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Automotive Engineering DISCUSSION
I understand that a differential lets the outer wheel turn faster than the inner wheel in a corner. What I do not understand is the stuck-car case: one driven wheel on ice spins freely and the wheel on dry tarmac does nothing, even though the engine is clearly running and in gear.
How does an open differential split speed and torque, and how do limited-slip differentials and brake-based traction control get around the problem?
An open differential obeys two rules. Speed: the carrier speed is the average of the two wheel speeds, so one wheel can speed up exactly as much as the other slows down. Torque: it always sends equal torque to both sides, whatever their speeds. The spider gears act as a balance beam between the two half-shafts.
In a corner the speed rule is what you want. With a 10 m turn radius to the centre of the axle and a 1.5 m track, the inner wheel follows a 9.25 m radius and the outer a 10.75 m radius, so the outer wheel must turn about 16 percent faster, and the differential allows it without tyre scrub.
The equal-torque rule is what strands the car. The torque in the axle can only be as large as the reaction the wheels provide. If the wheel on ice can transmit only 30 N·m before it spins, then 30 N·m is all that builds up on that side, and the differential passes the same 30 N·m to the wheel on tarmac. That is not enough to move the car, so that wheel stands still while the other spins at twice the carrier speed.
The engine is not producing much torque in that state either. It is revving against almost no load, so the power delivered is small because the torque is small.
A limited-slip differential adds friction or another coupling between the two outputs, so that a speed difference is resisted. The gripping wheel can then receive more torque than the slipping one. A torque-sensing type is described by its bias ratio; with a ratio of 3 to 1 the good wheel can get up to three times the torque of the slipping wheel, which in the example would be 90 N·m. That is still limited if one wheel has almost no grip at all.
A locking differential removes the limit by joining both shafts, at the cost of tyre scrub in corners, so it is engaged only off-road or at low speed.
Brake-based traction control uses the equal-torque rule instead of fighting it. The system applies the brake on the spinning wheel. Brake torque adds to that wheel's reaction, so the axle torque rises, and the differential sends the same higher torque to the other side. If the brake adds 300 N·m, the tarmac wheel receives about 330 N·m of drive.
It costs energy, since the engine is working against its own brake, and the brake heats up in prolonged use, but it needs no extra hardware beyond the ABS modulator. Many current cars with an open differential rely on it.
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