# Gear ratio

Gears on a piece of farm equipment, gear ratio 1:1.61

The gear ratio is the relationship between the number of teeth on two gears that are meshed or two sprockets connected with a common roller chain, or the circumferences of two pulleys connected with a drive belt.

In the picture to the right, the smaller gear has thirteen teeth, while the second, larger gear has twenty-one teeth. The gear ratio is therefore 13/21 or 1/1.61 (also written as 1:1.61)

The first number in the ratio is usually the gear that power is applied to. In an automobile the first number is the gear receiving power from the engine.

This means that for every one revolution of the smaller gear, the larger gear has made 1/1.61, or 0.62, revolutions. In practical terms, the larger gear turns more slowly.

Suppose the largest gear in the picture has 42 teeth, the gear ratio between the second and third gear then is; 21/42 = 1/2 and for every revolution of the smallest gear the largest gear has only turned 0.62/2 = 0.31 revolution, a total reduction of around 1:3.

Since the number of teeth is also proportional to the circumference of the gear wheel (the bigger the wheel the more teeth it has) the gear ratio can also be expressed as the relationship between the circumferences of both wheels (where d is the diameter of the smaller wheel and D is the diameter of the larger wheel):

$gr = (\pi \times d) / (\pi \times D) = d/D$

Since the diameter is equal to twice the radius;

$gr = d / D = (2 \times r) /( 2 \times R) = r / R$

as well.

But keep in mind that counting the teeth derives the exact gear ratio, regardless of any variations in the diameter measurement. In the picture, each time the 13 teeth of the smaller gear make a revolution, 13 teeth of the larger gear will have moved, i.e. made 13/21 of a revolution or 0.62 of a revolution. As long as the gears remain meshed, the accounting of teeth and revolutions will remain perfect. So for instance gears can be used to construct a clock in which the minute hand moves exactly sixty times faster than the hour hand, regardless of the overall accuracy of the watch.

Diameter measurements are useful for determining approximate gear ratios for non-gear linkages such as pulleys and belts. Smooth belts can slip, so even if exact pulley diameters are known quite exactly, the gear ratio may vary in operation, and may even depend on the load.

Belts can have teeth in them also and be coupled to gear-like pulleys. Special gears called sprockets can be coupled together with chains, as on bicycles and some motorcycles. Again, exact accounting of teeth and revolutions can be applied with these machines.

A belt with teeth, called the timing belt, is used in some internal combustion engines to exactly synchronize the movement of the camshaft, so that the valves open and close at the top of each cylinder at exactly the right time to the movement of each cylinder. From the time the car is driven off the lot, to the time the belt needs replacing thousands of kilometers later, it synchronizes the two shafts exactly. A chain, called a timing chain, is used on other automobiles for this purpose. In a few automobiles, the camshaft and crankshaft are coupled directly together through meshed gears.

## Example

Automobiles drivetrains generally have two or more areas where gearing is used: one in the transmission, which contains a number of different sets of gearing that can be changed to allow a wide range of vehicle speeds, and another at the differential, which contains one additional set of gearing that provides further mechanical advantage at the wheels. These components might be separate and connected by a shaft, or they might be combined into one unit called a transaxle

A Chevrolet Corvette Z06 with a six-speed manual transmission has the following gear ratios in the transmission:

 Gear Ratio 1st gear 2.97:1 2nd gear 2.07:1 3rd gear 1.43:1 4th gear 1.00:1 5th gear 0.84:1 6th gear 0.56:1 reverse 3.28:1

In 1st gear, the engine makes 2.97 revolutions for every revolution of the transmission’s output. In 4th gear, the gear ratio of 1:1 means that the engine and the transmission’s output are moving at the same speed. 5th and 6th gears are known as overdrive gears, in which the output of the transmission is revolving faster than the engine.

The above Corvette has a differential ratio of 3.42:1. This means that for every 3.42 revolutions of the transmission’s output, the wheels make one revolution. The differential ratio multiplies with the transmission ratio, so in 1st gear, the engine makes 10.16 revolutions for every revolution of the wheels.

The car’s tires can almost be thought of as a third type of gearing. The example Corvette Z06 is equipped with 295/35-18 tires, which have a circumference of 82.1 inches. This means that for every complete revolution of the wheel, the car travels 82.1 inches. If the Corvette had larger tires, it would travel farther with each revolution of the wheel, which would be like a higher gear. If the car had smaller tires, it would be like a lower gear.

With the gear ratios of the transmission and differential, and the size of the tires, it becomes possible to calculate the speed of the car for a particular gear at a particular engine speed.

For example, it is possible to determine the distance the car will travel for one revolution of the engine by dividing the circumference of the tire by the combined gear ratio of the transmission and differential.

$d = \frac{c_t}{gr_t \times gr_d}$

It is possible to determine a car’s speed from the engine speed by multiplying the circumference of the tire by the engine speed and dividing by the combined gear ratio.

$v_c = \frac{c_t \times v_e}{gr_t \times gr_d}$

 Gear Inches per engine revolution Speed per 1000 rpm 1st gear 8.1inches 11.2 mph 2nd gear 11.6 inches 16.1 mph 3rd gear 16.8 inches 23.3 mph 4th gear 24.0 inches 33.3 mph 5th gear 28.6 inches 39.7 mph 6th gear 42.9 inches 59.5 mph