Gear ratio

The gear ratio of a gear train is the ratio of the angular velocity of the input gear to the angular velocity of the output gear, also known as the speed ratio of the gear train.^[1] The gear ratio can be computed directly from the numbers of teeth of the various gears that engage to form the gear train. The torque ratio of the gear train, also known as its mechanical advantage, is defined by the gear ratio.^[2]

Illustration of gears of an automotive transmission

 General description

The input or drive gear in a gear train is generally connected to a power source, such as a motor or engine. Thus, the drive gear engages the remaining gears in the gear train, and transmits power through to the output or driven gear.

Two meshing gears transmit rotational motion.

 Simple gear train with two gears

The simplest gear train is a pair of meshing gears. The input gear drives the output gear. Gear teeth are designed so the pitch circles of the two gears roll on each other without slipping. The velocities v of the points of contact of the two pitch circles are the same, therefore

where input gear G_A has radius r_A and meshes with output gear G_B of radius r_B.
The number of teeth on a gear is proportional to the radius of its pitch circle, this means that the ratio of the radii equals the ratio of the number of teeth, that is

where N_A is the number of teeth on the input gear and N_B is the number of teeth on the output gear.
This shows that a simple gear train with two gears has the gear ratio R given by

This equation shows that if the number of teeth on the output gear G_B is larger than the number of teeth on the input gear G_A, then the input gear G_A must rotate faster than the output gear G_B.

 Simple gear train with an idler

If a simple gear train has three gears, so that the input gear G_A meshes with an intermediate gear G_I which in turn meshes with the output gear G_B , then the pitch circle of the intermediate gear rolls without slipping on both the pitch circles of the input and output gears. This yields the two relations

The speed ratio of this gear train is obtained by multiplying these two equations to obtain

Notice that this gear ratio is exactly the same as for the case when the gears G_A and G_B engaged directly. The intermediate gear provides spacing but does not otherwise affect the performance of the gear train. For this reason it is called an idler gear. The same gear ratio is obtained for a sequence of idler gears and hence idler gear is used to provide the same direction to rotate the driver and driven gear , as if the driver gear move in clockwise direction then the driven gear also moves in the clockwise direction with the help of idler gear.

Gears on a piece of farm equipment, total (3 gears) gear ratio 42/13 = 3.23

 Example

If we assume that in the photo the smallest gear is connected to the motor, then it is the driver gear. The somewhat larger gear on the upper left is called an idler gear—it is not connected directly to either the motor or the output shaft and serves only to transmit power between the input and output gears. There is a third gear in the upper-right corner of the photo. If we assume that gear is connected to the machine's output shaft, it is the output or driven gear.
The idler gear in this particular gear train has 21 teeth and the input gear has 13. Considering for the moment only those two gears, we can regard the idler as the driven gear. Therefore, the gear ratio is driven/driver = 21/13 = ~1.62 or 1.62:1.
The ratio means that the driver gear must make 1.62 revolutions to turn the driven gear 1 revolution. It also means that for every one revolution of the driver, the driven gear has made 1/1.62, or 0.62, revolutions. In practical terms, the larger gear turns more slowly.
Now suppose the third gear in the picture has 42 teeth. The gear ratio between the idler and third gear is thus 42/21, or 2:1, and hence the final gear ratio is 1.62x2=~3.23. For every 3.23 revolutions of the smallest gear, the largest gear turns one revolution, or for every one revolution of the smallest gear, the largest gear turns 0.31 (1/3.23) revolution, a total reduction of about 1:3.23 (Gear Reduction Ratio (GRR) = 1/Gear Ratio (GR)).
Since the intermediate (idler) gear contacts directly both the smaller and the larger gear it can be removed from the calculation, also giving a ratio of 42/13 = ~3.23.

Speed ratio

The teeth of a gear are distributed on the circumference of the pitch circle so that the thickness of each tooth t and the space between two teeth are the same. The pitch p of a gear, which is the distance between the equivalent points on two teeth, is equal to twice the thickness of a tooth,

The pitch of a gear G_A can be computed from the number of teeth N_A and the radius r_A of its pitch circle

In order to mesh smoothly two gears G_A and G_B must have the same sized teeth and therefore they must have the same pitch p, which means

This equation shows that the ratio of the circumference, the diameters and the radii of two meshing gears is equal to the ratio of their number of teeth,

The speed ratio of two gears rolling without slipping on their pitch circles is given by,

therefore

In other words, the gear ratio, or speed ratio, is inversely proportional to ratio of the radii of the pitch circles and the number of teeth of the two gears.

 Torque ratio

A gear train can be analyzed using the principle of virtual work to show that its torque ratio, which is the ratio of its output torque to its input torque, is equal to the gear ratio, or speed ratio, of the gear train.
This means that the input torque T_A applied to the input gear G_A and the output torque T_B" on the output gear G_B are related by the ratio

where R is the gear ratio of the gear train.
The torque ratio of a gear train is also known as its mechanical advantage, thus

 Belt drives

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.

Valve timing gears on a Ford Taunus V4 engine — the small gear is on the crankshaft, the larger gear is on the camshaft. The crankshaft gear has 34 teeth, the camshaft gear has 68 teeth and runs at half the crankshaft RPM.
(The small gear in the lower left is on the balance shaft.)

A belt with teeth, called the timing belt, is used in some internal combustion engines to exactly synchronize the movement of the camshaft with that of the crankshaft, so that the valves open and close at the top of each cylinder at exactly the right time relative to the movement of each piston. 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 some automobiles for this purpose, while in others, the camshaft and crankshaft are coupled directly together through meshed gears. But whichever form of drive is employed, on four-stroke engines the crankshaft/camshaft gear ratio is always 2:1, which means that for every two revolutions of the crankshaft the camshaft will rotate through one revolution.(In case of 4 stroke engines the valve cycle is repeated after every two rotations of the flywheel.)

Automotive applications

Automobile 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 speed reduction at the wheels. As well, the differential contains further gearing that splits torque equally between the two wheels while permitting them to have different speeds when traveling a curved path. The components might be separate and connected by a driveshaft, or they might be combined into one unit called a transaxle.
A 2004 Chevrolet Corvette C5 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	3.38:1

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 Corvette above has a differential ratio of 3.42:1. The ratio 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 RPM.
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.

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.

Gear	Distance per engine revolution	Speed per 1000 RPM
1st gear	8.1 in (210 mm)	7.7 mph (12.4 km/h)
2nd gear	11.6 in (290 mm)	11.0 mph (17.7 km/h)
3rd gear	16.8 in (430 mm)	15.9 mph (25.6 km/h)
4th gear	24.0 in (610 mm)	22.7 mph (36.5 km/h)
5th gear	28.6 in (730 mm)	27.1 mph (43.6 km/h)
6th gear	42.9 in (1,090 mm)	40.6 mph (65.3 km/h)

 Wide-ratio vs. close-ratio transmission

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A close-ratio transmission^[3] is a transmission in which there is a relatively little difference between the gear ratios of the gears. For example, a transmission with an engine shaft to drive shaft ratio of 4:1 in first gear and 2:1 in second gear would be considered wide-ratio when compared to another transmission with a ratio of 4:1 in first and 3:1 in second. This is because, for the wide-ratio first gear = 4/1 = 4, second gear = 2/1 = 2, so the transmission gear ratio = 4/2 = 2 (or 200%). For the close-ratio first gear = 4/1 = 4, second gear = 3/1 = 3 so the transmission gear ratio = 4/3 = 1.33 (or 133%), because 133% is less than 200%, the transmission with the 133% ratio between gears is considered close-ratio. However, not all transmissions start out with the same ratio in 1st gear or end with the same ratio in 5th gear, which makes comparing wide vs. close transmission more difficult.
Close-ratio transmissions are generally offered in sports cars and sport bikes, and especially in race vehicles, in which the engine is tuned for maximum power in a narrow range of operating speeds, and the driver or rider can be expected to shift often to keep the engine in its power band.
Factory 4-speed or 5-speed transmission ratios are good compromises for mixed street and moderate performance use, and are "staged" or "progressive", in that the engine speed loss on shifting from 1st to 2nd is higher than the loss on shifting from 2nd to 3rd and so on. The purpose is to keep the engine in its torque range at higher vehicle speed, where wind resistance requires more power for acceleration. Wider gaps between ratios will allow a "stronger" (higher numerically, e.g. 2.90:1 instead of 2.50:1) 1st gear for better manners in traffic, but increase the RPM lost on shifting. Narrowing the gaps will increase acceleration at speed, and potentially improve top speed under certain conditions, but acceleration from stopped and operation in traffic will suffer.
The 1st gear ratio for most 4-speed transmissions is about 2.50:1, and 4th is almost always .70:1. The ratios of 2nd and 3rd are placed in between these two, and are discretionary to best serve the weight, intended use, speed, engine tune, and other features of the vehicle.
"Range" is the torque multiplication difference between 1st and 4th gears; wider-ratio gear-sets have more, typically between 2.8 and 3.2. This is the single most important determinant of low-speed acceleration from stopped.
"Progression" is the next factor. This is the reduction or decay in the percentage drop in engine speed in the next gear (e.g. after shifting from 1st to 2nd). Most transmissions have some degree of progression in that the RPM drop on the 1-2 shift is larger than the RPM drop on the 2-3 shift, which is in turn larger than the RPM drop on the 3-4 shift. The progression may not be linear (continuously reduced) or done in proportionate stages for various reasons, including a special need for a gear to reach a specific speed or RPM for passing, racing and so on, or simply economic necessity that the parts were available.
The two factors are not mutually exclusive, but each limits the number of options for the other. A wide range, which gives a strong torque multiplication in 1st gear for excellent manners in low-speed traffic (especially with a smaller motor, heavy chassis or numerically low axle ratio such as 2.50) mean that the progression percentages must all be high. The amount of engine speed (and therefore power) that must be lost on each up-shift is higher than would be the case in a transmission with less range (but less power in 1st gear). A numerically low 1st gear (2.00, &c.) reduces available torque in 1st gear, but allows more choices of progression.
There is no choice of ratios that gives the "best" performance at all speeds, nor is there a choice of final drive (axle) ratio that gives the "best" performance at all speeds. It simply does not exist, all ratios are compromises, and not necessarily better than the original ratios for most use.
The advantage of a close ratio gear-set lies in the fact that the RPM loss at very high speed is reduced, allowing extra power to accelerate above 100 mph. However, of necessity, the torque multiplier in the lower gears is reduced by the same proportion, and performance at low speeds is much worse. Even for road racing, the closest possible ratio is not always the best choice since many races begin with a grid start (favoring slightly wider ratios with high progression, where 1st gear acceleration is very important) and some with a flying start (favoring close ratios, where 1st gear acceleration is less important).
In general, engines with smaller displacement, very long duration cams, ported heads, large carburetors and so on don’t pull well from low rpm, and when the 3-4 shift will benefit more from close ratios in the upper gears, and even more so as the maximum speed at a specific course increases. If the shift takes place at a speed where air resistance is high (70+ mph), closer ratios are better. If your engine has been specifically designed for a tuned RPM torque peak (or if that is how the engine behaves), the transmission ratios must be chosen to ensure that after each shift during a lap the engine speed recovers to a point above this peak at that specific track. From the negative viewpoint, the ratios must be arranged to avoid dropping the engine into a "hole" on an up shift, where power falls off disproportionately.
If the widest ratio change gives a 25% loss, the shift RPM is 7,000 RPM, and there is a torque increase at 5,000 RPM you’re safe: 7,000 - 25% = 5,250, the engine will be in this desirable range on acceleration.
If the widest ratio change is 30%, shift at 7,000, and torque at 5,500: 7,000 - 30% = 4,900, far below the power range and the acceleration (and perhaps the jetting) will be weak until you reach 5,500. You will definitely benefit from a closer gear set, or at least re-arranging the progression to reduce the 30% drop to a better number. Depending on the bike and the track, adding to the drop in the previous gear pair (i.e., problem with the 2-3 shift: add some drop to the 1-2 not the 3-4) is the 1st choice but results will vary.
Individual race tracks with combination of maximum speed and corner speed will require different intermediate (2nd & 3rd) gears to allow downshifting for a specific gear to enter a turn, or to use only one gear during a turn to avoid traction loss. The key to analysis here is whether the track has a spot where the engine is "flat" after shifting at an awkward moment in a turn, but better as it speeds up.

 Idler gear

In a sequence of gears chained together, the ratio depends only on the number of teeth on the first and last gear. The intermediate gears, regardless of their size, do not alter the overall gear ratio of the chain. However, the addition of each intermediate gear reverses the direction of rotation of the final gear.
An intermediate gear which does not drive a shaft to perform any work is called an idler gear. Sometimes, a single idler gear is used to reverse the direction, in which case it may be referred to as a reverse idler. For instance, the typical automobile manual transmission engages reverse gear by means of inserting a reverse idler between two gears.
Idler gears can also transmit rotation among distant shafts in situations where it would be impractical to simply make the distant gears larger to bring them together. Not only do larger gears occupy more space, the mass and rotational inertia (moment of inertia) of a gear is proportional to the square of its radius. Instead of idler gears, a toothed belt or chain can be used to transmit torque over distance.