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A gear train is a
mechanical system A machine is a physical system using power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecul ...
formed by mounting
gear A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic ...
s on a frame so 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 An epicyclic gear train (also known as a planetary gearset) consists of two gears mounted so that the center of one gear revolves around the center of the other. A carrier connects the centers of the two gears and rotates the planet and sun gea ...
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 A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
and belt drives are computed in the same way as gear ratios. See
bicycle gearing Bicycle gearing is the aspect of a bicycle drivetrain that determines the relation between the cadence, the rate at which the rider pedals, and the rate at which the drive wheel turns. On some bicycles there is only one gear and, therefor ...
. The transmission of rotation between contacting toothed wheels can be traced back to the
Antikythera mechanism The Antikythera mechanism ( ) is an Ancient Greek hand-powered orrery, described as the oldest example of an analogue computer used to predict astronomical positions and eclipses decades in advance. It could also be used to track the four-y ...
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.


Mechanical advantage

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. The velocity ''v'' of the point of contact on the pitch circles is the same on both gears, and is given by : v = r_A \omega_A = r_B \omega_B, \! where input gear ''A'' with radius ''rA'' and angular velocity ''ωA'' meshes with output gear ''B'' with radius ''rB'' and angular velocity ''ωB''. Therefore, : \frac = \frac = \frac. where ''NA'' is the number of teeth on the input gear and ''NB'' is the number of teeth on the output gear. The mechanical advantage of a pair of meshing gears for which the input gear has ''NA'' teeth and the output gear has ''NB'' teeth is given by : \mathrm = \frac. This shows that if the output gear ''GB'' has more teeth than the input gear ''GA'', 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.


Analysis using virtual work

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 size of the gears and the sequence in which they engage define the ratio of the angular velocity ''ωA'' of the input gear to the angular velocity ''ωB'' of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let ''R'' be the speed ratio, then : \frac = R. The input torque ''TA'' acting on the input gear ''GA'' is transformed by the gear train into the output torque ''TB'' exerted by the output gear ''GB''. 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. 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: : \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: : F_\theta = T_A \frac - T_B \frac= T_A - T_B/R = 0. The ''mechanical advantage'' of the gear train is the ratio of the output torque ''TB'' to the input torque ''TA'', and the above equation yields: : \mathrm = \frac = 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.


Gear trains with two gears

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.


Formula

The teeth on gears are designed so the gears can roll on each other smoothly (without slipping or jamming). In order for two gears to roll on each other smoothly, they must be designed so the velocity at the point of contact of the two pitch circles (represented by ''v'') is the same for each gear. Mathematically, if the input gear ''GA'' has the radius ''rA'' and angular velocity \omega_A \!, and meshes with output gear ''GB'' of radius ''rB'' and angular velocity \omega_B \!, then: : v = r_A \omega_A = r_B \omega_B, \! The number of teeth on a gear is proportional to the radius of its pitch circle, which means the ratios of the gears' angular velocities, radii, and number of teeth are equal. Where ''NA'' is the number of teeth on the input gear and ''NB'' is the number of teeth on the output gear, the following equation is formed: : \frac = \frac = \frac. This shows that a simple gear train with two gears has the gear ratio ''R'' given by: : R = \frac = \frac. This equation shows that if the number of teeth on the output gear ''GB'' is larger than the number of teeth on the input gear ''GA'', then the input gear ''GA'' must rotate faster than the output gear ''GB''.


Double reduction gear

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 A layshaft is an intermediate shaft within a gearbox that carries gears, but does not transfer the primary drive of the gearbox either in or out of the gearbox. Layshafts are best known through their use in car gearboxes, where they were a ubiquit ...
. 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.


Speed 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, : p=2t.\! The pitch of a gear ''GA'' can be computed from the number of teeth ''NA'' and the radius ''rA'' of its pitch circle : p = \frac. In order to mesh smoothly two gears ''GA'' and ''GB'' must have the same sized teeth and therefore they must have the same pitch ''p'', which means : p = \frac = \frac. 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, : \frac = \frac. The speed ratio of two gears rolling without slipping on their pitch circles is given by, : R =\frac=\frac, therefore : R = \frac = \frac. 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.


Torque ratio

A gear train can be analyzed using the principle of virtual work to show that its
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
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 ''ΤA'' applied to the input gear ''GA'' and the output torque ''ΤB'' on the output gear ''GB'' are related by the ratio : R = \frac, where ''R'' is the gear ratio of the gear train. The torque ratio of a gear train is also known as its mechanical advantage : \mathit = \frac.


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 The Idler refers to someone idle: * An idle game * A slacker, a person who habitually avoids work * Idler-wheel, a system used to transmit the rotation of the main shaft of a motor to another rotating device * Idler circuit, a circuit in a parame ...
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 A manual transmission (MT), also known as manual gearbox, standard transmission (in Canada, the United Kingdom, and the United States), or stick shift (in the United States), is a multi-speed motor vehicle transmission (mechanics), 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 The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
) 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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
over distance.


Formula

If a simple gear train has three gears, such that the input gear ''GA'' meshes with an intermediate gear ''GI'' which in turn meshes with the output gear ''GB'', 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 : \frac = \frac, \quad \frac = \frac. The speed ratio of this gear train is obtained by multiplying these two equations to obtain : R = \frac = \frac. Notice that this gear ratio is exactly the same as for the case when the gears ''GA'' and ''GB'' 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

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 The Idler refers to someone idle: * An idle game * A slacker, a person who habitually avoids work * Idler-wheel, a system used to transmit the rotation of the main shaft of a motor to another rotating device * Idler circuit, a circuit in a parame ...
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 In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...
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.


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
bicycle A bicycle, also called a pedal cycle, bike or cycle, is a human-powered or motor-powered assisted, pedal-driven, single-track vehicle, having two wheels attached to a frame, one behind the other. A is called a cyclist, or bicyclist. B ...
s and some
motorcycle A motorcycle (motorbike, bike, or trike (if three-wheeled)) is a two or three-wheeled motor vehicle Steering, steered by a Motorcycle handlebar, handlebar. Motorcycle design varies greatly to suit a range of different purposes: Long-distance ...
s. Again, exact accounting of teeth and revolutions can be applied with these machines. For example, a belt with teeth, called the timing belt, is used in some internal combustion engines to synchronize the movement of the
camshaft A camshaft is a shaft that contains a row of pointed cams, in order to convert rotational motion to reciprocating motion. Camshafts are used in piston engines (to operate the intake and exhaust valves), mechanically controlled ignition systems ...
with that of the
crankshaft A crankshaft is a mechanical component used in a piston engine to convert the reciprocating motion into rotational motion. The crankshaft is a rotating shaft containing one or more crankpins, that are driven by the pistons via the connecti ...
, 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 piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-t ...
. 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 engine A four-stroke (also four-cycle) engine is an internal combustion (IC) engine in which the piston completes four separate strokes while turning the crankshaft. A stroke refers to the full travel of the piston along the cylinder, in either directi ...
s, which means that for every two revolutions of the crankshaft the camshaft will rotate once.


Automotive applications

Automobile drivetrains generally have two or more major areas where gearing is used. Gearing is employed in the
transmission Transmission may refer to: Medicine, science and technology * Power transmission ** Electric power transmission ** Propulsion transmission, technology allowing controlled application of power *** Automatic transmission *** Manual 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 A transaxle is a single mechanical device which combines the functions of an automobile's transmission, axle, and differential into one integrated assembly. It can be produced in both manual and automatic versions. Engine and drive at the s ...
. The gear ratios in transmission and final drive are important because different gear ratios will change the characteristics of a vehicle's performance.


Example

A 2004 Chevrolet Corvette C5 Z06 with a six-speed
manual transmission A manual transmission (MT), also known as manual gearbox, standard transmission (in Canada, the United Kingdom, and the United States), or stick shift (in the United States), is a multi-speed motor vehicle transmission (mechanics), transmission ...
has the following gear ratios in the transmission: 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 rotate at the same speed, referred to as the "direct drive" ratio. 5th and 6th gears are known as
overdrive Overdrive may refer to: Organizations * OverDrive, Inc., a digital distributor of entertainment media ** OverDrive Media Console, a media player developed by OverDrive, Inc. * Overdrive PC, a subsidiary of Velocity Micro Technology * Overdrive ...
gears, in which the output of the transmission is revolving faster than the engine's output. The Corvette above has an axle ratio of 3.42:1, meaning that for every 3.42 revolutions of the transmission's output, the
wheel A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be ...
s 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
tire A tire (American English) or tyre (British English) is a ring-shaped component that surrounds a Rim (wheel), wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide Traction (engineering), t ...
s can almost be thought of as a third type of gearing. This car 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 . 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. d = \frac It is also 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 Note that the answer is in inches per minute, which can be converted to mph by dividing by 1056.


Wide-ratio vs. close-ratio transmission

A close-ratio transmission 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 the close-ratio transmission has less of a progression between gears. For the wide-ratio transmission, the first gear ratio is 4:1 or 4, and in second gear it is 2:1 or 2, so the progression is equal to 4/2 = 2 (or 200%). For the close-ratio transmission, first gear has a 4:1 ratio or 4, and second gear has a ratio of 3:1 or 3, so the progression between gears is 4/3, or 133%. Since 133% is less than 200%, the transmission with the smaller progression between gears is considered close-ratio. However, the difference between a close-ratio and wide-ratio transmission is subjective and relative. Close-ratio transmissions are generally offered in
sports car A sports car is a car designed with an emphasis on dynamic performance, such as handling, acceleration, top speed, the thrill of driving and racing capability. Sports cars originated in Europe in the early 1900s and are currently produced by ...
s, sport bikes, and especially in race vehicles, where 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 The power band of an internal combustion engine or electric motor is the range of operating speeds under which the engine or motor is able to output the most power, that is, the maximum energy per unit of time. This usually means that maximum ...
. Factory 4-speed or 5-speed transmission ratios generally have a greater difference between gear ratios and tend to be effective for ordinary driving and moderate performance use. Wider gaps between ratios allow a higher 1st gear ratio for better manners in traffic, but cause engine speed to decrease more when shifting. Narrowing the gaps will increase acceleration at speed, and potentially improve top speed under certain conditions, but acceleration from a stopped position and operation in daily driving will suffer. ''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 reduction or decay in the percentage drop in engine speed in the next gear, for example after shifting from 1st to 2nd gear. 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. Range and progression 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 vehicle, or numerically low axle ratio such as 2.50, means the progression percentages must be high. The amount of engine speed, and therefore power, lost on each up-shift is greater than would be the case in a transmission with less range, but less power in 1st gear. A numerically low 1st gear, such as 2:1, reduces available torque in 1st gear, but allows more choices of progression. There is no optimal choice of transmission gear ratios or a final drive ratio for best performance at all speeds, as gear ratios are compromises, and not necessarily better than the original ratios for certain purposes.


See also

* Machine (mechanical) *
Mechanism (engineering) In engineering, a mechanism is a device that transforms input forces and movement into a desired set of output forces and movement. Mechanisms generally consist of moving components which may include: * Gears and gear trains; * Belts and chai ...
* Powertrain *
Wheel train (horology) In horology, a wheel train (or just train) is the gear train of a mechanical watch or clock. Although the term is used for other types of gear trains, the long history of mechanical timepieces has created a traditional terminology for their gear ...
*
Outline of machines Machine – mechanical system that provides the useful application of power to achieve movement. A machine consists of a power source, or engine, and a mechanism or transmission for the controlled use of this power. The combination of for ...
*
Epicyclic gearing An epicyclic gear train (also known as a planetary gearset) consists of two gears mounted so that the center of one gear revolves around the center of the other. A carrier connects the centers of the two gears and rotates the planet and sun gea ...
- related to
turboprop A turboprop is a turbine engine that drives an aircraft propeller. A turboprop consists of an intake, reduction gearbox, compressor, combustor, turbine, and a propelling nozzle. Air enters the intake and is compressed by the compressor. ...
reduction gear boxes *
Continuously variable transmission A continuously variable transmission (CVT) is an automatic transmission that can change seamlessly through a continuous range of gear ratios. This contrasts with other transmissions that provide a limited number of gear ratios in fixed steps. ...
(CVT) * Virtual work


References


External links


Gear ratio at How Stuff Works

Online Motorcycle gear train calculator at Gearingcommander.com
{{DEFAULTSORT:Gear train Mechanisms (engineering) Gears