formulasearchengine - User contributions [en]

Dow Jones Industrial Average

2014-02-01T03:41:55Z

14.96.101.186: /* Components */

: ''For other uses of "atlas", see [[Atlas (disambiguation)]].''

In [[mathematics]], particularly [[topology]], one describes
a [[manifold]] using an '''atlas'''. An atlas consists of individual
''charts'' that, roughly speaking, describe individual regions
of the manifold. If the manifold is the surface of the Earth,
then an atlas has its more common meaning. In general,
the notion of atlas underlies the formal definition of a [[manifold]].

==Charts==

The definition of an atlas depends on the notion of a ''chart''.
A '''chart''' for a [[topological space]] ''M'' is a [[homeomorphism]] <math>\varphi</math> from an [[open set|open subset]] ''U'' of ''M'' to an open subset of [[Euclidean space]]. The chart is traditionally recorded as the ordered pair <math> (U, \varphi)</math>.

==Formal definition of atlas==

An '''atlas''' for a [[topological space]] ''M'' is a collection <math> \{(U_{\alpha}, \varphi_{\alpha})\}</math> of charts on ''M'' such that
<math> \bigcup U_{\alpha} = M</math>. If the codomain of each chart is the ''n''-dimensional [[Euclidean space]] and the atlas is connected, then ''M'' is said to be an ''n''-dimensional [[manifold]].

==Transition maps==

{{ Annotated image | caption=Two charts on a manifold
| image=Two coordinate charts on a manifold.svg
| image-width = 250
| annotations =
{{Annotation|45|70|<math>M</math>}}
{{Annotation|67|54|<math>U_\alpha</math>}}
{{Annotation|187|66|<math>U_\beta</math>}}
{{Annotation|42|100|<math>\varphi_\alpha</math>}}
{{Annotation|183|117|<math>\varphi_\beta</math>}}
{{Annotation|87|109|<math>\tau_{\alpha,\beta}</math>}}
{{Annotation|90|145|<math>\tau_{\beta,\alpha}</math>}}
{{Annotation|55|183|<math>\mathbf R^n</math>}}
{{Annotation|145|183|<math>\mathbf R^n</math>}}
}}
A transition map provides a way of comparing two charts of an atlas.
To make this comparison, we consider the composition of one chart
with the inverse of the other. This composition is not well-defined
unless we restrict both charts to the intersection of their domains
of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)

To be more precise, suppose that <math>(U_{\alpha}, \varphi_{\alpha})</math> and <math>(U_{\beta}, \varphi_{\beta})</math> are two charts for a manifold ''M'' such that <math>U_{\alpha} \cap U_{\beta}</math> is non-empty.
The '''transition map''' <math> \tau_{\alpha,\beta}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})</math> is the map defined by

: <math>\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.</math>

Note that since <math>\varphi_{\alpha}</math> and <math>\varphi_{\beta}</math> are both homeomorphisms, the transition map <math> \tau_{\alpha, \beta}</math> is also a homeomorphism.

==More structure==

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of [[differentiation (mathematics)|differentiation]] of functions on a manifold, then it is necessary to construct an atlas whose transition functions are [[differentiable]]. Such a manifold is called [[Differentiable manifold|differentiable]]. Given a differentiable manifold, one can unambiguously define the notion of [[tangent vectors]] and then [[directional derivative]]s.

If each transition function
is a [[smooth map]], then the atlas is called a
[[smooth structure|smooth atlas]], and the manifold itself is called [[Differentiable manifold#Definition|smooth]].
Alternatively, one could require that the transition maps
have only ''k'' continuous derivatives in which case the atlas is
said to be <math> C^k </math>.

Very generally, if each transition function
belongs to a [[pseudo-group]] <math> {\mathcal G} </math>
of [[homeomorphism]]s of [[Euclidean space]],
then the atlas is called a <math> {\mathcal G}</math>-atlas.

==References==
{{reflist}}
{{refbegin}}
*{{cite book | first = John M. | last = Lee | year = 2006 | title = Introduction to Smooth Manifolds | publisher = Springer-Verlag | isbn = 978-0-387-95448-6}}
*{{cite book | first = Mark R. | last = Sepanski | year = 2007 | title = Compact Lie Groups | publisher = Springer-Verlag | isbn = 978-0-387-30263-8}}
{{refend}}

==External links==
*[http://mathworld.wolfram.com/Atlas.html Atlas] by Rowland, Todd

[[Category:Differential topology]]

Schwarzschild radius

2014-01-01T14:51:15Z

14.96.92.207:

A '''differential''' is a particular type of simple [[planetary gear train]] that has the property that the angular velocity of its carrier is the average of the angular velocities of its sun and annular gears. This is accomplished by packaging the gear train so it has a [[epicyclic gearing#Fixed Carrier Train Ratio|fixed carrier train ratio]] ''R = -1'', which means the gears corresponding to the sun and annular gears are the same size. This can be done by engaging the planet gears of two identical and coaxial [[epicyclic gearing|epicyclic gear train]]s to form a ''spur gear differential''. Another approach is to use [[bevel gear]]s for the sun and annular gears and a bevel gear as the planet, which is known as a ''bevel gear differential''.

[[File:Spur gear differential (Manual of Driving and Maintenance).jpg||thumb|right|200px|A spur gear differential constructed by engaging the planet gears of two co-axial epicyclic gear trains. The casing is the carrier for this planetary gear train.]]

==Overview==
[[File:Differential_(Manual_of_Driving_and_Maintenance).jpg|thumb|upright=1|Automotive differential: The drive gear 2 is mounted on the carrier 5 which supports the planetary bevel gears 4 which engage the driven bevel gears 3 attached to the axles 1.]]
[[File:BAUMA 2004 ZF Differentialgetriebe.jpg|thumb|[[ZF Friedrichshafen AG|ZF]] Differential. The drive shaft enters from the front and the driven axles run left and right.]]
[[File:13-04-05-Skoda Museum Mladá Boleslav by RalfR-009.jpg|thumb|Car differential of a [[Škoda 422]].]]

In [[automobile]]s and other wheeled vehicles, a differential couples the drive shaft to half-shafts that connect to the rear driving wheels. The differential gearing allows the outer drive wheel to rotate faster than the inner drive wheel during a turn. 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. Average of the rotational speed of the two driving wheel equals the input rotational speed of the drive shaft. An increase in the speed of one wheel is balanced by a decrease in the speed of the other.

A differential consists of one input, the drive shaft, and two outputs which are the two drive wheels, however the rotation of the drive wheels are coupled by their connection to the roadway. Under normal conditions, with small tyre slip, the ratio of the speeds of the two driving wheels is defined by the ratio of the radii of the paths around which the two wheels are rolling, which in turn is determined by the track-width of the vehicle (the distance between the driving wheels) and the radius of the turn.

Non-automotive uses of differentials include performing [[Analog signal|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.<ref>Earlier uses of differentials have been postulated, but not proved. See [[Antikythera mechanism]] and [[South-pointing chariot]].</ref> In the 20th Century, large assemblies of many differentials were used as [[analog computer]]s, 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.<ref>Military uses may still exist. See [[Electromagnetic pulse]].</ref> Practically all the differentials that are now made are used in automobiles and similar vehicles.

==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:

*1050 BC–771 BC: The ''[[Book of Song]]'' (which itself was written between 502 and 557 A.D.) makes the assertion that the [[South Pointing Chariot]], which may have used a differential gear, was invented during the [[Western Zhou Dynasty]] in China.{{Citation needed|date=January 2012}}
*150 - 100 BC: The [[Antikythera mechanism]] has been dated to this period. It was discovered in 1902 on a shipwreck by [[Sponge diving|sponge divers]], and modern research suggests that it was designed to predict solar eclipses using differential gears.<ref>{{cite web|title = Discovering How Greeks Computed in 100 B.C. |first=John |last=Noble Wilford|work = [[The New York Times]]|url = http://www.nytimes.com/2008/07/31/science/31computer.html?_r=0|date=July 31, 2008|accessdate=2013-12-25}}</ref>
*30 BC - 20 BC: Differential gear systems possibly used in [[China]]
[[File:South-pointing chariot (Science Museum model).jpg|thumb|upright=0.75|[[South Pointing Chariot]] model]]
*227–239 AD: Despite doubts from fellow ministers at court, [[Ma Jun]] from the [[Kingdom of Wei]] in [[China]] invents the first historically verifiable [[South Pointing Chariot]], which provided [[cardinal direction]] as a non-[[magnetic]], mechanized [[compass]]. Some such chariots may have used differential gears.
*658, 666 AD: two Chinese Buddhist monks and engineers create South Pointing Chariots for [[Emperor Tenji]] of Japan.
*1027, 1107 AD: Documented Chinese reproductions of the South Pointing Chariot by Yan Su and then Wu Deren, which described in detail the mechanical functions and gear ratios of the device much more so than earlier Chinese records.
*1720: Joseph Williamson uses a differential gear in a clock.
*1810: [[Rudolph Ackermann]] of Germany invents a four-wheel steering system for carriages, which some later writers mistakenly report as a differential.
*1827: modern automotive differential patented by watchmaker [[Onésiphore Pecqueur]] (1792–1852) of the ''[[Conservatoire des Arts et Métiers]]'' in [[France]] for use on a [[steam cart]]. (Sources: Britannica Online and<ref>{{cite web|url=http://www.gmcanada.com/inm/gmcanada/english/about/OverviewHist/hist_auto.html |title=History of the Automobile |publisher=Gmcanada.com |date= |accessdate=2011-01-09}}</ref>)
*1832: [[Richard Roberts (engineer)|Richard Roberts]] of England patents 'gear of compensation', a differential for [[Traction engine#Road locomotive|road locomotive]]s.
*1874: [[Aveling and Porter]] of [[Rochester, Kent]] list a crane locomotive in their catalogue fitted with their patent differential gear on the rear axle.<ref>{{citation | last = Preston | first = J.M. | title = Aveling & Porter, Ltd. Rochester. | publisher = North Kent Books | year = 1987 | isbn = 0-948305-03-7 | pages = 13–14}} includes sectional drawing.</ref>
*1876: [[James Starley]] of [[Coventry]] invents chain-drive differential for use on [[bicycle]]s; invention later used on automobiles by [[Karl Benz]].
*1897: first use of differential on an Australian [[steam car]] by [[David Shearer (engineer)|David Shearer]].
*1958: [[Vernon Gleasman]] patents the [[Torsen]] dual-drive differential, a type of [[limited slip differential]] that relies solely on the action of gearing instead of a combination of clutches and gears.

==Epicyclic differential==
[[Image:Epicyclic gear ratios.png|thumb|280px|[[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 can use [[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 gear]]s 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 [[wiktionary:prime mover|prime mover]] or [[transmission (mechanics)|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 pinion 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.

A spur gear differential is constructed from two identical coaxial epicyclic gear trains assembled with a single carrier such that their planet gears are engaged. This forms a [[planetary gear train]] with a fixed carrier train ratio ''R = -1''.

In this case, the fundamental formula for the planetary gear train yields,
:<math>\frac{\omega_s-\omega_c}{\omega_a-\omega_c}=-1,</math>
or
:<math> \omega_c = \frac{1}{2}(\omega_s + \omega_a).</math>
Thus, the angular velocity of the carrier of a spur gear differential is the average of the angular velocities of the sun and annular gears.<ref>J. J. Uicker, G. R. Pennock and J. E. Shigley, 2003, '''Theory of Machines and Mechanisms,''' Oxford University Press, New York.</ref>

In discussing the spur gear differential, the use of the term ''annular gear'' is a convenient way to distinguish the sun gears of the two epicyclic gear trains. The second sun gear serves the same purpose as the annular gear of a simple planetary gear train, but clearly does not have the internal gear mate that is typical of an annular gear.

==Non-automotive applications==
Chinese [[south-pointing chariot]]s 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 clock]]s 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 computer]]s, called [[differential analyzer]]s, 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. [[Gristmill|Mill]]s 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.

==Application to vehicles==
A vehicle with two drive wheels has the problem that when it turns a corner the drive wheels must rotate at different speeds to maintain traction. The automotive differential is designed to drive a pair of wheels while allowing them to rotate at different speeds. In vehicles without a differential, such as [[kart racing|kart]]s, 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 travels a shorter distance than the outer wheel, so without a differential either the inner wheel rotates too fast or the outer wheel drags, which results in difficult and unpredictable handling, damage to [[tire]]s and roads, and strain on (or possible failure of) the entire [[powertrain|drivetrain]].

In rear-wheel drive automobiles the central drive shaft engages the differential through a [[hypoid gear]] mounted on the carrier of the planetary chain that forms the differential. This hypoid gear is a bevel gear that changes the direction of the drive rotation.
[[File:Sprocket35b.jpg|thumb|upright=.75|Hypoid gear pair that connects an automotive drive shaft to a differential.]]

==Functional description==
[[Image:Differential free.png|thumb|right|250px|Input torque is applied to the ring gear (blue), which turns the entire carrier (blue). The carrier is connected to both 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.]]
[[Image:Differential locked-2.png|thumb|right|250px|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 using bevel gears (these are not strictly necessary - see [[#Spur-gear differential|spur-gear differential]]):

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.

==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 (mechanics)|transmission]] and drive axles applying a twisting force against the resistance of the [[traction (engineering)|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 limited internal friction) torque to each side.<ref name="bare_url">Chocholek, S. E. (1988) [http://zhome.com/ZCMnL/tech/Torsen/Torsen.htm "The development of a differential for the improvement of traction control"]</ref> 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.<ref>Bonnick, Allan. (2001) [http://books.google.com/books?id=odUZ0O_OWZwC&lpg=PA22&pg=PA22#v=onepage&q&f=false "Automotive Computer Controlled Systems] p. 22</ref><ref>Bonnick, Allan. (2008). [http://books.google.com/books?id=78HoZb-ohbIC&pg=PA123&lpg=PA123&source=bl&ots=l0V7lmn3sI&sig=t-KtW3Pz5T5Lbe5X8xNW2-7W_LU&hl=en&ei=2CarTf7PLIKCgAeGz7WpBg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CC8Q6AEwBTgK#v=onepage&q&f=false "Automotive Science and Mathematics] p. 123</ref>

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 used 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.<ref name="bare_url" />

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,<ref>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</ref> but the various configurations seem to correspond either to the "sliding pins and cams" type, such as the [[ZF Friedrichshafen|ZF]] B-70 available on early VWs, or are a variation of the [[ball differential]].

Many newer vehicles feature [[traction control system|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==
{{Section OR|date=July 2009}}
[[Image:ARB Air Locking Differential (RLH).JPG|thumb|right|200px|ARB, air-locking differential]]
[[Image:Transmission diagram.JPG|thumb|right|200px|A [[cutaway drawing]] of a car's rear [[axle]], showing the [[Gear#Crown|crown wheel]] and [[pinion]] of the final drive, and the smaller differential gears]]
[[Image:Differentialgetriebe2.jpg|thumb|right|200px|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. [http://www.youtube.com/watch?v=Z9iPqIQ_8iM 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 system]]s 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<ref>{{cite web|url=http://www.autozine.org/911/911_9.htm |title=The Complete Story of Porsche 911 |publisher=Autozine.org |date= |accessdate=2011-01-09}}</ref>) the centre differential is an [[epicyclic gearing|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 (four wheel drive system)|quattro]] cars (with [[longitudinal engine]]s).

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 (mechanics)|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".

==Active differentials==
A relatively new technology is the electronically-controlled 'active differential'. An [[electronic control unit]] (ECU) uses inputs from multiple sensors, including [[yaw angle|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 [[Fédération Internationale de l'Automobile|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 Friedrichshafen|ZF]] is also being offered on the B8 chassis [[Audi S4#B8|Audi S4]] and [[Audi A4]].<ref name="zf">{{cite web|url=http://www.zf.com/corporate/en/press/press_releases/products_press/products_detail.jsp?newsId=21442669 |title=ZF Press release |publisher=Zf.com |date= |accessdate=2011-01-09}}</ref> The [[Golf_GTI#Seventh_generation_.28A7.2FTyp_5G.2C_2013.E2.80.93.29|Volkswagen Golf GTI Mk7]] in Performance trim also has an electronically controlled front-axle transverse differential lock, also known as VAQ.<ref name="golf_gti">{{cite web|url=http://www.pistonheads.com/news/default.asp?storyId=25303 |title=Golf VII GTI |publisher=pistonheads.com |date= |accessdate=2013-06-24}}</ref>

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 [[degrees of freedom (engineering)|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 most 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 [[freewheel]]s. 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 transmission]]s, such as the [[DAF Daffodil]]. The Daffodil, and other similar vehicles which were made until the 1970s by the Dutch company [[DAF trucks#Car business|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_car#Acceleration_and_drivetrain_design|Electric car]]s 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.

==See also==
*[[Ball differential]]
*[[Limited slip differential]]
*[[Locking differential]]
*[[Whippletree (mechanism)]], which evenly divides linear force as a differential divides torque.
*[[Hermann Aron#Electricity meters|Aron's electricity meter]], an early [[electricity meter]], relying on the use of a mechanical differential.
*[[Equation clock]]. One design uses a differential to add mean (clock) time and the equation of time to get solar (sundial) time.
*[[Torque Vectoring]]

==References and footnotes==
{{reflist}}

==External links==
{{Commons category|Automobile differentials}}
* [http://www.youtube.com/watch?v=vBm-SzO3ggE A video of a 3D model of an open differential]
* [http://www.drivingfast.net/technology/Differentials.htm An article explaining differentials with illustrations and video]
* [http://www.archive.org/details/Aroundth1937 "Around the Corner"] (1937), a [[Jam Handy]] film made for [[Chevrolet]] explaining very clearly how an open differential works.
* [http://books.google.com/books?id=kiEDAAAAMBAJ&pg=PA76&dq=popular+science+September+1941&hl=en&ei=8PqSTPfJF8aMnQeXkv2MCA&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEMQ6AEwBQ#v=onepage&q=popular%20science%20September%201941&f=true ''Popular Science'', May 1946, ''How Your Car Turns Corners''], a large article with numerous illustrations on how differentials work.

{{Gears}}
{{Powertrain}}

{{DEFAULTSORT:Differential (Mechanical Device)}}
[[Category:Automotive transmission technologies]]
[[Category:Mechanisms]]
[[Category:Vehicle technology]]
[[Category:Auto parts]]

Transmission delay

2013-10-21T17:37:55Z

14.96.38.90:

{{Japanese name|Takebe}}

{{nihongo|'''Takebe Katahiro'''|建部賢弘||1664 - August 24, 1739}}, also known as '''Takebe Kenkō''', was a [[Japan]]ese [[mathematician]] in the [[Edo period]].<ref name="smith146">Smith, David. (1914). {{Google books|J1YNAAAAYAAJ|''A History of Japanese Mathematics,'' p. 146. |page=146}}</ref>

==Biography==
Takebe was the favorite student of [[Seki Takakazu]]<ref name="smith146"/> Takebe is considered to have extended and disseminated Seki's work.<ref>[http://www.britannica.com/eb/article-9384143/Takebe-Katahiro "Takebe Katahiro," ''Encyclopædia Britannica'' online.]</ref>

In 1706, Takebe was offered a position in the [[Tokugawa shogunate]]'s department of ceremonies.<ref name="smith146"/>

In 1719, Takebe's new map of Japan was completed; and the work was highly valued for its quality and detail.<ref name="smith146"/>

[[Shogun]] [[Tokugawa Yoshimune|Yoshimune]] honored Takebe with rank and successively better positions in the shogunate.<ref>Jochi, Shigeru. (1997). "Takebe Katahiro," {{Google books|raKRY3KQspsC&dq|''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures,'' p. 932. |page=932}}</ref>

==Legacy==
Takebe played critical role in the development of the [[Enri]] ({{lang|ja-Hani|円理}}, "circle principle") - a crude analogon to the western [[calculus]]. He also created charts for trigonometric functions.<ref name="msj_takebe">[http://mathsoc.jp Mathematical Society of Japan], [http://mathsoc.jp/en/pamph/current/takebe_pr.html Takebe Prize]</ref>

He obtained power series expansion of (arcsin(x))^2 in 1722, 15 years earlier than Euler.
This was the first power series expansion obtained in Wasan. This result was first conjectured by heavy numeric computation.

He used [[Richardson extrapolation]], about 200 years earlier than Richardson.

He also computated 41 digits of <math>\pi </math>, based on polygon approximation and Richardson extrapolation.

===Takebe Prizes===
In the context of its 50th anniversary celebrations, the [[Mathematical Society of Japan]] established the Takebe Prize and the Takebe Prizes for the encouragement of young people who show promise as mathematicians.<ref name="msj_takebe"/>

==Selected works==
In a statistical overview derived from writings by and about Takebe Kenko, [[OCLC]]/[[WorldCat]] encompasses roughly 10+ works in 1o+ publications in 3 languages and 10+ library holdings.<ref>[http://www.oclc.org/research/activities/identities/default.htm WorldCat Identities]: [http://www.worldcat.org/identities/lccn-nr2004-22449 建部賢弘 1664-1739]</ref>
{{dynamic list}}
* 1683 — {{nihongo|''Kenki sanpō''|研幾算法}} [http://www.worldcat.org/search?q=no:%22056510086%22 OCLC 22056510086]
* 1685 — {{nihongo|'' Hatsubi sanpō endan genkai'|發微算法演段諺解}} [http://www.worldcat.org/search?q=no:%22056085721%22 OCLC 22056085721]

==See also==
* [[Sangaku]], the custom of presenting mathematical problems, carved in wood tablets, to the public in [[shinto shrines]]
* [[Soroban]], a Japanese [[abacus]]
* [[Japanese mathematics]] (''[[wasan]]'')
* [[Richardson extrapolation]]

==Notes==
{{reflist|2}}

==References ==
* Endō Toshisada (1896). {{nihongo|''History of mathematics in Japan''|日本數學史史 |Dai Nihon sūgakush}}. Tōkyō: _____. [http://www.worldcat.org/title/dai-nihon-sugakushi-history-of-mathematics-in-japan-by-endo-toshisada/oclc/122770600&referer=brief_results OCLC 122770600]
* Horiuchi, Annick. (1994). [http://books.google.com/books?id=qMnZHUSAYzMC&dq=History+of+Mathematics+in+Japan+1896&lr=lang_ja&as_brr=0&source=gbs_navlinks_s ''Les Mathematiques Japonaises a L'Epoque d'Edo (1600–1868): Une Etude des Travaux de Seki Takakazu (?-1708) et de Takebe Katahiro (1664–1739).''] Paris: Librairie Philosophique J. Vrin. 10-ISBN 2711612139/13-ISBN 9782711612130; [http://www.worldcat.org/title/mathematiques-japonaises-a-lepoque-dedo-1600-1868-une-etude-des-travaux-de-seki-takakazu-1708-et-de-takebe-katahiro-1664-1739/oclc/318334322 OCLC 318334322]
* [[Helaine Selin|Selin, Helaine]], ed. (1997). [http://books.google.com/books?id=raKRY3KQspsC&dq=Aida+Yasuaki&source=gbs_navlinks_s ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures.''] Dordrecht: [[Kluwer]]/[[Springer Science+Business Media|Springer]]. 10-ISBN 0792340663/13-ISBN 9780792340669; [http://www.worldcat.org/title/encyclopaedia-of-the-history-of-science-technology-and-medicine-in-non-western-cultures/oclc/186451909 OCLC 186451909]
* [[David Eugene Smith]] and [[Yoshio Mikami]]. (1914). [http://books.google.com/books?id=J1YNAAAAYAAJ&dq=Shiraishi+Chochu&source=gbs_navlinks_s ''A History of Japanese Mathematics.''] Chicago: Open Court Publishing. [http://www.worldcat.org/title/history-of-japanese-mathematics/oclc/1515528 OCLC 1515528] [http://www.archive.org/details/historyofjapanes00smitiala -- note alternate online, full-text copy at archive.org]
* {{MacTutor Biography|id=Takebe|title=Takebe Katahiro}}

{{Persondata
| NAME = Takebe Kenko
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Mathematician
| DATE OF BIRTH = 1664
| PLACE OF BIRTH =
| DATE OF DEATH = August 24, 1739
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Takebe, Kenko}}
[[Category:Japanese mathematics]]
[[Category:Japanese mathematicians]]
[[Category:17th-century mathematicians]]
[[Category:18th-century mathematicians]]
[[Category:Japanese writers of the Edo period]]
[[Category:18th-century Japanese people]]
[[Category:18th-century cartographers]]

Moment of inertia

2012-08-28T15:08:19Z

14.96.29.145:

== たぶん...年間約今」 ==

？前者は、ややバッフルを見つめて「混乱」 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ腕時計 gps]。同様シャオヤンの言葉を聞いて、シャオゆうはびっくりし、すぐに容疑者に驚いた」。 インストラクター大メイシェン「陰陽」のリン·組織の良い長い間、ちょうどうなずい点灯している場合 シャオヤンはそっと、「色」、厳粛な顔を見て、「それはあまりにもある場合さて、あなたが必要とするどのくらい、残すために長い言葉。私の権利と義務にするだけでなく、あなたが上で反撃するため。 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 電波腕時計カシオ] ' キャロラインはパステル目を光って見えた ペアは、シャオヤンは突然沈黙の後、彼はただ恥ずかしい本物の持っていた、いくつかの赤い肌を感じた： '。たぶん...年間約今」 この爆発を 、テントが急に静かに、道路仰天の目は瞬時に約一年、ティーンエイジャーであることの嘲笑になっていますか [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html カシオ腕時計スタンダード]？この瞬間。誰もが彼の耳には、問題となっている見たことがないされていないままに [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック腕時計]...しかし、これはただではカナンの設立以来、病院で...今年そのようなことを指示してくださいことを考えているようだ、頭思える
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== 「極端な崩壊 ==

低レベルの戦いの技術：グリーン風のスピン拳！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html カシオアナログ腕時計] ' 風音から鋭いブレークと空気中の 拳、巨大な圧力が、シャオヤンは実際に地面に破片の横に、すべてが飛んで持ち上げる [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計メンズ]。 マイクロ斜視、テレビドラマ対向車の怒りは、圧力、シャオヤンの顔「色」の沈黙次第に厳粛な瞬間体が急に向きを変えて戻って押し込んだ後に、壁の上に右足が猛烈に乗って感じる逆推力装置壁の力によって、深い約半インチのフットプリントを残して壁の巨大な運動量は、シャオヤン智玄体が空中で右足が奇妙な力に引き伸ばさ鋼は一般的に困難であるかのようにラジアン、この瞬間に、ソフトな足が、それはそうです [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオソーラー時計]。 「極端な崩壊！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html 時計メンズカシオ] ' タイトな非効率、シャオヤン面畏敬の念、右空中に完成完璧充電近く、最終的に完全なビューであり、嘉Lieaoパンチ、クロスH一緒 [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml]。
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== 'あなたは死の願望を主張するので、私はあなたの心を与える ==

彼らの残留魂の結束、いくつかの方法で、彼らはあなたを作成し、 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html casio 時計] 'シャオ玄は、穏やかな声が微笑んだが、それは巨大な顔がますます歪んでいることを得ることです、に見えますが、非常に厳しい見える。 'あなたは死の願望を主張するので、私はあなたの心を与える [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html 時計 casio]！' 巨大な顔の高騰、激しい魂の中で巨大な口は、一般的にコーンのような非常に巨大な嵐、嵐クレイジースピンを、放出されたことを、シャオ玄激しく離れて暴力「ショット」に対して [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ腕時計電波ソーラー]。 魂墓攻撃、シャオ玄はちょうどペースを行進した日、一歩一歩に直面し、テレビドラマでの巨人の顔は、一見獰猛な魂の嵐に行きましたが、これは体で汚染され、それが偽である脱ぎ履き、彼がどのような上海の一般的な原因としていないようです [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml]。 シャオ玄の体は、巨大な顔の下に妨げられることなく歩いて、彼の体は、突然、に実際にあるだけで、彼の魂のように、その気持ちのように、奇妙なスパークが登場
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== 空厚いHuoyun包ま ==

多くの強さ、ああ、私は彼がすでに高い参照しても、まだ彼は戦争の強力な位相でのケースを戦うために期待していなかったことが物事が中にここに戻っている場合、あなたは、ああ、やっとこの男は、ピーク戦闘力の王である知っているが、家庭、それでもソ連千長老たちは、それが非常に大きな驚きを感じるだろう。 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ腕時計電波ソーラー] 「クリフと劉清林秀談話は、フラストレーションの口連絡を保持し、それは彼らがシャオヤンの難易度を超えて行ってみたい、と拡大して無限のようです。 空厚いHuoyun包ま 」と彼火の波が、わかりませんが、いくつかの戦闘スキルを持つこの男死んでいるか生きている、でも彼は逃れることはできない [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ時計メンズ]。「研究宝石のような紫色の目を見て、彼の歯を粉砕するが、それは彼らの少し顔を心配する少しであってもよいように言った [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml]。 見上げ 林ヤンは、、目は眉をひそめ厚い雲、心は少し緊張している、反対者が直面している男が、本物の戦いの事件の三つの [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ腕時計メンズ] '色'光のかすかな痕跡に目を向ける強いああ
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Squaring the circle

2012-07-15T03:14:36Z

14.96.36.231: /* Impossibility */

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