The gear train in an automobile differential is a common application of gears, but often misunderstood by the lay public. Here we present a simplified explanation of how and why an automobile differential works.
The car is turning about a circle with nominal radius r_{n}. (For this discussion, we assume that the axis of the wheel axle for the driven (rear) wheels passes through the turn circle center. This is typically true only for a fairly large radius of turn.)
The outer wheel traverses an arc with radius r_{o} and the inner wheel traverses an arc with radius r_{i}. As illustrated, the lengths of the arcs traversed are s_{o}, s_{n}, and s_{i}. The outer arc s_{o} is obviously larger than the inner arc s_{i} for a given traversed angle theta. Some way of ensuring that the outer wheel is able to turn slightly faster than the inner wheel must be ensured in order to prevent binding and slippage of the tires on the road. For nondriven wheels which simply rotate freely independently of other machinery, this is not a problem. Driven wheels connected to the engine via the driveshaft, however, must both be turned by gearing and this gear train must allow for differential movement of the left wheel with respect to the right wheel. This is a difficult problem since for every turning circle the differential rotation of the left and right wheels is different. Fortunately, the automobile differential solves this problem with only one transmission and one drive shaft for both driven wheels.
Since s=r(THETA), the length of the arc traversed for a given theta is proportional to the radius. Since r_{o} is greater than r_{n} by the same amount that r_{i} is less than r_{n}, the right wheel center must travel further than the car center by the same amount that the left wheel center must travel less further than the car center. As its name implies, a differential allows the left and right drive wheels to turn differentially with respect to each other. As can be seen by turning the drive wheels of a car on a mechanic's lift, turning one drive wheel results in the opposite wheel turning at the same rate in the opposite direction.
