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# definition - Differential (mechanical device)

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# Differential (mechanical device)

A differential is a device, usually, but not necessarily, employing gears, which is connected to the outside world by three shafts, through which it transmits torque and rotation. The gears or other components make the three shafts rotate in such a way that $\scriptstyle a=pb+qc$, where $\scriptstyle a$, $\scriptstyle b$, and $\scriptstyle c$ are the angular velocities of the three shafts, and $\scriptstyle p$ and $\scriptstyle q$ are constants. Often, but not always, $\scriptstyle p$ and $\scriptstyle q$ are equal, so $\scriptstyle a$ is proportional to the sum (or average) of $\scriptstyle b$ and $\scriptstyle c$. Except in some special-purpose differentials, there are no other limitations on the rotational speeds of the shafts. Any of the shafts can be used to input rotation, and the other(s) to output it. See animation here of a simple differential in which $\scriptstyle p$ and $\scriptstyle q$ are equal. The shaft rotating at speed $\scriptstyle a$ is at the bottom-right of the image.

In automobiles and other wheeled vehicles, a differential allows the driving roadwheels to rotate at different speeds. This is necessary when the vehicle turns, making the wheel that is travelling around the outside of the turning curve roll farther and faster than the other. The engine is connected to the shaft rotating at angular velocity $\scriptstyle a$. The driving wheels are connected to the other two shafts, and $\scriptstyle p$ and $\scriptstyle q$ are equal. If the engine is running at a constant speed, the rotational speed of each driving wheel can vary, but the sum (or average) of the two wheels' speeds can not change. An increase in the speed of one wheel must be balanced by an equal decrease in the speed of the other.

It may seem illogical that the speed of one input shaft can determine the speeds of two output shafts, which are allowed to vary. Logically, the number of inputs should be at least as great as the number of outputs. However, the system has another constraint. The ratio of the speeds of the two driving wheels equals the ratio of the radii of the paths around which the two wheels are rolling, which is determined by the track-width of the vehicle (the distance between the driving wheels) and the radius of the turn. Thus the system does not have one input and two independent outputs. It has two inputs and two outputs.

A different automotive application of differentials is in epicyclic gearing. A gearbox is constructed out of several differentials. In each differential, one shaft is connected to the engine (through a clutch or functionally similar device), another to the driving wheels (through another differential as described above), and the third shaft can be braked so its angular velocity is zero. (The braked component may not be a shaft, but something that plays an equivalent role.) When one shaft is braked, the gear ratio between the engine and wheels is determined by the value(s) of $\scriptstyle p$ and/or $\scriptstyle q$ for that differential, which reflect the numbers of teeth on its gears. Several differentials, with different gear ratios, are permanently connected in parallel with each other, but only one of them has one shaft braked so it can not rotate, so only that differential transmits power from the engine to the wheels. (If the transmission is in "neutral" or "park", none of the shafts is braked.) Shifting gears simply involves releasing the braked shaft of one differential and braking the appropriate shaft on another. This is a much simpler operation to do automatically than engaging and disengaging gears in a conventional gearbox. Epicyclic gearing is almost always used in automatic transmissions, and is nowadays also used in some hybrid and electric vehicles.

Non-automotive uses of differentials include performing analog arithmetic. Two of the differential's three shafts are made to rotate through angles that represent (are proportional to) two numbers, and the angle of the third shaft's rotation represents the sum or difference of the two input numbers. An equation clock that used a differential for addition, made in 1720, is the earliest device definitely known to have used a differential for any purpose.[1] In the 20th Century, large assemblies of many differentials were used as analog computers, calculating, for example, the direction in which a gun should be aimed. However, the development of electronic digital computers has made these uses of differentials obsolete.[2] Practically all the differentials that are now made are used in automobiles and similar vehicles. This article therefore emphasizes automotive uses of differentials.

## Purpose

A vehicle's wheels rotate at different speeds, mainly when turning corners. The differential is designed to drive a pair of wheels while allowing them to rotate at different speeds. In vehicles without a differential, such as karts, both driving wheels are forced to rotate at the same speed, usually on a common axle driven by a simple chain-drive mechanism. When cornering, the inner wheel needs to travel a shorter distance than the outer wheel, so with no differential, the result is the inner wheel spinning and/or the outer wheel dragging, and this results in difficult and unpredictable handling, damage to tires and roads, and strain on (or possible failure of) the entire drivetrain.

## Functional description

Input torque is applied to the ring gear (blue), which turns the entire carrier (blue). The carrier is connected to both the sun gears (red and yellow) only through the planet gear (green). Torque is transmitted to the sun gears through the planet gear. The planet gear revolves around the axis of the carrier, driving the sun gears. If the resistance at both wheels is equal, the planet gear revolves without spinning about its own axis, and both wheels turn at the same rate.
If the left sun gear (red) encounters resistance, the planet gear (green) spins as well as revolving, allowing the left sun gear to slow down, with an equal speeding up of the right sun gear (yellow).

The following description of a differential applies to a "traditional" rear-wheel-drive car or truck with an "open" or limited slip differential combined with a reduction gearset:

Torque is supplied from the engine, via the transmission, to a drive shaft (British term: 'propeller shaft', commonly and informally abbreviated to 'prop-shaft'), which runs to the final drive unit that contains the differential. A spiral bevel pinion gear takes its drive from the end of the propeller shaft, and is encased within the housing of the final drive unit. This meshes with the large spiral bevel ring gear, known as the crown wheel. The crown wheel and pinion may mesh in hypoid orientation, not shown. The crown wheel gear is attached to the differential carrier or cage, which contains the 'sun' and 'planet' wheels or gears, which are a cluster of four opposed bevel gears in perpendicular plane, so each bevel gear meshes with two neighbours, and rotates counter to the third, that it faces and does not mesh with. The two sun wheel gears are aligned on the same axis as the crown wheel gear, and drive the axle half shafts connected to the vehicle's driven wheels. The other two planet gears are aligned on a perpendicular axis which changes orientation with the ring gear's rotation. In the two figures shown above, only one planet gear (green) is illustrated, however, most automotive applications contain two opposing planet gears. Other differential designs employ different numbers of planet gears, depending on durability requirements. As the differential carrier rotates, the changing axis orientation of the planet gears imparts the motion of the ring gear to the motion of the sun gears by pushing on them rather than turning against them (that is, the same teeth stay in the same mesh or contact position), but because the planet gears are not restricted from turning against each other, within that motion, the sun gears can counter-rotate relative to the ring gear and to each other under the same force (in which case the same teeth do not stay in contact).

Thus, for example, if the car is making a turn to the right, the main crown wheel may make 10 full rotations. During that time, the left wheel will make more rotations because it has further to travel, and the right wheel will make fewer rotations as it has less distance to travel. The sun gears (which drive the axle half-shafts) will rotate in opposite directions relative to the ring gear by, say, 2 full turns each (4 full turns relative to each other), resulting in the left wheel making 12 rotations, and the right wheel making 8 rotations.

The rotation of the crown wheel gear is always the average of the rotations of the side sun gears. This is why, if the driven roadwheels are lifted clear of the ground with the engine off, and the drive shaft is held (say leaving the transmission 'in gear', preventing the ring gear from turning inside the differential), manually rotating one driven roadwheel causes the opposite roadwheel to rotate in the opposite direction by the same amount.

When the vehicle is traveling in a straight line, there will be no differential movement of the planetary system of gears other than the minute movements necessary to compensate for slight differences in wheel diameter, undulations in the road (which make for a longer or shorter wheel path), etc.

## History

There are many claims to the invention of the differential gear but it is possible that it was known, at least in some places, in ancient times. Some historical milestones of the differential include:

Note: The Antikythera mechanism (150 BC100 BC), discovered on an ancient shipwreck near the Greek island of Antikythera, was once suggested to have employed a differential gear. This has since been disproved. Other possible uses of differentials prior to Joseph Williamson's clock of 1720 are hypothetical.

## Loss of traction

One undesirable side effect of a conventional differential is that it can limit traction under less than ideal conditions. The amount of traction required to propel the vehicle at any given moment depends on the load at that instant—how heavy the vehicle is, how much drag and friction there is, the gradient of the road, the vehicle's momentum, and so on.

The torque applied to each driving wheel is a result of the engine, transmission and drive axles applying a twisting force against the resistance of the traction at that roadwheel. In lower gears and thus at lower speeds, and unless the load is exceptionally high, the drivetrain can supply as much torque as necessary, so the limiting factor becomes the traction under each wheel. It is therefore convenient to define traction as the amount of torque that can be generated between the tire and the road surface, before the wheel starts to slip. If the torque applied to one of the drive wheels exceeds the threshold of traction, then that wheel will spin, and thus only provide torque at each other driven wheel limited by the sliding friction at the slipping wheel. The reduced nett traction may still be enough to propel the vehicle.

A conventional "open" (non-locked or otherwise traction-aided) differential always supplies close to equal (because of internal friction) torque to each side.[4] To illustrate how this can limit torque applied to the driving wheels, imagine a simple rear-wheel drive vehicle, with one rear roadwheel on asphalt with good grip, and the other on a patch of slippery ice. It takes very little torque to spin the side on slippery ice, and because a differential splits torque equally to each side, the torque that is applied to the side that is on asphalt is limited to this amount.[5][6]

Based on the load, gradient, et cetera, the vehicle requires a certain amount of torque applied to the drive wheels to move forward. Since an open differential limits total torque applied to both drive wheels to the amount utilized by the lower traction wheel multiplied by a factor of 2, when one wheel is on a slippery surface, the total torque applied to the driving wheels may be lower than the minimum torque required for vehicle propulsion.[4]

A proposed way to distribute the power to the wheels, is to use the concept of gearless differential, of which a review has been reported by Provatidis,[7] but the various configurations seem to correspond either to the "sliding pins and cams" type, such as the ZF B-70 available for early VWs, or are a variation of the ball differential.

Many newer vehicles feature traction control, which partially mitigates the poor traction characteristics of an open differential by using the anti-lock braking system to limit or stop the slippage of the low traction wheel, increasing the torque that can be applied to both wheels. While not as effective in propelling a vehicle under poor traction conditions as a traction-aided differential, it is better than a simple mechanical open differential with no electronic traction assistance.

## Traction-aiding devices

ARB, air-locking differential
A cutaway drawing of a car's rear axle, showing the crown wheel and pinion of the final drive, and the smaller differential gears
A cutaway view of an automotive final drive unit which contains the differential

There are various devices for getting more usable traction from vehicles with differentials.

• One solution is the Positive Traction (Posi), the most well-known of which is the clutch-type. With this differential, the sun gears are coupled to the carrier via a multi-disc clutch which allows extra torque to be sent to the wheel with higher resistance than available at the other driven road wheel when the limit of friction is reached at that other wheel. Below the limit of friction more torque goes to the slower (inside) wheel.
• A limited slip differential (LSD) or anti-spin is another type of traction aiding device that uses a mechanical system that activates under centrifugal force to positively lock the left and right spider gears together when one wheel spins a certain amount faster than the other. This type behaves as an open differential unless one wheel begins to spin and exceeds that threshold. While positraction units can be of varying strength, some of them with high enough friction to cause an inside tire to spin or outside tire to drag in turns like a spooled differential, the LSD will remain open unless enough torque is applied to cause one wheel to lose traction and spin, at which point it will engage. A LSD can use clutches like a posi when engaged, or may also be a solid mechanical connection like a locker or spool. It is called limited slip because it does just that; it limits the amount that one wheel can "slip" (spin).
• A locking differential, such as ones using differential gears in normal use but using air or electrically controlled mechanical system, which when locked allow no difference in speed between the two wheels on the axle. They employ a mechanism for allowing the axles to be locked relative to each other, causing both wheels to turn at the same speed regardless of which has more traction; this is equivalent to effectively bypassing the differential gears entirely. Other locking systems may not even use differential gears but instead drive one wheel or both depending on torque value and direction. Automatic mechanical lockers do allow for some differentiation under certain load conditions, while a selectable locker typically couples both axles with a solid mechanical connection like a spool when engaged.
• A high-friction 'Automatic Torque Biasing' (ATB) differential, such as the Torsen differential, where the friction is between the gear teeth rather than at added clutches. This applies more torque to the driven roadwheel with highest resistance (grip or traction) than is available at the other driven roadwheel when the limit of friction is reached at that other wheel. When tested with the wheels off the ground, if one wheel is rotated with the differential case held, the other wheel will still rotate in the opposite direction as for an open differential but there will be some frictional losses and the torque will be distributed at other than 50/50. Although marketed as being "torque-sensing", it functions the same as a limited-slip differential. 3D Animation of a Torsen Differential
• A very high-friction differential, such as the ZF "sliding pins and cams" type, so that there is locking from very high internal friction. When tested with the wheels off the ground with torque applied to one wheel it will lock, but it is still possible for the differential action to occur in use, albeit with considerable frictional losses, and with the road loads at each wheel in opposite directions rather than the same (acting with a "locking and releasing" action rather than a distributed torque).
• Electronic traction control systems usually use the anti-lock braking system (ABS) roadwheel speed sensors to detect a spinning roadwheel, and apply the brake to that wheel. This progressively raises the reaction torque at that roadwheel, and the differential compensates by transmitting more torque through the other roadwheel—the one with better traction. In Volkswagen Group vehicles, this specific function is called 'Electronic Differential Lock' (EDL).
• A spool is just what it sounds like. It may replace the spider gears within the differential carrier, or the entire carrier. A spool locks both axle shafts together 100% for maximum traction. This is typically only used in drag racing applications, where the vehicle is to be driven in a straight line while applying tremendous torque to both wheels.
• In a four-wheel drive vehicle, a viscous coupling unit can replace a centre differential entirely, or be used to limit slip in a conventional 'open' differential. It works on the principle of allowing the two output shafts to counter-rotate relative to each other, by way of a system of slotted plates that operate within a viscous fluid, often silicone. The fluid allows slow relative movements of the shafts, such as those caused by cornering, but will strongly resist high-speed movements, such as those caused by a single wheel spinning. This system is similar to a limited slip differential.

A four-wheel drive (4WD) vehicle will have at least two differentials (one in each axle for each pair of driven roadwheels), and possibly a centre differential to apportion torque between the front and rear axles. In some cases (e.g. Lancia Delta Integrale, Porsche 964 Carrera 4 of 1989[8]) the centre differential is an epicyclic differential (see below) to divide the torque asymmetrically, but at a fixed rate between the front and rear axle. Other methods utilise an 'Automatic Torque Biasing' (ATB) centre differential, such as a Torsen—which is what Audi use in their quattro cars (with longitudinal engines).

4WD vehicles without a centre differential should not be driven on dry, paved roads in four-wheel drive mode, as small differences in rotational speed between the front and rear wheels cause a torque to be applied across the transmission. This phenomenon is known as "wind-up", and can cause considerable damage to the transmission or drive train. On loose surfaces these differences are absorbed by the tire slippage on the road surface.

A transfer case may also incorporate a centre differential, allowing the drive shafts to spin at different speeds. This permits the four-wheel drive vehicle to drive on paved surfaces without experiencing "wind-up".

## Epicyclic differential

Epicyclic gearing is used here to apportion torque asymmetrically. The input shaft is the green hollow one, the yellow is the low torque output, and the pink is the high torque output. The force applied in the yellow and the pink gears is the same, but since the arm of the pink one is 2× to 3× as big, the torque will be 2× to 3× as high.

An epicyclic differential uses epicyclic gearing to split and apportion torque asymmetrically between the front and rear axles. An epicyclic differential is at the heart of the Toyota Prius automotive drive train, where it interconnects the engine, motor-generators, and the drive wheels (which have a second differential for splitting torque as usual). It has the advantage of being relatively compact along the length of its axis (that is, the sun gear shaft).

Epicyclic gears are also called planetary gears because the axes of the planet gears revolve around the common axis of the sun and ring gears that they mesh with and roll between. In the image, the yellow shaft carries the sun gear which is almost hidden. The blue gears are called planet gears and the pink gear is the ring gear or annulus.

## Spur-gear differential

This is another type of differential that was used in some early automobiles, more recently the Oldsmobile Toronado, as well as other non-automotive applications. It consists of spur gears only.

A spur-gear differential has two equal-sized spur gears, one for each half-shaft, with a space between them. Instead of the Bevel gear, also known as a miter gear, assembly (the "spider") at the centre of the differential, there is a rotating carrier on the same axis as the two shafts. Torque from a prime mover or transmission, such as the drive shaft of a car, rotates this carrier.

Mounted in this carrier are one or more pairs of identical pinions, generally longer than their diameters, and typically smaller than the spur gears on the individual half-shafts. Each pinion pair rotates freely on pins supported by the carrier. Furthermore, the pinions pairs are displaced axially, such that they mesh only for the part of their length between the two spur gears, and rotate in opposite directions. The remaining length of a given pinion meshes with the nearer spur gear on its axle. Therefore, each pinion couples that spur gear to the other pinion, and in turn, the other spur gear, so that when the drive shaft rotates the carrier, its relationship to the gears for the individual wheel axles is the same as that in a bevel-gear differential.

## Non-automotive applications

The oldest known example of a differential was once thought to be in the Antikythera mechanism. It was supposed to have used such a train to produce the difference between two inputs, one input related to the position of the sun on the zodiac, and the other input related to the position of the moon on the zodiac; the output of the differential gave a quantity related to the moon's phase. It has now been proven that the assumption of the existence of a differential gearing arrangement was incorrect.[9][original research?]

Chinese south-pointing chariots may also have been very early applications of differentials. The chariot had a pointer which constantly pointed to the south, no matter how the chariot turned as it travelled. It could therefore be used as a type of compass. It is widely thought that a differential mechanism responded to any difference between the speeds of rotation of the two wheels of the chariot, and turned the pointer appropriately. However, the mechanism was not precise enough, and, after a few miles of travel, the dial could have very well been pointing in the complete opposite direction.

The earliest definitely verified use of a differential was in a clock made by Joseph Williamson in 1720. It employed a differential to add the Equation of Time to local mean time, as determined by the clock mechanism, to produce solar time, which would have been the same as the reading of a sundial. During the 18th Century, sundials were considered to show the "correct" time, so an ordinary clock would frequently have to be readjusted, even if it worked perfectly, because of seasonal variations in the Equation of Time. Williamson's and other equation clocks showed sundial time without needing readjustment. Nowadays, we consider clocks to be "correct" and sundials usually incorrect, so many sundials carry instructions about how to use their readings to obtain clock time.

In the first half of the twentieth century, mechanical analog computers, called differential analyzers, were constructed that used differential gear trains to perform addition and subtraction. The U.S. Navy Mk.1 gun fire control computer used about 160 differentials of the bevel-gear type.

A differential gear train can be used to allow a difference between two input axles. Mills often used such gears to apply torque in the required axis. Differentials are also used in this way in watchmaking to link two separate regulating systems with the aim of averaging out errors. Greubel Forsey use a differential to link two double tourbillon systems in their Quadruple Differential Tourbillon.

## Active differentials

A relatively new technology is the electronically-controlled 'active differential'. An electronic control unit (ECU) uses inputs from multiple sensors, including yaw rate, steering input angle, and lateral acceleration—and adjusts the distribution of torque to compensate for undesirable handling behaviours like understeer. Active differentials used to play a large role in the World Rally Championship, but in the 2006 season the FIA has limited the use of active differentials only to those drivers who have not competed in the World Rally Championship in the last five years.

Fully integrated active differentials are used on the Ferrari F430, Mitsubishi Lancer Evolution, and on the rear wheels in the Acura RL. A version manufactured by ZF is also being offered on the latest Audi S4 and Audi A4.[10]

The second constraint of the differential is passive—it is actuated by the friction kinematics chain through the ground. The difference in torque on the roadwheels and tires (caused by turns or bumpy ground) drives the second degree of freedom, (overcoming the torque of inner friction) to equalise the driving torque on the tires. The sensitivity of the differential depends on the inner friction through the second degree of freedom. All of the differentials (so called “active” and “passive”) use clutches and brakes for restricting the second degree of freedom, so all suffer from the same disadvantage—decreased sensitivity to a dynamically changing environment. The sensitivity of the ECU controlled differential is also limited by the time delay caused by sensors and the response time of the actuators.

## Automobiles without differentials

Although the vast majority of automobiles in the developed world use differentials, there are a few that do not. Several different types exist:

• Vehicles with a single driving wheel. Besides motorcycles, which are generally not classified as automobiles, this group includes most three-wheeled cars. These were quite common in Europe in the mid-20th Century, but have now become rare there. They are still common in some areas of the developing world, such as India. Some early four-wheeled cars also had only one driving wheel to avoid the need for a differential. However, this arrangement led to many problems. The system was unbalanced, the driving wheel would easily spin, etc.. Because of these problems, few such vehicles were made.
• Vehicles using two freewheels. A freewheel, as used on a pedal bicycle for example, allows a road wheel to rotate faster than the mechanism that drives it, allowing a cyclist to stop pedalling while going downhill. Some early automobiles had the engine driving two freewheels, one for each driving road wheel. When the vehicle turned, the engine would continue to drive the wheel on the inside of the curve, but the wheel on the outside was permitted to rotate faster by its freewheel. Thus, while turning, the vehicle had only one driving wheel. Driving in reverse is also impossible as is engine braking due to the freewheels.
• Vehicles with continuously variable transmissions, such as the DAF Daffodil. The Daffodil, and other similar vehicles which were made until the 1970s by the Dutch company DAF, had a type of transmission that used an arrangement of belts and pulleys to provide an infinite number of gear ratios. The engine drove two separate transmissions which ran the two driving wheels. When the vehicle turned, the two wheels could rotate at different speeds, making the two transmissions shift to different gear ratios, thus functionally substituting for a differential. The slower moving wheel received more driving torque than the faster one, so the system had limited-slip characteristics. The duplication also provided redundancy. If one belt broke, the vehicle could still be driven.
• Light vehicles with closely spaced rear wheels, such as the Isetta and Opperman Unicar, or very low mass vehicles.
• Vehicles with separate motors for the driving wheels. Electric cars can have a separate motor for each driving wheel, eliminating the need for a differential, but usually with some form of gearing at each motor to get the large wheel torques necessary. Hybrid vehicles in which the final drive is electric can be configured similarly.

## References and footnotes

1. ^ Earlier uses of differentials have been postulated, but not proved. See Antikythera mechanism and South-pointing chariot.
2. ^ Military uses may still exist. See Electromagnetic pulse.
3. ^ "History of the Automobile". Gmcanada.com. Retrieved 2011-01-09.
4. ^ a b Chocholek, S. E. (1988) "The development of a differential for the improvement of traction control"
5. ^ Bonnick, Allan. (2001) "Automotive Computer Controlled Systems p. 22
6. ^ Bonnick, Allan. (2008). "Automotive Science and Mathematics p. 123
7. ^ Provatidis, Christopher, G. (2003). "A critical presentation of Tsiriggakis’ gearless differential". Mobility & Vehicles Mechanics 29 (4): 25–46; also: http://users.ntua.gr/cprovat/index_en.htm
8. ^ "The Complete Story of Porsche 911". Autozine.org. Retrieved 2011-01-09.
9. ^ Wright, M T. (2005). "The Antikythera Mechanism and the early history of the Moon Phase Display". Antiquarian Horology 29 (3 (March 2006)): 319–329.
10. ^ "ZF Press release". Zf.com. Retrieved 2011-01-09.

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