Differentials are a variety of gearbox, almost always used in one of two ways. In one of these, it receives one input and provides two outputs; this is found in every automobile. In the other, less commonly encountered, it combines two inputs to create an output that is the sum (or difference) of the inputs.
In an automobile and other wheeled vehicles, the differential allows each of the driving wheels to rotate at different speeds, while supplying equal torque to each of them. In automotive applications, the differential and its housing are sometimes collectively called a "pumpkin" (because the housing resembles a pumpkin).
The following description of a differential applies to a "traditional" rear- or front-wheel-drive car or truck: Power is supplied from the engine, via the transmission or gearbox, to a drive shaft (British term: propeller shaft, more commonly abbreviated to "prop-shaft"), which runs to the differential. A spiral bevel pinion gear at the end of the propeller shaft is encased within the differential itself, and it meshes with the large spiral bevel ring gear (British term: crown wheel). (The ring and pinion may mesh in hypoid orientation, not shown.) The ring gear is attached to a carrier, which holds a what is sometimes called a spider, a cluster of four bevel gears in a rectangle, so each bevel gear meshes with two neighbors and rotates counter to the the third, that it faces and does not mesh with. Two of these spider gears are aligned on the same axis as the ring gear and drive the half shafts connected to the vehicle's driven wheels. These are called the side gears. The other two spider gears are aligned on a perpendicular axis which changes orientation with the ring gear's rotation. These two gears are just called pinion gears, not to be confused with the main pinion gear. (Other spider designs employ different numbers of pinion gears depending on durability requirements.) As the carrier rotates, the changing axis orientation of the pinion gears imparts the motion of the ring gear to the motion of the side gears by pushing on them rather than turning against them (that is, the same teeth stay in contact), but because the spider gears are not restricted from turning against each other, within that motion the side gears can counter-rotate relative to the ring gear and to each other under the same force. Thus, for example, if the car is making a turn to the right, the main ring gear 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 side gears 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 ring gear is always the average of the rotations of the side gears. This is why if the wheels are lifted off the ground with the engine off, and the drive shaft is held (preventing the ring gear from turning inside the differential), manually rotating one wheel causes the other 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.
The torque on each wheel is a result of the engine and transmission applying torsion, a twisting force, against the resistance of the traction at that wheel. Unless the load is exceptionally high, the engine and transmission can usually supply as much torque as necessary, so the limiting factor is usually 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 ground before the wheel starts to slip. If the total traction under all the driven wheels exceeds the threshold torque, the vehicle will be driven forward; if not, then one or more wheels will simply spin.
To illustrate how a differential can limit overall torque, imagine a simple rear-wheel-drive vehicle, with one rear wheel on asphalt with good grip, and the other on a patch of slippery ice. With the load, gradient, etc., the vehicle requires, say, 2000 N·m of torque to move forward (i.e. the threshold torque). Let us further assume that the non-spinning traction on the ice equates to 400 N·m, and the asphalt to 3000 N·m.
If the two wheels were driven without a differential, each wheel would push against the ground as hard as possible. The wheel on ice would quickly reach the limit of traction (400 N·m), but would be unable to spin because the other wheel has good traction. The traction of the asphalt plus the small extra traction from the ice exceeds the threshold requirement, so the vehicle will be propelled forward.
With a differential, however, as soon as the "ice wheel" reaches 400 N·m, it will start to spin, and then develop less traction ~300 N·m. The planetary gears inside the differential carrier will start to rotate because the "asphalt wheel" encounters greater resistance. Instead of driving the asphalt wheel with more force, the differential will allow the ice wheel to spin faster, and the asphalt wheel to remain stationary, compensating for the stopped wheel by extra speed of the spinning ice wheel. The torque on both wheels will be the same - limited to the lesser traction of 300 N·m each. Since 600 N·m is less than the required threshold torque of 2000 N·m, the vehicle will not be able to move.
An observer will simply see one stationary wheel and one spinning wheel. It will not be obvious that both wheels are generating the same torque (i.e. both wheels are in fact pushing equally, despite the difference in rotational speed). This has led to a widely held misconception that a vehicle with a differential is really only "one-wheel-drive". In fact, a normal differential always provides equal torque to both driven wheels (unless it is a locking, torque-biasing, or limited slip type).
There are various devices for getting more usable traction from vehicles with differentials.
A four-wheel-drive vehicle will have at least two differentials (one for each pair of wheels) and possibly a center differential to apportion power between the front and rear axles. In many cases (eg. Lancia Delta Integrale, Porsche 964 Carrera 4 of 1989 ) the center differential is an epicyclic differential to divide the torque asymmetrically between the front and rear axle. Vehicles without a center 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 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 center 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".
An epicyclic differential uses epicyclic gears to split 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.
A spur-gear differential has two equal-sized spur gears, one for each half-shaft, with a space between them. Instead of the miter gear assembly (the "spider") at the center of the differential, there is a rotating carrier on the same axis as the two shafts. Power 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. 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.
When the drive shaft rotates the carrier, its relationship to the gears for the individual wheel axles is the same as that in a miter-gear differential.
A differential gear train can also be used to give the difference between two input axles. Mills often used such gears to apply torque in the required axis. It's also used in fine mechanical watches with a hand to show the amount of reserve power in the mainspring.
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.
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 miter gear type.
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 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 computer controlled differential is also limited by the time delay caused by sensors and the response time of the actuators.