There is a fundamental design flaw with British roads: they are not straight. If they were straight, life would be simple. We could have cars with solid axles and we could drive all day with no problems. But as soon as we start to go round corners - problem.
And that means, of course, that the wheels on the outside of the bend have further to go than those on the inside - which is fine for whichever wheels are not being driven - they can be like a supermarket trolley's castors and go at different speeds, though hopefully not different directions as supermarket trolley castors generally do - but for the driven wheels it's a different matter. You have to drive both of them simultaneously, while letting them turn at different speeds.
'But how is zis possible?', you ask in your best impression of Bela Lugosi.
The answer is called a differential, so-called because it allows for differential movement of the two driven wheels. Grease-monkeys call it a 'diff', because that's shorter and easier to remember (which is also the reason they are all called Dave).
In its simplest form a differential works like this: there is a large ring gear with teeth all the way round the outside, called the crown wheel. This is driven round by another gear, the pinion, connected to the engine in some way (on rear-wheel drive cars this will usually be a worm gear, like a screw resting on the outside gear ring). It's not that important to understand how, just accept that we can make the crown wheel go round.
Add a single spoke going all the way across the inside of the crown wheel. It's fixed to the crown wheel so it goes round with it. We need to be able to remove the spoke, so we will probably put some sort of simple mounting offset to one side. Even real differentials are remarkably crude - it doesn't need to be anything fancy.
Now get four identical cones. They need to have a base angle of 45°. If you put the points of the cones together and looked down from on top you would see a cross. Cut teeth into the cones, running from top to bottom. They will look like lemon squeezers when you've finished. Cut the tops off the cones, maybe about halfway down.
Take two of these cones and weld them to the ends of the driveshafts, one for each wheel. Drill holes though the other two and thread them onto the spoke attached to the crown wheel, facing inwards, so they can spin freely. Bring the four cones back together. You should now have an arrangement with the crown wheel driving round, the driveshafts at right angles to the crown wheel (one each side), and the teeth on the four cones meshing.
When the crown wheel goes round, it drives the spoke. When the spoke goes round it takes its two cones with it. When the cones go round, because they are meshed with the cones on the driveshafts, the driveshafts will go round as well.
And now the magic. One driveshaft wants to go faster than the other. This is easiest to imagine if we think of the rather extreme situation where one wheel is stopped completely. The crown wheel drives round, but one driveshaft doesn't turn. The two conical gears on the spoke begin to turn, in opposite directions - they have to, because the driveshaft is stationary. They are rolling round the outside of the now stationary cone. But what happens to the other cone? It's got two spinning gears attached to it - it will turn, at twice the speed of the crown wheel. You are going round the bend!
In the above example, one wheel is stuck and the other goes at twice the speed. This is, in reverse, what happens when one wheel is on mud or ice and the other isn't. The wheel on the slippery stuff will whizz round happily and the other wheel will stay right where it is.
Differentials are designed to take up small amounts of movement between wheels moving at broadly similar speeds. Spinning one wheel like this will subject the diff to stresses it was never designed for, and diffs are not cheap to replace. So, if your wheels are slipping, keep the revs down.
In a four-wheel drive vehicle both axles are driven, so there have to be differentials on both. Also, the front wheels could be moving at slightly different speeds from the back, so there has to be a third differential there. All those whining pinions on their crown wheels, no wonder 4x4s have bad transmission noise.
But great in the slippery stuff, eh? Well, not necessarily. With differentials left to right and front to back, it's just as easy for one wheel to spin and all the others stay still. This doesn't usually happen because 4x4s are generally fitted with limited slip differentials. These have an arrangement (usually some form of fluid coupling) which limits the difference in shaft speeds, so you get some traction on the wheels which are not spinning. And if you have a proper off-roader, you will be able to lock the differentials entirely. Do this on road and you'll wear out tyres in minutes, but on soft ground it's brilliant.
What Goes Wrong?
These days it is getting quite hard to buy a bad car, so we tend to forget the things which used to plague motorists. Diff trouble was one such plague.
The diff on a rear-wheel drive car is part of the rear axle. Older cars had a solid rear axle with one large housing holding the diff and driveshafts. The problem with this is the diff is quite hard to get to, so it tended not to receive quite as much care and attention as it should. The casing could get cracked by stones off the road, and loss of lubricant through worn seals and corroded casings was a constant source of problems. Once a diff runs dry it will wear out in very short order.
Differentials are pretty crude things and the axle on which the conical gears run - the ones attached to the crownwheel you will remember - is a case in point. The gears might have oilite bushes (a solid ring of phosphor bronze which is slightly porous so absorbs oil) but certainly no proper bearings, and their axle would generally have some grooves in to let oil flow - but that's about it. These components soon wear, allowing slack to build up. The normal process of acceleration and deceleration will add to this, and you end up with a situation where there is 'lash' in the drivetrain - the car can move backwards and forwards sometimes several inches without the engine turning.
The pinion gear which runs on the crownwheel also takes a tremendous hammering. Once this arrangement becomes worn there will be a constant whining from the back axle, a sure sign of impending failure.
... and Tear
And finally, all those bits thrashing around are subject to the usual stresses and strains. Metal fatigue can set in, and the whole assembly can simply disintegrate. This can be quite spectacular, and can lock the whole back axle solid requiring removal of the driveshafts before you can even be towed.
Front-wheel drive cars have the differential unit in the gearbox, so it is less prone to mechanical damage. Modern cars also typically use higher grade components, and front-wheel drive diffs will use a direct drive from a helical gear, rather than the worm gear used in rear-wheel drive. This is less prone to wear. So thankfully diff troubles are uncommon these days.
A top tip for any devotees of The Italian Job who are thinking of looking under the back end of their Mini to find the diff: it's at the other end.