It is cheaper but I don't know that "cheaper" is always the reason. BMW, perhaps. Cayman... well I think that's as much about making sure people can still find a reason to pony up the extra cash for a 911 vs a Cayman. The Cayman is perhaps the better 2WD platform so we need to hobble it so it doesn't out shine the premier model of the Porsche lineup. I don't buy the cheaper claim with the Lotus. When the Elise came out they didn't have a higher end model to protect like Porsche and they also knew that unlike BMW, the Elise wasn't going to be an every day car for most owners. It was a car about handling. Lotus didn't fit the fattest tires nor the most powerful motor. They also didn't make a cheap car. I figure in that case they probably really did feel the LSD's help in tight corner exits wasn't worth the decrease in handling feel. Of course if your objective is autocross times you might disagree.

 

I wasn't with him at the track but I did hear comments from another driver that he was quite quick. It did cost him on the autocross but he didn't feel it was a significant issue on the race track and was happy with the open diff and more so when he realized the negatives the LSD can have on the handling of the car when you have less than sticky tires or the roads aren't dry and clean.

However, to back what you have said, I have heard that the various Miata class racers want the LSD to lower their lap times. Of course the Miata has a relatively short wheel base and track and consequently the ratio of CG to footprint might be less favorable than many realize.

 

Exactly what are the negatives to th handling. I have heard this before, but I don't buy it. Since the install, the only real difference that I have noticed other than being able to put the power down (yes all 100 of it) coming out of corners, especially tight corners, is that on/off throttle rotation has been greatly improved. I would say that for someone not comfortable with oversteer, that it could be an issue. But for a better driver, I see everything as a win-win. All of the people who have driven/ridden in my car at the track have all been surprised by how easy it is to drive, and how well it turns in and rotates without over rotating. Most of that my be due to my suspension setup, but I noticed good improvements after the diff went in.

That said, there are times that I wish that I had a bit more rear end compliance for rain races. But that can be solved by me not being lazy and un-hooking the rear sway bar.

 

First, let me state that the degree to which a LSD will negatively affect handling is dependent on many things. In some cases it's going to be transparent in others it will not. Don't take my handling points to mean ALL cars with a LSD will have problems. Also, remember that not all LSDs are equal. The BMW M diff would likely have basically no ill effect on handling as it is a speed dependent diff. A torque sensing diff is more likely to have some of these issues while a fixed force spring clutch pack is most likely. The setup of an individual car will also make a big difference. I first noticed the issue with my car because the previous owner had installed economy tires with little grip or compliance on the car. Replacing the junk tires with decent, grippy tires largely fixed the issue but got me thinking about the original issue. Also, when I say issue I don't mean the car handles "badly". My personal car had a LSD and I don't think it handles badly. However, I do notice the negative side effects of the LSD from time to time.

The fundamental sources of a handling issue due to two things we have mentioned in this thread. One, the diff will basically behave like a spool in conditions where the difference in forces on the left and right tires are less than sufficient to allow the wheels to move independently. Two, the diff sends torque to the inside wheel of a turn under normal conditions.

One becomes apparent when driving on say gravel or other low grip surfaces. It manifests as the rear end wants to step out very easily. Sure that's fun if your intent is to drift around every corner but mine often is not. In snow this can be particularly problematic as the rear end tends to shift sideways very very easily.

The other is a sometimes noticeable tenancy to understeer. Most people think about power oversteer with a LSD. That is an issue and certainly had lead to a few Miata's smacking their rear wheels into curbs. If you read one of the "it just stepped out on me" stories on the Miata forum you will find that often the car had a LSD. The open diff cars don't surprise as much because they just spin the inside tire and slow down. So I think we can all see the oversteer part. The understeer comes from the torque transfer I've been talking about. I drive on a number of gravel roads. With my open diff Miata I never noticed a tendency for the rear to want to come out under power. With my LSD car the car clearly understeers into the corner as the rear axle applies more torque to the inside tire and tries to keep me going straight. Once the turn gets going it's easy for the car to transition into oversteer as is common with a LSD. This basically makes the whole turn less fluid than with the open diff car. The open diff car doesn't put the power down as well but the rear axle doesn't fight the front wheels.

With hard tires or say in the rain, this same issue can be felt. However, in the dry with good tires the extra grip of all 4 tires masks this behavior and all is fine. I have theorized about this though I haven't run the numbers. Here is my thought. The behaviors I'm talking about and noticing happen at less than 100%. For instance, I might take a particular turn at 35 mph. That is a safe speed in rain or shine. Well in rain I might be at say 85% of my tire's grip. In sun perhaps only 40%. What does the diff see? Well it takes basically the same power to push me through the turn rain or shine. That means the diff see's the same input torque and so long as the wheels aren't slipping the wheels deliver the same torque. However, in one case we are far from the limits of the tires. In the other case we aren't that far from the limits. We start feeling the behavior as the tires get to their limits due to the transfers of torque and as the front tires get to their limits while trying to rotate a rear axle that is fighting their action. Net result the rain makes the feeling obvious at low speeds because we have limited the performance of the tires.

I did a search for other sources mentioning the same things I'm discussing. Here is part of a book about the creation of the GT-R mentioning much of the same stuff.
http://books.google.com/books?id=0JW...page&q&f=false
Note that the understeer is caused by the mechanisms I mentioned on page 1.

The emphasis is mine and is exactly what I have been telling Nunco, the slower wheel gets more torque!
The article goes on but I have to hand copy it. Basically it agrees with everything I've been saying and also explains why Nissan elected to go with active diffs rather than traditional passive/mechanical LSDs for the GT-R.

 

See, here's one issue. If you experience understeer at the beginning of a corner, the way you are driving and the way the car is set up are not compatible. One might more accurately say that open diffs are slower because they make the car harder to rotate and cannot put power down around a corner. The fact that they require slightly different driving is indicative of their function.

In addition, much of what you just quoted is inaccurate or an over-simplicfication. As you know, there are numerous LSD types and variations of design with a particuilar type. So any statement such as "LSDs (except 1-way clutch-type LSDs, but including helical torque-biasing LSDs) always create understeer..." (3rd quote) are just ludicrous. One can adjust the behavior of clutch type designs. It is the degree of those adjustments that make a diff a 1-, 1.5-, or 2-way, and one would expect those adjustments to be intentional. So if understeer was an issue because of your adjustments, you'd fix it.

Furthermore, if you consider the LSD's behavior in a straight line a "trivial case" despite the fact that a stright line is perhaps the most common situation, how can you consider driving around a curve with both wheels having equal traction anything other than a trivial case? I would like to see the super-common situation you are imagining whereby the inside wheel, despite being unweighted, has more traction than the outside wheel. How often does that happen in real life?

 

We aren't talking about the total car's behavior. We are talking about the diff's contribution to the car's behavior. The diff will typically add an understeer moment to the chassis. That doesn't mean the chassis goes straight, only that the front wheels have to work harder to turn the car. I can set the chassis up so it will always over or understeer. However, in this case if the chassis is otherwise neutral the LSD will make it understeer in most cases then oversteer more easily in the power on case.

Actually no that statement is not ludicrous. Any passive LSD when applying it's limiting toque while the wheels are not actively slipping, and not in the trivial equal wheel speed case, is applying more torque to the slower wheel. With a viscous LSD that delta in torque might be very small and effectively makes no difference in almost all cases (that's BMW's intent with their speed sensing diff). However, it is true. The 1-way was excluded because it is effectively open in one direction.

Note that the article said exactly what I said about that, the slower wheel gets the torque. With that in mind if you give the slower wheel the torque that means you create an understeer moment about the rear axle. Thus the statement is accurate. You seem to have trouble separating the behavior of the chassis from the forces the diff applies to the car (via the wheels).

In math the "trivial" case is often the one that is easy to think of or solve. The trivial case might be the one were something is multiplied by zero or 1 or the negative of some other variable or using Real vs Complex numbers. That doesn't mean it's a case that doesn't occur. In the straight case we aren't worried about one wheel pushing harder because we aren't trying to work against the uneven forces the tires are applying to the road. So now that we understand the context of the word "trivial" lets move on. When did I ever say we would have a common case where the inside wheel has more traction? Sure it could happen if we put say sand on the outside of the corner. However, I don't recall making that a condition of any of my claims. No, I said the inside wheel is generally the wheel with less grip due to the unloading you mentioned. However, as the article says, most of the time in a street car we aren't driving so hard as to have the inside wheel actually slipping. That means, as I've said and as the article said, the inside tire is getting more torque than the outside tire.

I find it disappointing that the moment I present some article that backs 100% of what I have said it's "inaccurate". What article do you have that suggests otherwise?

Do you now agree based on both what I have said AND the claims of this article that a passive LSD will transfer torque to the slower of the two wheels any time we have a difference in wheel speeds? I think that is a critical question since I believe it is one we disagreed on.

 

While I can't disagree that an lsd will allow the rear end to be more free, I will say that having driving my miata quite a lot in the snow this past winter, I prefer it with the LSD. It allows much better acceleration, and I feel more control in the corners as well. But again, this is coming from someone who is very comfortable with a little movement in the rear end. I have said for quite a while that for the average driver, an open diff is safer due to the fact that most people can't drive very well. As for wet conditions, I can't stand to drive an open diff car anymore. It is so frustrating to try to make a quick right turn from a stop at a light and just spin up the inside tire with any small amount of throttle.

As for understeer dynamics of an lsd, I guess that there maybe some small amount that this does effect, but on a track, I would still have my money on the one of two identical cars an drivers that had an LSD rear end. I am guessing that there is a reason that pretty much every form of professional racing uses LSD's.

So if I were to agree with someone on something, it would be that for someone who is not comfortable with any rear end movement, an open diff is better, For a better driver, an LSD is better for most scenarios.

 

The free action of the rear is something that I would agree is personal preference. Sometimes I really like it but I also agree with the book that I quoted that with a neutral chassis the LSD can have less than ideal side effects. I had a long debate with someone on the Miata forum about the handling of a LSD equipped car. He just couldn't buy what I was saying as he had never noticed it in his car and suggested I might just be noticing power off oversteer. I knew this wasn't the case and actually recall noticing some of the LSD's affects even before I bought my current car. The discussion took a turn when the guy drove a friend's Miata that was not setup for track day events. After that he completely could see all the effects I had noted. It's not immediately obvious but once you know what you are looking for it's not hard to spot. I do appreciate having some limiting on the car but I do wish it was something along the lines of the BMW M diff vs a torque sensing diff.

I would alter your last sentence to say the LSD is faster. Better is subjective. I often prefer the more fluid feeling of an open diff. A carefully balanced car will allow you to feel the action of the LSD. Now that I know what to look for I notice it rather quickly and don't care for it... unless hanging the tail out is my objective in which case I love the LSD.

 

Nunco,
I did some additional digging on the subject. I hope the following papers and presentation will convince you that my description was correct. The papers are by the same group out of U of Minnesota. One is an SAE paper the other is an IEEE paper. Both papers are peer reviewed. The presentation is by Eaton Corp though it also is in connection with the same group of researchers. The researchers were looking at electronically controlled LSDs and center couplings for AWD vehicles. The materials include information about rear LSDs which would be relevant to our discussion. Given the parties involved this should be considered authoritative.
http://ghost.engin.umich.edu/tcac/8_workshop06_lew.pdf
http://www.me.umn.edu/~qhyuan/public...TorqueBias.pdf
http://www.me.umn.edu/~qhyuan/public...06-01-1963.pdf

The first link includes a great diagram and the equations that define the torque flow through the diff. On slide 8 we have a model of a clutch based LSD. In this case they model the clutch as only a single unit rather than a pair. At first this may seem odd. However, because the left and right outputs are coupled via the spider gears a resistance on one is applied to both. So even though I have been describing things with a pair of clutches there really is no reason to have two nor would the two need to be equal in size. The last paper (the SAE paper) shows a LSD which has a hydraulically controlled clutch on just one side of the diff. I think it's safe to say that if Eaton Corp felt they didn't need to put clutches on both sides of the diff we can assume that the locking torque of the pair of clutches is the sum of the individual clutches. I have typically referred to T_clutch as the torque that is needed to rotate a single clutch so what the presentation calls T_tf is what I would have called 2*(T_clutch).

So on page 8 along with a schematic of a diff the presentation has some equations for the torque applied to each wheel. Those equations follow mine though the actual names of the variables are different. Let's identify the parallel terms:

• I have said T_diff is the torque rotating the whole diff housing (eg the torque from the engine). They call the torque from the engine T_p for propshaft. They also note an assumed 1:1 final drive ratio to keep the math easier to follow.
• The torque transfered by the clutch system is T_tf. In there system T_tf is controlled by a computer where as we have been talking about passive systems where T_ft would be controlled as a function of relative wheel speeds (speed sensing) or as a ratio of T_p as a torque sensing diff. The fact that in this case the magnitude of the clamping force on the diff is computer controlled does change things in that the computer can choose to release the clutch when the properties of an open diff are more appropriate. However it could also be programmed to perfectly mirror a Torsen diff. We would simply measure the torque into the diff and apply clutch clamping pressure accordingly. Net result, even though the control strategy for this diff could be more complex than say a Torsen, the torque flows will be the same and its affects on the handling of the car would be the same if programmed to behave as a torque sensing diff. In short, don't claim that because this is an eLSD the information doesn't apply.
• What they call T_diff I was calling in effect 2*T_spider. That is the torque the open differential gears apply to the outputs.
• Because their model has a clutch only on one side they have to show how the one sided clutch can affect both outputs. They show that math on the same slide. The final result of the math is T_l=(T_p +T_tf)/2 and T_r=T_p -T_tf)/2. I said T_l=T_spider+T_clutch and T_r=T_spider-T_clutch.

Note that since I had effectively two T_clutches and since I called the T_diff = 2*T_spider the peer reviewed papers and I ended up saying the exact same thing! The torque delivered to each wheel is the sum of the torque from the spider gears (which is in the same direction) and the torque from the clutching system which is added to one output and subtracted from the other!

Now look at slide 9. Note the following two points.

• Transfers torque from the faster speed wheel to the the lower speed wheel.
• Increases vehicle yaw understeer tendency on a constant mu surface

These are the points I was talking about all along. Yes, my musings were only those of a random car enthusiast but it's clear they parallel the understandings of experts at both Eaton corp and U of Minnesota. I would say that is pretty darn conclusive. While I found some of this discussion trying as I struggled to try to explain these concepts in the and I think this was a good discussion and hopefully others came away with a new understanding of what a LSD is good and not good for. I certainly came away with a much better understanding of how to describe the inherent function of the system.

 

You should examine the papers again. All three use T_diff for the torque transferred by the driveshaft, T_CT_r for the torque transferred by the clutches, and T_lr and T_rr for the torque at each axle.

In their math, T_lr = T_CT_r + (T_diff/2), while T_rr = 9T_diff/2). This clearly shows the clutches imparting torque to the side axle they are mounted on, and their system is not the same as one with clutches on both axles. One with two sets of clutches would feature T_CT_lr and T_CT_rr. According to their math, a system with a set of clutch stacks on both sides would be T_lr = T_CT_lr + (T_diff/2) and T_rr = T_CT_rr + (T_diff/2).

They clearly show that the people writing the papers feel that torque is transferred based on how tightly the clutches are bound. If you put clutches on both sides, then the side with more tightly bound clutch stacks would be transferring more torque.

I do see slide 9. And yet again, the slide is doing what you did - only addressing one half of the equation. It leaves out entirely the forces countering the engine torque. The engine cannot generate torque if there is no conteracting force. For instance, if the wheels are off the ground, a gasoline engine cannot generate any more torque than is required to spin the wheels. As such, if one is going to model the behavior of a system such as a limited slip differential, one needs to include both the torque of the engine and the reaction torque of friction, et al.

In other words, when you leave out half the equation, you get a wrong answer. I don't care if it is peer-reviewed or not, my physics professor would never accept that work.

The inside wheel is unweighted. It cannot transfer as much torque as the outside wheel. If it tries it will slip. Luckily with an LSD it will only slip at the same speed the outside wheel is moving, making perhaps one unnecessary revolution around the course of the turn. Not much of a slip, and doubtful that an observer would even notice.

Again, papers are paper. Show me a situation where the inside wheel has the same traction as the outside wheel despite being unweighted. Because the outside wheel can only transfer more torque than the outside wheel when it has sufficient traction. And we know even on a Miata that is going to be a rare case, otherwise nobody would spin the inside rear wheel with an open diff.

What I am saying is that a situation where an LSD actually causes understeer during a corner while under acceleration due to the inside axle tranferring more torque to the wheel is so remarkably rare it is not a consideration. The inside tire is going slip slightly far more often.

Just because those people wrote a paper does not mean the real world matches their simulations. Also, if you read the second paper, near the beginning it highlights the lack of relevant works at this level. Which is the situtation I also found.

IMHO, the circumstances described in the papers as "high friction surface" is a rarity, not a common situation. I think many non-engineers have already disputed what you and the papers claim. I'm not sure how long one is expected to insist on papers trumping observed reality, but I think it's past time to concede the issue.

I appreciate you finding and posting the papers, and have saved them for future reference, but I also take them with a grain of salt.