Differential (mechanical device)

A differential is a device, usually, but not necessarily, employing gears, which is connected to the outside world by three shafts, chains, or similar, through which it transmits torque and rotation. The gears or other components make the three shafts rotate in such a way that , where , , and  are the angular velocities of the three shafts, and  and  are constants. Often, but not always,  and  are equal, so  is proportional to the sum (or average) of  and . Except in some special-purpose differentials, there are no other limitations on the rotational speeds of the shafts, apart from the usual mechanical/engineering limits. Any of the shafts can be used to input rotation, and the other(s) to output it. 

In automobiles and other wheeled vehicles, a differential is the usual way to allow the driving road wheels 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 . The driving wheels are connected to the other two shafts, and  and  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. (If one wheel is rotating backward, which is possible in very tight turns, its speed should be counted as negative.)

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. Under normal conditions (i.e. only small tyre slip), 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  and/or  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 transmission, 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. 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. Practically all the differentials that are now made are used in automobiles and similar vehicles. This article therefore emphasizes automotive uses of differentials.