Gears are practically everywhere. They're in cars, both in the transmission and in the windshield wipers. They're in bicycles, in such kitchen utensils as the egg beater and even in watches – or at least they used to be. A gear is basically a set of toothed wheels coupled together to increase or decrease the speed of rotation of a motor drive shaft.

The amount a gear system can alter rotational speed is a function of the relative sizes of the gear wheels, and it's known as the gear ratio. The gear ratio formula turns out to be fairly simple. You basically count the number of teeth on the driven wheel and divide that by the number of teeth on the driver wheel, which is the one attached to the motor. It's a straightforward calculation, even when the gear system consists of several intermediate wheels called idlers.

## It's Easier to Calculate Gear Ratio Than You Think

When you couple two gear wheels, their relative sizes determine how fast each will spin. If the driver wheel is smaller than the driven wheel, it will spin more often than the larger one. If the driver wheel is larger, the driven wheel will spin faster.

You could calculate the amount of speeding up and slowing down that a simple gear system produces by comparing the radii of the wheels, but there's an easier way. Because the teeth of both gear wheels interlock, they have to be the same size on both wheels, so you can simply compare the number of teeth on the two wheels. This is actually how you calculate gear ratio. You count the number of teeth on both the driver wheel and on the driven wheel and express these numbers as a ratio, or a fraction.

For example, if the driver wheel has 20 teeth, and the driven wheel has 40, calculate the gear ratio as 40/20, which simplifies to 2/1, or 2:1. (The tooth count on the driven wheel always goes on top of the fraction or first in the ratio). This tells you that, for every rotation of the driven wheel, the driver wheel makes two rotations. Similarly, a ratio of 1/2 tells you that the driven wheel rotates twice for every rotation of the driver wheel -- in other words, the driven wheel spins faster than the motor shaft.

## How to Apply the Gear Ratio Equation to Complex Systems

Many gear systems incorporate one or more idler wheels, which are often there to ensure that the driver wheel and driven wheel spin in the same direction or to change the plane of rotation. You can apply the gear ratio formula to each pair of wheels in the gear system in succession to arrive at a final gear ratio for the system, but you don't need to do that. If you do, you'll find that the product of all the gear ratios is the same as the ratio between the driver wheel and the driven wheel.

In other words, the driver wheel and the driven wheel are the only two that matter. No matter how many idlers there are in the system, the final gear ratio is the ratio between the driver wheel and the driven wheel. This is true for all types of gears, including spur gears, bevel gears and worm gears.

## Using Gear Ratio to Calculate Speed

If you know the rotational speed of the driver wheel, which is usually measured in revolutions per minute (rpm), the gear ratio tells you the speed of the driven wheel. For example, consider a system with a gear ratio of 3:1, which means the driver wheel spins three times as fast as the driven wheel. If the speed of the driver wheel is 300 rpm, the speed of the driven wheel is 100 rpm.

In general, you can calculate rotational speed using the following gear ratio equation:

S_{1} • T_{1} = S_{2} • T_{2}, where

S_{1} is the speed of the driver wheel and T_{1} is the number of teeth on that wheel.

S_{2} and T_{2} are the speed and tooth count of the driven wheel.

If you're designing a gear system, you will find a gear ratio chart handy. You can find the rpm of the motor in the specifications and use the chart to design a gear system that will produce whatever rotational speed in the driven wheel you require.

References

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.

Photo Credits

gears image by Stanisa Martinovic from Fotolia.com