Regression analysis for kg/hp vs. Ring laptimes

[lucid]

Originally Posted by swamp2

Thanks, files seem to be garbled. Have a look at the data in a text editor. Can you post in .csv format? I thought you were going to try a multi-variable regression? If you get me the data in a readable format I will certainly do it.

Data are fine. It's tab delimeted text. Excel should be able to read it fine if you tell it that. I can try tonight.

[bruce.augenstein@comcast.]

Originally Posted by enigma

Hmm, another good grip aproximation is:

(Avg rim width) / (LBS)

The idea being its not simply weight but weight per area of tire on the ground that deterimins grip. The Nissan, Lotus, Z06 and some Porsche have really wide wheels for their weights.

The weight per area of tire on the ground is equal to the pressure in the tire divided into the weight on that tire. Wheel diameter, width and tire size doesn't matter. If the tire is supporting, say, 800 pounds, and is inflated to 40 psi, the contact patch will be 20 square inches with 40 pounds on each square inch. This will be true regardless of whether it's a 185/60X14 or a 285/30X19. There will be minor variations due to the vagaries of tread and sidewall stiffness, but you can take this to the bank.

What changes is the shape of the contact patch, and this is where your idea has merit. Generally speaking, you want the long axis of the contact patch to be at right angles to where you expect the most force to be encountered. For drag racing, you want the contact patch to be lo-o-o-ng. For the 'Ring, you want it to be wide.

Therfore using tire width over weight (at both ends of the car) would be better than using rim width.

Bruce

[enigma]

Originally Posted by bruce.augenstein@comcast.

Therfore using tire width over weight (at both ends of the car) would be better than using rim width.

Bruce

In theory I agree, in practice it probably not possible. Not all 265 tires are the same width, nor do they have similar contact patches. The reason I would use rim width is its a known and available figure. Getting specific tire sizes then translating those into contact patch widths is a much harder exercise.

Using the rim width assumes that each test was done with a proper sized tire for the rim which is generally safe on production cars.

[bruce.augenstein@comcast.]

Originally Posted by enigma

In theory I agree, in practice it probably not possible. Not all 265 tires are the same width, nor do they have similar contact patches. The reason I would use rim width is its a known and available figure. Getting specific tire sizes then translating those into contact patch widths is a much harder exercise.

Using the rim width assumes that each test was done with a proper sized tire for the rim which is generally safe on production cars.

Variation in tread width between various manufacturers of a given tire size is *much* less than the variation in rim widths that car manufacturers use for that size tire.

Tread width on a 265 tire may vary as much as an inch (probably less), but rim widths may vary at least twice that amount. Rule of thumb says ideal rim width is about the same as (max) tire width, but you will very often find a given car manufacturer go under that figure, often to improve the ride. Porsche, as you point out, is usually generous with rim width, but other manufacturers are often less generous. My last Vette, for instance, had 9" rims with 285 40s. On the other hand, I've seen SEMA cars with rim widths a bit wider than max tire width. They say this gives a more direct feel to the car, with less rollover under load.

Maybe that's why Porsche is often generous on the rim widths? Perhaps.

Bruce

PS - All tires with a given tread width will have the same contact patches at a given pressure and load, again allowing for *very* slight differences due to tread and sidewall stiffness. Therefore, minor variations in tread width at a given 265 size tire will result in equally minor variations in the shape of the contact patch. It's 20 square inches at 40 psi supporting an 800 pound load, so a tire with a ten inch tread width will have a 10 by 2 footprint, while a 9.5 inch treadwidth will dictate a 9.5 by 2.11 footprint. Not a huge difference.

[paulyd]

holy fascinating

[swamp2]

Originally Posted by bruce.augenstein@comcast.

The weight per area of tire on the ground is equal to the pressure in the tire divided into the weight on that tire. Wheel diameter, width and tire size doesn't matter. If the tire is supporting, say, 800 pounds, and is inflated to 40 psi, the contact patch will be 20 square inches with 40 pounds on each square inch. This will be true regardless of whether it's a 185/60X14 or a 285/30X19. There will be minor variations due to the vagaries of tread and sidewall stiffness, but you can take this to the bank.

Bruce, absolutely not true.

This is a hugely simplified veiw of a tire, modeling it as a very large linear balloon (no such "linear" balloon actually exisits although a regular childs balloon will be quite close to behaving as you suggest) or as something like the fluid in a perfect hydraulic piston. Tires are stiff, as much solid as balloon like, non linear, full of steel, carry load on their sidewalls, have a non contacting area (the grooves), are more rectangular in cross section than round, etc. So they are highly non-linear from both a material and geometry perspective and are anisotropic as well. All of these things make a tire not behave as you suggest. Contact patch size depends on tire width for automotive tires.

Have a look at this very interesting write up disproving this commonly held "fact" here. The write up does so by providing actual test data that shows your claim is not the case.

Back on topic. I have lucids modified list, the new data provided by jaiman and am now scouring the web for
1. US skidpad results (200 or 300 ft constant radius test)
2. EU (Sport Auto) skidpad test results (using an actual road cornering situation)
3. Cd and frontal area results.

I will post my analysis as soon as I have enough data points.

[sdiver68]

Originally Posted by swamp2

Bruce, absolutely not true.

This is a hugely simplified veiw of a tire, modeling it as a very large linear balloon (no such "linear" balloon actually exisits although a regular childs balloon will be quite close to behaving as you suggest) or as something like the fluid in a perfect hydraulic piston. Tires are stiff, as much solid as balloon like, non linear, full of steel, carry load on their sidewalls, have a non contacting area (the grooves), are more rectangular in cross section than round, etc. So they are highly non-linear from both a material and geometry perspective and are anisotropic as well. All of these things make a tire not behave as you suggest. Contact patch size depends on tire width for automotive tires.

Have a look at this very interesting write up disproving this commonly held "fact" here. The write up does so by providing actual test data that shows your claim is not the case.

Back on topic. I have lucids modified list, the new data provided by jaiman and am now scouring the web for
1. US skidpad results (200 or 300 ft constant radius test)
2. EU (Sport Auto) skidpad test results (using an actual road cornering situation)
3. Cd and frontal area results.

I will post my analysis as soon as I have enough data points.

Gee, I seem to have recalled this argument from 1984 (when I first started scouring the libraries for anything and everything car related), 1990 (internet, ditto), and every so often since

Swamp is right, the balloon model does not hold.

[bruce.augenstein@comcast.]

Originally Posted by swamp2

Bruce, absolutely not true.

This is a hugely simplified veiw of a tire, modeling it as a very large linear balloon (no such "linear" balloon actually exisits although a regular childs balloon will be quite close to behaving as you suggest) or as something like the fluid in a perfect hydraulic piston. Tires are stiff, as much solid as balloon like, non linear, full of steel, carry load on their sidewalls, have a non contacting area (the grooves), are more rectangular in cross section than round, etc. So they are highly non-linear from both a material and geometry perspective and are anisotropic as well. All of these things make a tire not behave as you suggest. Contact patch size depends on tire width for automotive tires.

Have a look at this very interesting write up disproving this commonly held "fact" here. The write up does so by providing actual test data that shows your claim is not the case...

Wow! That is terrific stuff!

I'm scratching my head over some of the anomalies in the data, and still working on the implications in regard to handling dynamics, but thank you!

Bruce

[swamp2]

Originally Posted by swamp2

Back on topic. I have lucids modified list, the new data provided by jaiman and am now scouring the web for
1. US skidpad results (200 or 300 ft constant radius test)
2. EU (Sport Auto) skidpad test results (using an actual road cornering situation)
3. Cd and frontal area results.

I will post my analysis as soon as I have enough data points.

Posted is my work on gathering a decent data set for the multi variable regression. Really busy at work so making good progress here but not great. Anyone want to contribute some internet sleuthing time to a very interesting project? Simply fill in blanks where ever you can find some data to help. Post your results in a zipped .csv format (comma delimited text) and bump the revision number in the file name as well. Thanks in advance to all those who can help.

P.S. Hint: Preliminary analysis shows a large skidpad effect and almost no Cd effect. Speed in corners is key, acceleration and speed in straights and out of corners is covered by power to weight, speed in corners (suspension, tires, etc.) is covered by the skidpad performance. Again little to no effect based on Cd! Note: Despite my better judgement I used Cd without frontal area. Just not enough data points when you need complete rows and columns for the regression and require skidpad and frontal area both.

[jaiman]

swamp2 why do you say cd adds nothing? when i run it without frontal area i get a cd coefficient that is significant above 97% confidence.

[swamp2]

Originally Posted by jaiman

swamp2 why do you say cd adds nothing? when i run it without frontal area i get a cd coefficient that is significant above 97% confidence.

I simply did the following single and multi-variable regressions

1. Full data set lap time vs. power to weight
2. Reduced data set same as above
3. Reduced data set, time vs. p/w and Cd
4. Reducced data set, time vs p/w and skidpad

-1 and 2 were similar.
-3 did not improve the fit or R^2 and there is simply a visual scatter of data on the time vs. Cd plot. Clearly visually uncorrelated.
-2 vs. 4 showed better R^2 redcued residuals and a nice clear trend shown vs. skidpad

[jaiman]

Originally Posted by swamp2

I simply did the following single and multi-variable regressions

1. Full data set lap time vs. power to weight
2. Reduced data set same as above
3. Reduced data set, time vs. p/w and Cd
4. Reducced data set, time vs p/w and skidpad

-1 and 2 were similar.
-3 did not improve the fit or R^2 and there is simply a visual scatter of data on the time vs. Cd plot. Clearly visually uncorrelated.
-2 vs. 4 showed better R^2 redcued residuals and a nice clear trend shown vs. skidpad

try time vs p/w, cd and skidpad, best numbers i've seen yet.

[swamp2]

Originally Posted by jaiman

try time vs p/w, cd and skidpad, best numbers i've seen yet.

You can add variables all day and if each has any known physical mechanism for predicting car performance then regression should improve. Some examples would be tire width, redline, engine weight, ground clearance, brake rotor diameter, etc. These are known mechanisms bit not critical factors. The real question is are each of these variables really contributing in a significant fashion and are they cross correlated? Again my preliminary analysis shows Cd is not. Does the attached plots for a two variable regression, time vs. Cd and W/P (simultaneously) make it clear that one is a very strong factor and the other not? It does not take a math background an in depth look at ANOVA nor other esoteric regression results to see this. Note raw data in blue, predictions of model in pink, regression line in black. Furthermore the adjusted R^2 in my case 1 vs. case 3 (post #143) actually went from down from .86 to .85! In this limited data set Cd made the basic results (very strong correlation to begin with) weaker! Like I said long ago you should use Cd x area!.

[jaiman]

whether cd is as large a factor as p/w, is irrelevant. adding it to the model improves adjusted r square, and the standard error. the individual t stat for cd is also significant above 95% I'm sure adding frontal area would improve things, but cd alone does improve the model.

put another way, if someone came to you with a cars weight, horsepower, cd and skidpad numbers and you wanted to predict its ring time which model would you use?

[ajj]

Originally Posted by swamp2

Furthermore the adjusted R^2 in my case 1 vs. case 3 (post #143) actually went from down from .86 to .85! In this limited data set Cd made the basic results (very strong correlation to begin with) weaker! Like I said long ago you should use Cd x area!.

holy crap....0.86 to 0.85! is this statistically significant??

you're a smart, yet CRAZY hombre swamp...now i remember why i went to med school instead of becoming an engineer...will leave all the serious brainpower stuff to you guys!

[sdiver68]

Originally Posted by swamp2

You can add variables all day and if each has any known physical mechanism for predicting car performance then regression should improve. Some examples would be tire width, redline, engine weight, ground clearance, brake rotor diameter, etc. These are known mechanisms bit not critical factors. The real question is are each of these variables really contributing in a significant fashion and are they cross correlated? Again my preliminary analysis shows Cd is not. Does the attached plots for a two variable regression, time vs. Cd and W/P (simultaneously) make it clear that one is a very strong factor and the other not? It does not take a math background an in depth look at ANOVA nor other esoteric regression results to see this. Note raw data in blue, predictions of model in pink, regression line in black. Furthermore the adjusted R^2 in my case 1 vs. case 3 (post #143) actually went from down from .86 to .85! In this limited data set Cd made the basic results (very strong correlation to begin with) weaker! Like I said long ago you should use Cd x area!.

Relax, no one is arguing that if you have it, Cd x frontal area is better.

[Sedan_Clan]

Fuck! Some of you guys are serious engineering buffs. Good stuff!

[swamp2]

Originally Posted by jaiman

whether cd is as large a factor as p/w, is irrelevant. adding it to the model improves adjusted r square, and the standard error. the individual t stat for cd is also significant above 95% I'm sure adding frontal area would improve things, but cd alone does improve the model.

put another way, if someone came to you with a cars weight, horsepower, cd and skidpad numbers and you wanted to predict its ring time which model would you use?

With the limited data set R^2 DECREASED in going from time vs. w/p to time vs. w/p and Cd! Clearly with this limited data set (33 points) I would choose the single (compound) variable w/p! Hmmm why are you missing this?

[swamp2]

Originally Posted by sdiver68

Relax, no one is arguing that if you have it, Cd x frontal area is better.

Ugh, I am not "unrelaxed". My bolds were not ALL CAPS. I was simply emphasizing some key points. I am simply passioniate about clear applications of math, physics and engineering to understanding performance cars. Even if I was unrelaxed it was not about Cd x Area being better than Cd (that is what I call trivial), it would be that Cd is not relevant in this preliminary analysis.


Carry on.

[jaiman]

Originally Posted by swamp2

With the limited data set R^2 DECREASED in going from time vs. w/p to time vs. w/p and Cd! Clearly with this limited data set (33 points) I would choose the single (compound) variable w/p! Hmmm why are you missing this?

i'm sure you're not trying to come off as arrogant, but could it be that we are comparing apples and oranges? the set i'm using consists of 22 cars, and i posted the raw data earlier. On those cars, the model improved with cd. with your 33 cars it doesn't.

[Expired]

I have a solution.
We make a World Racing Time Foundation, gather all of these cars, get a car company to produce tires with the same treads for each cars tire dimensions.
Make them all race the Nurburg on the same day. (Each manufacturer can use whomever they choose as a driver.)

-Nathan

[swamp2]

Originally Posted by jaiman

i'm sure you're not trying to come off as arrogant, but could it be that we are comparing apples and oranges? the set i'm using consists of 22 cars, and i posted the raw data earlier. On those cars, the model improved with cd. with your 33 cars it doesn't.

We are comparing apples and oranges but then again not really. I expect the result to be mostly independent of the sub sample set of the master set in lucids revised list. Don't you? The physics and math don't really care too much in cased like this on your random sampling technique (yours or mine!)

Just for kicks I ran the same sensitivity analysis I did in my post #143/145 and viola I found the same thing. Perhaps we are comparing apples and oranges as well as far as interpretation of results. R^2 is the most basic measure of the quality of the fit of the regression to the data (or data to regression). Agreed or disagree? With your data set R^2 went from .80 for time vs. w/p alone to .81 for time vs. w/p and Cd. Cd adds (more or less) NOTHING to the quality of the prediction (again...). Comparing to the case time vs. w/p and skidpad the R^2 goes from .81 to .84. Not much but significant. The graphs do not look all that different than the ones I posted earlier to show the effect visually. I think the visual display is key for the non-math/non-science folks to really be convinced. Time vs. Cd is nothing but a random cloud of points, your data or mine!

Conclusion again, on the different sub sample sets w/p is dominant, Cd alone no effect, skidpad small but measureable effect.

Do you still disagree?