Gear train

Gear teeth are designed so the number of teeth on a gear is proportional to the radius of its pitch circle, and so the pitch circles of meshing gears roll on each other without slipping. The speed ratio for a pair of meshing gears can be computed from ratio of the radii of the pitch circles and the ratio of the number of teeth on each gear.

Two meshing gears transmit rotational motion.

The velocity v of the point of contact on the pitch circles is the same on both gears, and is given by

${\displaystyle v=r_{A}\omega _{A}=r_{B}\omega _{B},\!}$

where input gear A with radius r_A and angular velocity ω_A meshes with output gear B with radius r_B and angular velocity ω_B. Therefore,

${\displaystyle {\frac {\omega _{A}}{\omega _{B}}}={\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}.}$

where N_A is the number of teeth on the input gear and N_B is the number of teeth on the output gear.

The mechanical advantage of a pair of meshing gears for which the input gear has N_A teeth and the output gear has N_B teeth is given by

${\displaystyle \mathrm {MA} ={\frac {N_{B}}{N_{A}}}.}$

This shows that if the output gear G_B has more teeth than the input gear G_A, then the gear train amplifies the input torque. And, if the output gear has fewer teeth than the input gear, then the gear train reduces the input torque.

If the output gear of a gear train rotates more slowly than the input gear, then the gear train is called a speed reducer. In this case, because the output gear must have more teeth than the input gear, the speed reducer amplifies the input torque.

For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear.

The size of the gears and the sequence in which they engage define the ratio of the angular velocity ω_A of the input gear to the angular velocity ω_B of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio, then

${\displaystyle {\frac {\omega _{A}}{\omega _{B}}}=R.}$

The input torque T_A acting on the input gear G_A is transformed by the gear train into the output torque T_B exerted by the output gear G_B. If we assume the gears are rigid and there are no losses in the engagement of the gear teeth, then the principle of virtual work can be used to analyze the static equilibrium of the gear train.

Let the angle θ of the input gear be the generalized coordinate of the gear train, then the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear:

${\displaystyle \omega _{A}=\omega ,\quad \omega _{B}=\omega /R.\!}$

The formula for the generalized force obtained from the principle of virtual work with applied torques yields:[2]

${\displaystyle F_{\theta }=T_{A}{\frac {\partial \omega _{A}}{\partial \omega }}-T_{B}{\frac {\partial \omega _{B}}{\partial \omega }}=T_{A}-T_{B}/R=0.}$

The mechanical advantage of the gear train is the ratio of the output torque T_B to the input torque T_A, and the above equation yields:

${\displaystyle \mathrm {MA} ={\frac {T_{B}}{T_{A}}}=R.}$

The speed ratio of a gear train also defines its mechanical advantage. This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And if the input gear rotates slower than the output gear, the gear train reduces the input torque.

The simplest example of a gear train has two gears. The "input gear" (also known as drive gear) transmits power to the "output gear" (also known as driven gear). The input gear will typically be connected to a power source, such as a motor or engine. In such an example, the output of torque and rotational speed from the output (driven) gear depend on the ratio of the dimensions of the two gears.

Double reduction gears

A double reduction gear comprises two pairs of gears, as single reductions, in series. In the diagram, the red and blue gears give the first stage of reduction and the orange and green gears give the second stage of reduction. The total reduction is the product of the first stage of reduction and the second stage of reduction.

It is essential to have two coupled gears, of different sizes, on the intermediate layshaft. If a single intermediate gear was used, the overall ratio would be simply that between the first and final gears, the intermediate gear would only act as an idler gear: it would reverse the direction of rotation, but not change the ratio.

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

${\displaystyle p=2t.\!}$

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

${\displaystyle p={\frac {2\pi r_{A}}{N_{A}}}.}$

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

${\displaystyle p={\frac {2\pi r_{A}}{N_{A}}}={\frac {2\pi r_{B}}{N_{B}}}.}$

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,

${\displaystyle {\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}.}$

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

${\displaystyle R={\frac {\omega _{A}}{\omega _{B}}}={\frac {r_{B}}{r_{A}}},}$

therefore

${\displaystyle R={\frac {\omega _{A}}{\omega _{B}}}={\frac {N_{B}}{N_{A}}}.}$

In other words, the gear ratio, or speed ratio, is inversely proportional to the radius of the pitch circle and the number of teeth of the input gear.

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 the input torque Τ_A applied to the input gear G_A and the output torque Τ_B on the output gear G_B are related by the ratio

${\displaystyle R={\frac {T_{B}}{T_{A}}},}$

where R is the gear ratio of the gear train.

The torque ratio of a gear train is also known as its mechanical advantage

${\displaystyle {\mathit {MA}}={\frac {T_{B}}{T_{A}}}.}$

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.

Formula

If a simple gear train has three gears, such 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

${\displaystyle {\frac {\omega _{A}}{\omega _{I}}}={\frac {N_{I}}{N_{A}}},\quad {\frac {\omega _{I}}{\omega _{B}}}={\frac {N_{B}}{N_{I}}}.}$

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

${\displaystyle R={\frac {\omega _{A}}{\omega _{B}}}={\frac {N_{B}}{N_{A}}}.}$

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

Example

2 gears and an idler gear on a piece of farm equipment, with a ratio of 42/13 = 3.23:1

In the photo, assuming the smallest gear is connected to the motor, it is called the drive gear or input gear. The somewhat larger gear in the middle is called an idler gear. It is not connected directly to either the motor or the output shaft and only transmits power between the input and output gears. There is a third gear in the upper-right corner of the photo. Assuming that gear is connected to the machine's output shaft, it is the output or driven gear.

The input gear in this gear train has 13 teeth and the idler gear has 21 teeth. Considering only these gears, the gear ratio between the idler and the input gear can be calculated as if the idler gear was the output gear. Therefore, the gear ratio is driven/drive = 21/13 ≈1.62 or 1.62:1.

At this ratio, it means the drive gear must make 1.62 revolutions to turn the driven gear once. It also means that for every one revolution of the driver, the driven gear has made 1/1.62, or 0.62, revolutions. Essentially, the larger gear turns slower.

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 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. The idler gear serves to make both the drive gear and the driven gear rotate in the same direction, but confers no mechanical advantage.

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.)

For example, a belt with teeth, called the timing belt, is used in some internal combustion engines to 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. 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. Regardless of which form of drive is employed, the crankshaft-to-camshaft gear ratio is always 2:1 on four-stroke engines, which means that for every two revolutions of the crankshaft the camshaft will rotate once.

Illustration of gears of an automotive transmission

Automobile drivetrains generally have two or more major areas where gearing is used. Gearing is employed in the transmission, which contains a number of different sets of gears that can be changed to allow a wide range of vehicle speeds, and also in the differential, which contains the final drive to provide further speed reduction at the wheels. In addition, the differential contains further gearing that splits torque equally between the two wheels while permitting them to have different speeds when travelling in a curved path. The transmission and final drive might be separate and connected by a driveshaft, or they might be combined into one unit called a transaxle. The gear ratios in transmission and final drive are important because different gear ratios will change the characteristics of a vehicle's performance.

• Gear ratio at How Stuff Works
• Online Motorcycle gear train calculator at Gearingcommander.com