Please forgive me for bringing back this thread. This is actually one of the better diff discussions I've seen and thought I would add a bit. I apologize because not all of this is in order.
First, a bit about the open diff. Someone included a link to the old GM video on the open diff. That is a GREAT video and does an excellent job explaining what and why. It was also correctly mentioned by several that an open diff sends equal torque to both wheels. This of course is also true. I was debating with someone who didn't agree. Here is how I justified this view.
Consider the spider gears. These are the gears between the two axle outputs. Each output gear applies a load to the spider gear that would cause it to turn. We all know that if you have a force applied to an object it will want to move. We also know that if you don't have an opposing force the object will accelerate forever. Since we know the spider gear doesn't accelerate forever we can assume there is an equal but opposite force also acting on the gear. Well that equal but opposite force is the other output gear. Hence the spider pushes equally hard on both output gears thus equal torque.
What about relative speeds? Even though we normally don't think of it as such, a differential can be thought of as a mechanical math statement. The average speed of the two outputs equals the speed of the input. That's In=1/2 (Leftout+Rightout).
Tea Jay was discussing having traction and some confusion there. Really we might say "sufficient" traction. We can assume sufficient to be enough traction such that the wheel doesn't slip given the torque applied to the wheel. We can also think of how much torque the ground can apply to the tire. Even if the tire is slipping we know there is some resistance which will be seen as a torque resisting the turning of the wheel.
On to the LSD.
I think the thing that is hard to get about the LSD is the operating equation is actually quite easy. The sum of the torque applied to the two axle shafts is equal to the torque applied to the diff housing (the input torque). The speeds of the two wheels again follows the equation above. The catch is the equation for right and left torque. This is actually a two part equation. In all the stuff I posted below I describe the case when there is relative movement between the left and right wheel, the cornering case. There is also the straight line case.
In the cornering case torque is sent to each wheel via both the spider gears and the clutches (or other limiting mechanism). I have the equations and derivation below but the critical take away, which others have said is a LSD causes a tenancy to understeer. That's because it transfers torque to the SLOWER wheel, ie the inside wheel.
I didn't cover the straight case. In the case of going straight case we don't know exactly how much torque went to each wheel. If we know the torque required to get the clutches in the diff to slide then we can know the difference in wheel torque was less than what was needed to break the clutches free. So for a diff that can send say 70% power to one wheel, we can know that if the total torque sent to the diff was 100 ft lb, the lesser tire had enough traction to resist at least 30 ft lbs of torque applied to the axle.
OK, enough bathering, here is more blathering... with pictures!
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Just a few things starting with the open diff:
1. Torque is like force, it's how hard you twist something.
2. Power is force times velocity or in this case torque times rotational speed. If you push just as hard but move twice as fast you have twice the power.
3. An open diff always sends equal TORQUE to each wheel. ALWAYS.
4. Since an open diff sends equal torque to both wheels the wheel that spins faster gets more power. So that one wheel on ice gets the same torque as the wheel on dry ground but the ice wheel is the only one getting power since it's the only one spinning.
5. The torque applied to the rear wheels (not the power) determines how hard the tires are pushing the car forward.
So at this point we can figured out that with an open diff each rear wheel pushes just as hard. If one is one ice, neither can push that hard.
Limited slip diffs come in a few basic types. You have the old spring loaded clutch pack type. Those have a fixed locking torque. Locking torque is what makes the output shafts try to spin at the same speed as the diff housing. For a clutch pack diff the locking torque is the torque needed say turn the wheel when the car is on jacks and the transmission is in gear. Next you have speed sensitive diffs. These are ones where the locking torque is related to the difference in speed between the axles and the diff housing. They act like an open diff if both wheels are spinning at about the same speed but act like a LSD when one wheel spins up. Finally we have ramp clutch, torque sensing clutch packs, Torsen, Quaifee (sp?) and other torque sensing diffs. In these diffs the locking torque is proportional to the torque into the diff. So when you are on the gas the diff has a lot of locking torque. When you are off the gas the diff is rather open. Even those all these torque sensing diffs have different guts, as a black box they all work about the same way. Since the clutch pack version is the easiest to see I used it in these pictures.
Now on to the pictures...
1. A basic clutch pack diff. This diff uses the separation force of the output gears from the spider gears to load up the clutch discs. That means this example is using a torque sensing diff. Note that because the clutches are between the housing and output any time the housing is spinning faster than the output the output will feel a torque in the forward direction. The opposite is true in reverse. There are two ways torque can be applied from the housing to the output; 1. via the spider gears, 2. via the clutch plates. The total torque to the output is ALWAYS the sum of those two torques.
2. The open diff. Without the clutches the torque sent to each wheel is ALWAYS equal. The wheel on ice can't handle any torque so no torque gets sent to the wheel with grip. I stressed this because EVEN when we add clutches the SPIDER GEARS always send equal torque to each wheel. The clutches make up the difference.
3.Now we turn to the right (imagine we are looking at the back of the car). The right wheel slows, the left wheel speeds up. We have relative motion in the clutches. Since this is a torque sensing, not speed sensing, diff it doesn't mater if the speed difference is 1/2 RPM or 100 RPM, the torque the clutches apply to the output shaft is still just a function of input torque.
The right wheel is spinning slower than the diff. That means the torque on the right output shaft is forward (or trying to make the right wheel spin faster). The torque from the spider gears is also forward. So we have two forward torques added together. Call that
T_spider + T_clutches.
The left wheel is going faster so the torque the clutch applies to the output shaft is in the reverse direction. The equation looks like this:
T_spider - T_clutches. So in one case we have addition and in the other subtractions. It's pretty clear that the addition case (ie the inside wheel) is going to get more torque. More torque at the wheel means more force on the ground.
Since the inside wheel is pushing harder than the outside wheel the rear axle is creating an understeer moment. If the front wheels were on casters the car would try to straighten out! Of course the front wheels over power this with their own steering moment but this is why a LSD (and of course a spool which is like an LSD with REALLY REALLY high bias!) tend to make a car understeer. Its just most of the time the front wheels can compensate.
5. The above is true until the inside tire slips. This is the case where everyone says, "See the LSD sends the power to the wheel with grip!" Well yes but that's because it sends more power to the slower wheel. When the inside tire slips it speeds up. Makes sense, it doesn't have the road to slow it down. Of course all the math above holds true. The inside tire is now the fast tire. That means the
torque to the inside tire is T_spider - T_clutch. Since the outside tire is now the slow one it's torque equation is:
T_spider+T_clutch. So the non-slipping tire gets more torque and everyone is happy.
But there is a problem in that last bit. When you are cornering hard the outside tire is doing all it can to keep the rear end of the car from sliding out. When the inside tire loses grip it speeds up which means we get a lot of torque quickly transferring to the outside tire. The poor outside tire had just enough grip to handle pushing the car forward and keeping the tail in line. So now we spin thanks to the quick transfer of torque from the inside tire to the tire with grip! Fortunately this typically happens slowly and progressively enough that we can balance things and drive the car on the edge. Make no mistake, the LSD makes the edge sharper even though it doesn't always make it faster.
