Gear train

A gear train or gear set is a machine element of a mechanical system formed by mounting two or more gears on a frame such that the teeth of the gears engage.
Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, providing a smooth transmission of rotation from one gear to the next. Features of gears and gear trains include:

The gear ratio of the pitch circles of mating gears defines the speed ratio and the mechanical advantage of the gear set.
A planetary gear train provides high gear reduction in a compact package.
It is possible to design gear teeth for gears that are non-circular, yet still transmit torque smoothly.
The speed ratios of chain and belt drives are computed in the same way as gear ratios. See bicycle gearing.

The transmission of rotation between contacting toothed wheels can be traced back to the Antikythera mechanism of Greece and the south-pointing chariot of China. Illustrations by the Renaissance scientist Georgius Agricola show gear trains with cylindrical teeth. The implementation of the involute tooth yielded a standard gear design that provides a constant speed ratio.

Gear ratio

Dimensions and terms

The pitch circle of a given gear is determined by the tangent point contact between two meshing gears; for example, two spur gears mesh together when their pitch circles are tangent, as illustrated.
The pitch diameter is the diameter of a gear's pitch circle, measured through that gear's rotational centerline, and the pitch radius is the radius of the pitch circle. The distance between the rotational centerlines of two meshing gears is equal to the sum of their respective pitch radii.
The circular pitch is the distance, measured along the pitch circle, between one tooth and the corresponding point on an adjacent tooth.
The number of teeth per gear is an integer determined by the pitch circle and circular pitch.

Relationships

The circular pitch of a gear can be defined as the circumference of the pitch circle using its pitch radius divided by the number of teeth :
The thickness of each tooth, measured through the pitch circle, is equal to the gap between neighboring teeth to ensure the teeth on adjacent gears, cut to the same tooth profile, can mesh without interference. This means the circular pitch is equal to twice the thickness of a tooth,
In the United States, the diametral pitch is the number of teeth on a gear divided by the pitch diameter; for SI countries, the module is the reciprocal of this value. For any gear, the relationship between the number of teeth, diametral pitch or module, and pitch diameter is given by:
Since the pitch diameter is related to circular pitch as
this means
Rearranging, we obtain a relationship between diametral pitch and circular pitch:

Gear or speed ratio

For a pair of meshing gears, the angular speed ratio, also known as the gear ratio, can be computed from the ratio of the pitch radii or the ratio of the number of teeth on each gear. Define the angular speed ratio of two meshed gears A and B as the ratio of the magnitude of their respective angular velocities:
Here, subscripts are used to designate the gear, so gear A has a radius of and angular velocity of with teeth, which meshes with gear B which has corresponding values for radius, angular velocity, and teeth.
When these two gears are meshed and turn without slipping, the velocity of the tangent point where the two pitch circles come in contact is the same on both gears, and is given by:
Rearranging, the ratio of angular velocity magnitudes is the inverse of the ratio of pitch circle radii:
Therefore, the angular speed ratio can be determined from the respective pitch radii:
For example, if gear A has a pitch circle radius of and gear B has a pitch circle radius of, the angular speed ratio is 2, which is sometimes written as 2:1. Gear A turns at twice the speed of gear B. For every complete revolution of gear A, gear B makes half a revolution.
In addition, consider that in order to mesh smoothly and turn without slipping, these two gears A and B must have compatible teeth. Given the same tooth and gap widths, they also must have the same circular pitch, which means
This equation can be rearranged to show the ratio of the pitch circle radii of two meshing gears is equal to the ratio of their number of teeth:
Since the angular speed ratio depends on the ratio of pitch circle radii, it is equivalently determined by the ratio of the number of teeth:
In other words, the speed ratio is inversely proportional to the radius of the pitch circle and the number of teeth of gear A, and directly proportional to the same values for gear B.

Torque ratio analysis using virtual work

The gear ratio also determines the transmitted torque. The torque ratio of the gear train is defined as the ratio of its output torque to its input torque. Using the principle of virtual work, the gear train's torque ratio is equal to the gear ratio, or speed ratio, of the gear train. Again, assume we have two gears A and B, with subscripts designating each gear and gear A serving as the input gear.
For this analysis, 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 input torque acting on the input gear A is transformed by the gear train into the output torque exerted by the output gear B.
Let be the speed ratio, then by definition
Assuming 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. Because there is a single degree of freedom, the angle θ of the input gear completely determines the angle of the output gear and serves as the generalized coordinate of the gear train.
The speed ratio of the gear train can be rearranged to give the magnitude of angular velocity of the output gear in terms of the input gear velocity.
Rewriting in terms of a common angular velocity,
The principle of virtual work states the input force on gear A and the output force on gear B using applied torques will sum to zero:
This can be rearranged to:
Since is the gear ratio of the gear train, the input torque applied to the input gear A and the output torque on the output gear B are related by the same gear or speed ratio.

Mechanical advantage

The torque ratio of a gear train is also known as its mechanical advantage; as demonstrated, the gear ratio and speed ratio of a gear train also give its mechanical advantage.
The mechanical advantage of a pair of meshing gears for which the input gear A has teeth and the output gear B has teeth is given by
This shows that if the output gear B has more teeth than the input gear A, then the gear train amplifies the input torque. In this case, the gear train is called a speed reducer and since the output gear must have more teeth than the input gear, the speed reducer amplifies the input torque. When the input gear rotates faster than the output gear, then the gear train amplifies the input torque. Conversely, if the output gear has fewer teeth than the input gear, then the gear train reduces the input torque; in other words, when the input gear rotates slower than the output gear, the gear train reduces the input torque.

Hunting and non-hunting gear sets

A hunting gear set is a set of gears where the gear teeth counts are relatively prime on each gear in an interfacing pair. Since the number of teeth on each gear have no common factors, then any tooth on one of the gears will come into contact with every tooth on the other gear before encountering the same tooth again. This results in less wear and longer life of the mechanical parts.
A non-hunting gear set is one where the teeth counts are insufficiently prime. In this case, some particular gear teeth will come into contact with particular opposing gear teeth more times than others, resulting in more wear on some teeth than others.

Implementations

Gear trains with two gears

The simplest example of a gear train has two gears. The input gear transmits power to the output 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 gear depend on the ratio of the dimensions of the two gears or the ratio of the tooth counts.

Idler gears

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 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.