ways to cut a really accurate circle out of wood. You can't do it by hand unless you're far more skilled than I am. I've used two methods to make circles.

• I've used a table saw. It doesn't look anywhere near as dangerous in the pictures as it does in person. One very important thing to remember: put pulley on the LEFT side of the blade and to use a screw to hold it in place - that way if the blade grabs the wood, it will screw itself down tight instead of unscrewing itself and flying around the workshop. Ask me how I know...
• The other method I've used is to rough cut the wood into a circle, drill a 5/8" diameter hole through the center, and mount it in my table saw where the blade goes. Then I just held a sanding block against the edge until the wood was nice and round.

Be creative - there's got to be a million ways to cut a circle.

To be honest, I haven't managed to cut a v-grove in a circle of wood, but here are the two methods I am going to try.

• Make a special sanding block with the correct cross section and sand the groove into the wood while it is still mounted in the table saw.
• Mount the pulley vertically above the table saw blade with the axes more or less parallel, with the pulley axis canted at an angle of about 4 degrees about the vertical, and rotate it slowly as the blade cuts. The angle will cause the blade to cut a parabolic groove - with enough fiddling with the angle and the depth, I think this grove will be a pretty close approximation of a v-groove, at least where the belt contacts it. Note: this sounds dangerous as hell. Even I am wary of trying this...

Once you have a wooden pulley, you can do two things with it - use it as is, or cast it in metal. If you're going to use it as is, be advised that epoxying the pulley to the shaft isn't sufficient.

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