MAIN FUNCTIONAL REQUIREMENT(s): Distribute power from car transmission shaft to a pair of Left-Right wheels (1ST FUNCTIONAL REQUIREMENT) while allowing wheels to rotate at different speeds (SECOND FUNCTIONAL REQUIREMENT).
DESIGN PARAMETER: Differential
HISTORY: The differential was first invented in China, in the third century, A.D.
Gear Teeth: The crown wheel and drive pinion teeth are helical, allowing for up-and-down movement in rough or uneven road conditions.
EXPLANATION OF HOW IT WORKS/ IS USED:
Why use a Differential?: When a car turns a corner, one wheel is on the "inside" of a turning arc, and the other wheel is on the "outside." Consequently, the outside wheel has to turn faster than the inside one in order to cover the greater distance in the same amount of time. Thus, because the two wheels are not driven with the same speed, a differential is necessary. A car differential is placed halfway between the driving wheels, on either the front, rear, or both axes (depending on whether its a front-, rear-, or 4-wheel-drive car). In rear-wheel drive cars, the differential converts rotational motion of the transmission shaft which lies parallel to the cars motion to rotational motion of the half-shafts (on the ends of which are the wheels), which lie perpendicular to the cars motion.
How it works: Assuming the wheels do no slip and spin out of control, the following two examples of car motion describe how the differential works when the car is going forward and when it is turning. (see Limited Slip Differential section for wheel slipping).
When the car is traveling straight, both wheels travel at the same speed. Thus, the free-wheeling planet pinions do not spin at all. Instead, as the transmission shaft turns the crown wheel, the rotary motion is translated directly to the half-shafts, and both wheels spin with the angular velocity of the crown wheel (they have the same speed).
When the car is turning, the wheels must move at different speeds. In this situation, the planet pinions spin with respect to the crown wheel as they turn around the sun gears. This allows the speed of the crown gear to be delivered unevenly to the two wheels.
Ratios: The ratio of speeds between gears is dependent upon the ratio of teeth between the two adjoining gears such that
w1 x N1 = w2 x N2,
where w is the respective angular velocity, and N = the number of teeth on the gear.
win = (w1 + w2) / 2
Pin = Pout1 + Pout2, or Pin = (T1 x w1) + (T2 x w2),
where Pin is the power input from the transmission to the differential, and Pout is the power output from the differential to the wheels. T is the torque supplied to each half-shaft, respectively.
Things that might limit or disrupt the behavior of the differential include contact stresses between the gears, which limits the torque transmission, as well as fatigue and losses due to friction between the gears.
If one of the wheels attached to a differential decides to hit some ice, for example, it slips and spins with all of the speed the differential has to distribute. Thus, a locking mechanism, or "limited slip differential" allows one wheel to slip or spin freely while some torque is delivered to the other wheel (hopefully on dry land!).
WHERE TO FIND DIFFERENTIALS:
In the rear axles of most cars and trucks.
Macaulay, David. The Way Things Work, pg. 49
Lecture notes, course 2.72
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