Shorter Stopping Distance for ultralight vehicles?

[mender]

viperblues450, the next issue is controlling the transients while the car is doing its thing on the track. The shocks have to smooth out the body motions in such a way as to minimize instantaneous loading of any of the tires, again because of the load to grip curve of the tires.

 

[viperblues450]

Viperblues, I have to reply to this post just because you are completely misinterpreting what has been argued over in the past 4 pages. My argument can be summed up as such:
...There isn't any fundamental reason an ultra-light car can stop faster than a heavy one, as long as the brakes and tires on each car are sized appropriately.

It could be argued that it is easier and cheaper to make a light car stop quickly, but that's about it (and it's easier and cheaper to do most anything performance-based in a lightweight car).
What you're trying to argue is not what this thread is about, period.

Untrue. The name of the thread is "Shorter distances for ultralight vehicles?"
The answer is yes. And everything I've talked about is applicable to understanding why the answer is "yes."

The answer is of course that it takes more than mass to determine how quickly a vehicle can stop. The primary factors that will determine how quickly a vehicle can stop are the friction between the road surface and the tire (tire compound) and the power dissipation capacity of the brakes. This has been repeated over and over for 5 pages now.

You are leaving out a very important factor, the frictional coefficient of the brake pads to the rotors. This is much more important over a single stop than the size or heat capacity of the rotors. This is also a reason why the Saleen could not stop as fast as the Lambo. Looking at the size of the rotors is simply not enough. The bite of the brake pad material is much more important (until the discs heat up and brakes fade).

That's not what's being argued here. Given two cars that have been designed by two different manufacturers, the lighter one will not automatically be able to stop more quickly than the heavier one. Heavier cars tend to have heavier capacity brakes, and as such they will tend to be able to stop as quickly as lighter cars. This is especially true in sports cars, which I covered in extreme detail.

It is all about the relationship between the brake/tire/road system and the mass of the vehicle. The lighter car will be able to stop quicker assuming the maximum braking and tire grip is achieved with both vehicles.

Actually, it does to a first-order approximation, and the VERY simplified math was presented on page 4 by NateTG.

As I explained, the first order approx is invalid in this application because the "m" term in the F=mu*m*g equation for normal force is not actually equal to m. See the tire grip chart level to understand this. I know that its a weird concept, but try understand why those simple equations don't work here.

The dynamics of vehicle braking are indeed quite complex, but your vehement argument is completely missing the point of this thread. Ironically, your argument assumes that a car with more people in it will ALWAYS stop slower than one with less people in it, which isn't true either.

If a car is carrying 4 people, and the brakes were sized appropriately during the vehicle's design phase to take this extra weight into account (read- the tires can still lock up on a full stop and the ABS system engages), the car will stop very close to as quickly as the same car with only one person in it. Any difference in stopping distance will not have to do with the increased weight, it will instead probably be due to minute shifts in the vehicle's center of gravity or weight distribution. If we assume the vehicle's brakes can always lock the tires (properly sized brakes for the vehicles estimate operating weight, the vehicle is not overloaded), the extra momentum from any extra weight in the vehicle is offset by the fact that there is more available frictional force available to decelerate that extra weight as well.

This is all true except for the part where you say " the extra momentum from any extra weight in the vehicle is offset by the fact that there is more available frictional force"
I don't mean to keep repeating, but the nonlinearity tire characteristics invalidates this statement. Does the additional force provide additional grip? Yes, but it does not fully cancel out the additional mass.

While your argument is interesting (I'd be interested to see a curve that documents the phenomenea you are mentioning) the fact is that we are talking about basic, first order braking with respect to vehicle weight. The point is, an ultra-light vehicle cannot necessarily stop faster than a standard weight vehicle; there are of course MANY more things that have to be taken into account. Case closed..

Case unclosed, check the link. The lighter vehicle will stop quicker.

Sure, it's in theory possible that given the perfect driver and perfect conditions, ABS might not be necessary for a perfect stop. But the fact is they were using professional drivers in the test and they couldn't get the Saleen to stop anywhere close to as quickly as say the Porsche, even though it was lighter.

ABS NEVER gives you the perfect stop, because it has to wait for tires to lock up before it engages, where a perfect brake w/o ABS could brake right on the limit the whole time.

No, since the Lamborghini and the Saleen had the exact same tires, it's likely that at best the Saleen would have been able to stop as fast as the Lamborghini, but not faster. Slight differences in their stopping distances would have been due to differences in f/r weight distribution and center of gravity, but not overall weight.

The Saleen most definately should be able to stop quicker than the Lambo. Its poor performance here is probably due to the driver (pro or not) being unable to give a perfect braking run. This is very hard to do, and even a pro driver might not be able to do it. Other factors that the Saleen might not have been able to stop well are the conditions of the tires, tread, tire temp, tire pressure, brake pad material (as I previously explained, probably the most important factor besides tires and weight). The feel of the brakes may have been numb, not allowing the driver to get a good feel. Your argument that the Saleen SHOULD not be able to stop faster than the Lambo is just wrong. This assumes that the Saleen and Lambo has similar weight distribution, wheelbase, and CG height. If the Lambo has significantly advantagous geometery, its possible that also helped it, however I doubt this is the case.
The Porsche does so well because it starts with more weight over the rear wheels, and transfers weight to a near 50/50 distribution under braking. Having a equal load on all tires provides the greatest overall grip and braking force.
Using your incorrect understanding of tire grip, the extra weight on the front tires of say the Lambo would completely cancel out the weight lost on the rear wheels. This is not the case, and it shows up in how rear engined Porsche consistantly have some of the shortest braking distances.

 

[mheslep]

Note that the grip vs normal force graph mender linked to was for one tire. That by itself is not helpful, as obviously the heavier vehicles can, and do, use tires with a higher coefficient of friction. So VP450, you are changing the scenario I posed above - vehicle A vs vehicle B, to the scenario of vehicle A vs vehicle A with more or less load. That's not useful for my problem.

To illustrate, I prepared a naive family of curves extension for the graph mender linked. It shows how, for example, one could maintain the same coefficient of friction (slope) at say, 600, 1000, and 1400 lbs per tire in three different vehicles by selecting different tires in the design process.

 

[viperblues450]

Note that the grip vs normal force graph mender linked to was for one tire. That by itself is not helpful, as obviously the heavier vehicles can, and do, use tires with a higher coefficient of friction. So VP450, you are changing the scenario I posed above - vehicle A vs vehicle B, to the scenario of vehicle A vs vehicle A with more or less load. That's not useful for my problem.

To illustrate, I prepared a naive family of curves extension for the graph mender linked. It shows how, for example, one could maintain the same coefficient of friction (slope) at say, 600, 1000, and 1400 lbs per tire in three different vehicles by selecting different tires in the design process.

I'm really not understanding why you think its an unfair comparison. High coefficient of friction tires are not reserved for heavy cars! Any car can use a hard or soft (grippy) tire. In order to compare the effect of mass on braking, you have to use the same tires. You can ONLY compare vehicle A to vehicle A with more load.

If you are comparing car A to car B which have both been DESIGNED to have similar braking performance, then you are comparing nothing! Oh course they will have similar performance, that's how they are designed! The only thing you could say from that is that the heavier vehicle needed larger brakes, high friction pads, larger tires to have comparable performance. If both cars are designed with max braking, the lighter car sport better braking performance.

 

[viperblues450]

Note that the grip vs normal force graph mender linked to was for one tire. That by itself is not helpful, as obviously the heavier vehicles can, and do, use tires with a higher coefficient of friction. So VP450, you are changing the scenario I posed above - vehicle A vs vehicle B, to the scenario of vehicle A vs vehicle A with more or less load. That's not useful for my problem.

To illustrate, I prepared a naive family of curves extension for the graph mender linked. It shows how, for example, one could maintain the same coefficient of friction (slope) at say, 600, 1000, and 1400 lbs per tire in three different vehicles by selecting different tires in the design process.

Also, about your graph, choose the highest grip tire for lighter car and it will brake the best. You seem to be trying to change two variables at once and I'm not sure why.

 

[Stingray]

Viper,
I think that you're overstating the significance of these nonlinear tire effects. mheslep should probably have used an example where different tires "saturate" at different loads, but have the same initial friction coefficient. Plenty of examples like that exist in the real world, which explains why braking performance does not strongly correlate to weight in (say) street-legal sports cars.

So why not put the more linear tire on a small car? I'm guessing that it would probably be too big. There could be packaging or weight constraints as well as issues with the suspension geometry. Rolling resistance could be made much worse, or maybe other properties of the tire aren't very desirable.

 

[Mech_Engineer]

Untrue. The name of the thread is "Shorter distances for ultralight vehicles?"
The answer is yes. And everything I've talked about is applicable to understanding why the answer is "yes."

There are of course many cases of cars that can stop more quickly than other cars that are lighter than them, so your broadly-sweeping generalization is invalid.

You are leaving out a very important factor, the frictional coefficient of the brake pads to the rotors. This is much more important over a single stop than the size or heat capacity of the rotors.

The frictional coefficient between the brake pads and the brake rotor is unimportant, as long as the tires have enough braking power to lock up. Brake heat fade is beyond the scope of this thread.

As I explained, the first order approx is invalid in this application because the "m" term in the F=mu*m*g equation for normal force is not actually equal to m. See the tire grip chart level to understand this. I know that its a weird concept, but try understand why those simple equations don't work here.

I understand what you're arguing, and I agree that if there are frictional nonlinearities in a tire when braking they can be a factor in a car's overall stopping distance.

This is all true except for the part where you say " the extra momentum from any extra weight in the vehicle is offset by the fact that there is more available frictional force"
I don't mean to keep repeating, but the nonlinearity tire characteristics invalidates this statement. Does the additional force provide additional grip? Yes, but it does not fully cancel out the additional mass.

You have a point there. However I'm curious, all of these charts are measuring "lateral grip" in the tires, but do they also apply to acceleration and braking? Are these charts basically taking into account tire side wall roll-over and tread deformation in lateral cornering, which are not present when braking?

ABS NEVER gives you the perfect stop, because it has to wait for tires to lock up before it engages, where a perfect brake w/o ABS could brake right on the limit the whole time.

If you do a perfect brake right on the limit, the tires will never actually lock up and therefore ABS would never even come into play. Regradless, ABS equipped vehicles are able to stop very quickly regardless of driver skill, and I'm seriously doubtful ANY driver could stop a car "perfectly" without ABS.

The Saleen most definately should be able to stop quicker than the Lambo.

"Should" and "did" are miles apart in this case, sorry. It's probable the Saleen would have stopped slightly more quickly than the Lamborghini if it had ABS brakes, but it's likely the difference would have been small (less than 5%) while their weight difference is far larger than that.

Its poor performance here is probably due to the driver (pro or not) being unable to give a perfect braking run. This is very hard to do, and even a pro driver might not be able to do it.

In the article, the driver says that he was unable to properly modulate the brakes in the Saleen to get a braking run as an ABS equipped vehicle. It is my suspicion that given two identical cars, one with ABS and one without, the ABS equipped vehicle will stop more quickly 99 times out of 100.

Other factors that the Saleen might not have been able to stop well are the conditions of the tires,

They were the same exact tires as on the Lamborghini. Same brand, model, size, and wheel size. They were installed that day for that series of tests, same as the Lamborghini. All of the cars had the same brand and model of tire installed that day to ensure Tires were not a variable in the test (other than the car's required sizing).

tread,

AFAIK, all the tires on all of the cars were installed brand new for that series of tests so that all of the cars had a level playing field in that respect.

brake pad material (as I previously explained, probably the most important factor besides tires and weight).

Brake pad material doesn't matter, as long as the brakes are stong enough to lock up the tires. The brakes were clearly fine because the driver had to try and modulate the pedal because it way very easy to lock up the tires.

Your argument that the Saleen SHOULD not be able to stop faster than the Lambo is just wrong. This assumes that the Saleen and Lambo has similar weight distribution, wheelbase, and CG height. If the Lambo has significantly advantagous geometery, its possible that also helped it, however I doubt this is the case.

Look, the point of this thread is that just because a car is lighter doesn't mean it can stop more quickly. The Saleen is much lighter than the Lamborghini, yet it takes longer to stop. This is not a good comparison because the Saleen doesn't have ABS, but it proves the point of the thread.

We obviously agree on the core topic in this thread- it takes a lot more than just the weight of the vehicle to determine how quickly it will stop. Saying that a superlight car is fundamentally able to stop faster than a "standard" car is a shady hedge-bet. Now saying a well tuned sports car will stop faster than a minivan, that's obviously a safer bet.

 

[viperblues450]

Yeah Stingray I agree with you. I may be overstating the nonlinearity, but I wanted to bring it up because I wanted future readers (like myself) to know that tire characteristics are more complex than the simple physics equations that were being used in the thread.
In reality you're right, there is not a huge advantage in braking to decrease the mass because the braking components can usually make up most of the difference--but not all.
I am an armature racer and am used to competition where the "small" difference becomes a "large" competitive edge. Things can get down into such detail that adding larger brakes increases the rotational inertia and works against braking.
Also ultralight cars are not often outfitted with max performance brakes and tires. In a quick search I did find that found a Ferrari Enzo stopping 70mph-0 in 151ft
http://www.supercarstats.com/cars/ferrari/enzo_ferrari/
Where a $10k Locost 7 kit car (1297lbs) can brake from 70-0 in 141ft (no ABS I think)
http://www.caranddriver.com/reviews..._winterhalter_s_locost_sports_car+page-5.html
That Seven didn't even have super tires like the Ferrari would have had. I know these we separate tests, but I'm trying to show that even all the money and technology Ferrari can throw at their ultimate supercar cannot match simply "adding lightness." The Ferrari is almost there, but not quite, and that's all I've been trying to say. Shorter stopping distance for ultralight vehicles? Yes.

 

[jimgram]

I haven' seen any posts that relate net contact force (I.E. vehicle weight) to the maximum lateral force (braking) that can be applied without skidding. The standard fomula is Net (braking) force=Cf(coeff. of friction)X Vehicle weight. A linear relationship. So, if all other factors remained equal (e.g. contact area, surface, etc.) a heavy vehicle and a light vehicle will stop in the same distance.

 

[viperblues450]

There are of course many cases of cars that can stop more quickly than other cars that are lighter than them, so your broadly-sweeping generalization is invalid.

Broadly sweeping generalization in engineering discussions assume other variables constant.

The frictional coefficient between the brake pads and the brake rotor is unimportant, as long as the tires have enough braking power to lock up. Brake heat fade is beyond the scope of this thread.
Brake pad material doesn't matter, as long as the brakes are stong enough to lock up the tires. The brakes were clearly fine because the driver had to try and modulate the pedal because it way very easy to lock up the tires.

Well its very important as different compounds take different about of time to heat up to the temperature where they have the coefficient of friction to lock the wheels. The time that takes to heat up is short, but important if your comparing a brake test.
Formula 1 car brakes and tires can performance an INCREDIBLE 5 g's of braking! However, they have like a .5 second delay where there is almost no braking force from the "pads." It's interesting and makes the driver apply the brakes before they need them.

I understand what you're arguing, and I agree that if there are frictional nonlinearities in a tire when braking they can be a factor in a car's overall stopping distance.

You have a point there. However I'm curious, all of these charts are measuring "lateral grip" in the tires, but do they also apply to acceleration and braking? Are these charts basically taking into account tire side wall roll-over and tread deformation in lateral cornering, which are not present when braking?

I am also very curious about this question. I will consult my automobile dynamics textbook or do further research. If you find anything, please post.

If you do a perfect brake right on the limit, the tires will never actually lock up and therefore ABS would never even come into play. Regradless, ABS equipped vehicles are able to stop very quickly regardless of driver skill, and I'm seriously doubtful ANY driver could stop a car "perfectly" without ABS.

Thats true. What I learned in my automobile dynamics class was that tires actually provide the most grip with about 10% slip (both acceleration and braking). I'm not sure why that is, but if that's true, it would mean that an optimal braking performance would have the tires rotating about 90% of the car's speed.

"Should" and "did" are miles apart in this case, sorry. It's probable the Saleen would have stopped slightly more quickly than the Lamborghini if it had ABS brakes, but it's likely the difference would have been small (less than 5%) while their weight difference is far larger than that.

In the article, the driver says that he was unable to properly modulate the brakes in the Saleen to get a braking run as an ABS equipped vehicle. It is my suspicion that given two identical cars, one with ABS and one without, the ABS equipped vehicle will stop more quickly 99 times out of 100.

I agree, I think the limiting factor here is tires, and if the Saleen had ABS, I could see it only stopping about 5% shorter. Which was my point, less weight helps, not hurts or cancels out, even though it might be only 5-10% when both cars have high performance braking systems.

They were the same exact tires as on the Lamborghini. Same brand, model, size, and wheel size. They were installed that day for that series of tests, same as the Lamborghini. All of the cars had the same brand and model of tire installed that day to ensure Tires were not a variable in the test (other than the car's required sizing).

AFAIK, all the tires on all of the cars were installed brand new for that series of tests so that all of the cars had a level playing field in that respect.

Good to know.

Look, the point of this thread is that just because a car is lighter doesn't mean it can stop more quickly. The Saleen is much lighter than the Lamborghini, yet it takes longer to stop. This is not a good comparison because the Saleen doesn't have ABS, but it proves the point of the thread.

I think we are agreeing on most things now, but then our conclusions are opposite. I still retain that a car being lighter will stop more quickly. Since the Saleen doesn't have ABS, it does not prove the point you are making. It is only making the case for ABS. Look at the Ferrari Enzo and Locost 7 Kit car I posted.

We obviously agree on the core topic in this thread- it takes a lot more than just the weight of the vehicle to determine how quickly it will stop. Saying that a superlight car is fundamentally able to stop faster than a "standard" car is a shady hedge-bet. Now saying a well tuned sports car will stop faster than a minivan, that's obviously a safer bet.

I will agree that the difference is not large, but there is an fundamental advantage to a lighter car with braking performance. And yes, it is complicated.

 

[viperblues450]

jimgram what we have been saying here that it is not a linear relationship, and that simple formula doesn't hold up.
Think of the extreme case of a car braking so hard the rear wheels are just about leaving the ground. The front wheels do not have twice the grip to make up the difference. The front wheels have more with with the increased weight, but they do not have enough extra grip to make up for the rears (which have zero). The net grip of the 4 tires is less than if they all had an equal share.
Also, I think you meant longitudinal grip for braking.

 

[mender]

To clarify where I'm coming from, I have a fairly extensive practical background to go with my book learning. I don't usually deal in formulas and equations; I mostly go by lap times instead. I have been reading about chassis dynamics for the last twenty years and have been working on street and race cars for about thirty. Not to brag, just to let you know what my background is. Some things I won't be able to point to a specific page in a book for.

The slip angle that the tire assumes at maximum grip is about the same in all directions. The tire tread needs to be held flat to the surface to work effectively; when that is achieved, the suspension of the car can be tuned to either maximize the car's performance or the driver's. Note that these are not always the same! Getting the most out of the software (driver) is the more important of the two. You want the driver driving the car, not the car driving the driver.

The tire patch is essentially an oval that twists and moves around a bit in response to the forces placed upon it.

"Note that the grip vs normal force graph mender linked to was for one tire."

This is true: however this was only to show the trend. All tires will show this trend, with individual characteristics determining the exact curve. That's why there are different brands and construction techniques, to offer a tire that can be better matched to a car's needs and the driver's wants.

The Saleen very likely suffered from rear wheel lock-up. By adjusting the brake balance it would have outbraked the Lambo with the same tires just because of the difference in weight. Straight forward. A pro driver cannot get maximum braking on a car that is locking up the rear brakes without flatspotting the tires and introducing sliding friction instead of rolling friction. It has nothing to do with the driver's ability.

This is a link to one of the best books that I have on the subject. It has more than enough material to satisfy most engineers and is also very helpful for others:

http://www.powells.com/cgi-bin/biblio?inkey=4-0973432004-0

 

[Mech_Engineer]

Broadly sweeping generalization in engineering discussions assume other variables constant.

Well, the problem with holding all other variables constant is that would mean you are taking away or adding weight to the same car, not comparing two completely different cars.

Think of it as a scientific hypothesis. You state your hypothesis "if a car is lighter, it will stop faster"; that hypothesis is easily disproven because there are many examples to the contrary, so obviously the hypothesis is too generalized to be true. Perhaps if the hypotheis were "given two identical cars in every respect save weight, the lighter one will stop faster" it would be closer to the truth.

I guess I'll just leave it at in general, its a bad idea to make generalizations

Well its very important as different compounds take different about of time to heat up to the temperature where they have the coefficient of friction to lock the wheels. The time that takes to heat up is short, but important if your comparing a brake test.

I seriously doubt that would have any measurable effect when compared to all of the other far more significant variables in the test.

Formula 1 car brakes and tires can performance an INCREDIBLE 5 g's of braking! However, they have like a .5 second delay where there is almost no braking force from the "pads." It's interesting and makes the driver apply the brakes before they need them.

I'm willing to bet Formula cars only achieve those kind of decelerations at high speeds where aerodynamic dorwnforce is very large, so they have more downforce on the tires and hence more frictional force available. The same is true for cornering, where they can only achieve maximum cornering ability once they have reached sufficient speed.

Thats true. What I learned in my automobile dynamics class was that tires actually provide the most grip with about 10% slip (both acceleration and braking). I'm not sure why that is, but if that's true, it would mean that an optimal braking performance would have the tires rotating about 90% of the car's speed.

I think that's probably true from what I've seen in high performance speed stops.

I think we are agreeing on most things now, but then our conclusions are opposite. I still retain that a car being lighter will stop more quickly. Since the Saleen doesn't have ABS, it does not prove the point you are making. It is only making the case for ABS. Look at the Ferrari Enzo and Locost 7 Kit car I posted.

The difference is significant, but the problem is this ends up being an apples to oranges test. The tests were not done at the same time in the same location. In addition the Enzo probably had street legal touring tires fitted, while it's likely the kit race car had racing stickies, and the race car may have even been tested on a "sticky" race track... The only way to get a really definitive comparison is if they both have the same tires, and are tested on the same surface at the same time.

 

[viperblues450]

Well, the problem with holding all other variables constant is that would mean you are taking away or adding weight to the same car, not comparing two completely different cars.

Think of it as a scientific hypothesis. You state your hypothesis "if a car is lighter, it will stop faster"; that hypothesis is easily disproven because there are many examples to the contrary, so obviously the hypothesis is too generalized to be true. Perhaps if the hypotheis were "given two identical cars in every respect save weight, the lighter one will stop faster" it would be closer to the truth.

I think the latter hypothesis is the one that everyone has been referring to throughout the thread. I hope anyways because you are correct, the first one means nothing.

I seriously doubt that would have any measurable effect when compared to all of the other far more significant variables in the test.

Its true that there probably wasn't much difference between the Saleen and Lambo due to this effect, but this effect is still significant. Most race pads/rotors provide almost no bite until they are warm, requiring several easy stops to get them up to temperature. This is a consideration when racers are selecting pads depending on what type of racing they're doing. If the testers did the run on cold brakes, they would have gotten worse results. I'm sure R&T knows what they're doing though.

I'm willing to bet Formula cars only achieve those kind of decelerations at high speeds where aerodynamic dorwnforce is very large, so they have more downforce on the tires and hence more frictional force available. The same is true for cornering, where they can only achieve maximum cornering ability once they have reached sufficient speed.

Yes that's true, they require the downforce, but they still have amazing brake technology and tire technology to provide the braking and grip to the road. Think about that, that's more than 4X the hardest braking you could ever get from a street car.

The difference is significant, but the problem is this ends up being an apples to oranges test. The tests were not done at the same time in the same location. In addition the Enzo probably had street legal touring tires fitted, while it's likely the kit race car had racing stickies, and the race car may have even been tested on a "sticky" race track... The only way to get a really definitive comparison is if they both have the same tires, and are tested on the same surface at the same time.

From the pictures and several other tests on the 7 kit car, they just use street rubber, in the same league as the Ferrari tires. I know you can't compare these numbers exactly, but I think it shows the idea that low tech and lightweight can out perform or at least keep up with one of the best road cars ever. And for what its worth the 151ft of the Ferrari Enzo was also from a Car&Driver test I think.

 

[viperblues450]

mender thanks for the great post, its great to hear someone with a lot of experience confirm what's being said here.
Good thought about the Saleen too, if the bias isn't dialed it, there is nothing the driver can do.

So are you saying that the lateral tire curve is about the same shape as the longitudinal?

 

[mender]

I'll quote from that book:

"A tire's coefficient of friction, when considering longitudinal forces, is measured as percentage shear rather than shear angle. The "Percentage shear vs Coefficient of Friction" graph is much the same as the "Shear Angle vs Coefficient of Friction" graph. However, the longitudinal graph generally reaches its peak faster. It has a small bubble at the peak and a short flat area before it quickly drops off."

Most of my racing and vehicle dynamics books start off with the tire performance curve because almost all the suspension design and tuning revolves around that concept.

If you want to read some more from Warren's book, look on this page:

http://www.rowleyrace.com/publications.htm

No, I am not selling it but I consider it one of the best books out there and strongly recommend it despite the price.Here's a site that shows the lateral and longitudinal traction combined into another important concept:

http://www.teamassociated.com/racerhub/techhelp/marc/Handling.2.html

The traction circle helps drivers understand what is being demanded of them. I have several data acquisition programs that will map each part separately or together as needed. I use them when allowed (not in NASCAR) to "tune" the driver.