0-60 simplified wheel physics and garfield's wheel test

[flyboy2160]

This is a belated reply to an old thread that briefly mentioned how much energy is really used in spinning up the tires to 60 mph, to garfield’s lightweight wheel road test, and to those estimating the accelleration improvement from removing a pound from the wheels compared removing a pound from the chassis.

Some, but not all, of the work in getting the car to 60 is giving it translational kinetic energy (TKE). This is the energy of the whole car plunging straight along at 60 ignoring anything rotating, say like it was skidding on frictionless ice with the engine off. Some more of the work, but not all the rest, is giving rotational kinetic energy (RKE) to the wheels and tires. (I’m also going to do these calculations for engine flywheels and post on the performance forum.) Since work = power x time, if you reduce the work required by reducing these energies, you get to 60 faster.

First, I’ll compare 1 lb on the car with 1 lb on a wheel:

The TKE = 1/2 x the mass x (velocity squared). For a 1 pound mass (lbm) on the car going 60 mph (= 88 ft/sec), the required TKE = [1/2] x 1 lbm x (88 x 88) ft ft / sec sec = 3872 lbm ft ft / sec sec

The RKE = 1/2 x the rotational inertia x angular velocity squared. The rotional inertia (RI) = mass x (radius squared). The angular velocity (w) is in radians = rev/sec x 2 pi radians/rev.

Assuming that the wheel and tire mass is lumped at a diameter of 18” (= radius of .75 ft), a 1 lbm on the wheel has an RI = 1 x (.75 squared) = .562 lbm ft ft.

With a tire circumference of 76” (= 6.33 ft), 88 ft/sec is 13.9 rev/sec. w = 2 x pi x 13.9 = 87.3 rad/sec

So RKE = 1/2 x .562 lbm ft ft x 87.3 x 87.3 / sec sec = 2143 lbm ft ft / sec sec.

Taking 1 pound off a wheel saves both TKE and RKE. But the ratio compared to taking a pound off the chassis is only 1.6 = (3872 + 2143) / 3872. That is, taking 1 pound off a wheel is the same as taking 1.6 pounds off the chassis FOR THE PURPOSE OF ACCELLERATION. Sorry, but guys hoping for 5 sec. 0-60 times by lightening the wheels and tires just aren’t gonna make it...

As a sanity check, let’s see how this works with Garfield’s lightweight wheel test.

Assuming a car weight of 2800 lb and a weight of 200 lb for his set of 17" s-lites and runflats, the calculations above give that the wheel and tire RKE is an additional 4% above the TKE to get the car to 60. (For just doing a ratio, you can use the weight in pounds instead of the pounds in mass. The pounds weight ratio is 430,000/10,842,000. And no cracks from the Newton-kilogram-meters/sec guys)

Garfield saved 11 lb per wheel and, I’m guessing, 2 lb per tire, for a total savings of 52 lb. This reduces the TKE required by 52/2800 = .019 = 1.9% He also saved 52/200 of the 4% wheel and tire RKE, for another 1% savings. With a total savings of 2.9%, the reduction from his 'stock' 6.9 sec time should be about .029 x 6.9 sec = .2 sec.

Which is just what he did.... I guess he's not a national level driver(champ?) for nothing, since “he got all that Mother Nature offered”

An aerospace coworker once told me "You may choose to ignore the laws of physics, but the laws of physics never ignore you." I add "But sometimes you just do the math wrong or pick the wrong law...."

[gowest]

SCC did a story about this, wheel wt. vs. chassis wt., about two years ago. I think the ratio they came up with was about 2.0 vs. your 1.6 but I'm not sure.

What everybody failed to notice about Garfields test was he didn't just change wheels, he changed tires, and it was a big change. The Falkens he did his fastest runs on give about the best grip of any street tire out there AND are 23.5" in diameter, much smaller diameter than the factory tires and this made way more difference than the change in wheel weight.

You may want to take a look at the numbers again.

[flyboy2160]

i noticed that he changed tires, but i relied on his driver feedback [ which is what the engineer is supposed to do: listen to the test driver or the test pilot]. to quote him:

"But, in doing this test, I definitely think the compound of the tires isn't that far off. The Pirellis are definitely soft, just not quite as soft as the Azenis'."

"It's obvious to me that the rotational mass was the factor to get moving quicker (this is also why the first run on the 17's was significantly slower, it bogged a little at just above 3000rpm)."

i just tried to estimate what effect the weight reduction would have, all other things being equal.

i never said .2 isn't significant. hey, i may buy centerlines and lighter tires myself for the street just to get it.....but there are some guesstimates around here about how much 0-60 improvement you can get just by lightening the wheels and tires. and the answer isn't 2 seconds (to get you down to 5 seconds 0-60.)

you'll have to take weight out of the chassis and/or add horsepower.

[DaveNagy]

Nice write up. Flyboy, have you had the chance to take a look at the spreadsheet I linked to in this other thread? I was wondering if the gentleman who created it is using the same formulas as those you are relying on. It would appear from his figures that the RKE/TKE ratio can be up around 4:1. Perhaps you can shed some light on where your two models diverge from each other. (I have a hard time following either of the methods, so any light you could shed would be appreciated.)

Here's some factors that might come into play:

The spreadsheet models tire weight seperately from wheel weight. This might be important because it's easy to drop 7 lbs. off the weight of the tire alone. (Runflats are heavy!) Garfield's tires aren't particularly light either, but he probably saved 3-4 lbs. That weight could probably be "lumped" out around the 21" diameter. (Your method of lumping the total weight out aound 17" seems plausible too.)

Also, exactly how quickly the tire/wheel is accelerating is important, isn't it? Much of the "perceived" benefits of lighter wheels are reported during high acceleration at relatively low speeds. (That's mostly because high accelerations are really only possible at low speeds. In first gear, in other words.) This makes me suspect that heavy wheels "consume" a greater chunk of power when the car (and wheel) is accelerating quickly.

Yes, I plugged a couple different 0-60 times into the spreadsheet, and got very different RKE/TKE ratios. A 10 second run gave a ratio of around 3.5:1, whereas a 5 second 0-60 run spit out something closer to 6:1. I can replicate your 1.6:1 ratio by assuming a very slow 0-60 time. (40 seconds) What rate of acceleration did you use? Would your ratio come out different if you assumed a different rate of acceleration? Say, the very fast 5-15mph rate, as opposed to the much slower 50-60mph rate?

I will soon be able to give you some more real-world data. I'll be measuring/graphing my car's acceleration at various speeds with both the stock 17" runflats, and with a tire/wheel package weighing about 33 lbs. I will seek to minimise traction and driver-technique factors, so it should give us more to look at than Garfield's 0-60 times, which were also determined by traction and launch technique. I should have something to post in another couple weeks. (Getting my car tomorrow!)

-Dave

[flyboy2160]

OL I knew somebody would ask that. The energy ratio stays the same as the speed varies. The TKE depends on the square of the speed. The angular velocity varies directly as the speed and it, too, gets squared.

Just to be sure here are the numbers for 10 mi/hr (= 14.7 ft/sec = 2.3 rev/sec)

TKE = 1/2 X 1 lbm X (14.7 squared) ft ft/ sec/sec = 108 lbm ft ft /sec sec

W= 2 X pi radians/rev X 2.3 rev/sec = 14.6 rad/sec

RKE = 1/2 X .562 lbm ft ft X (14.6 squared) /sec sec = 59.8 lbm ft ft/sec sec

So the ratio is (59.8 + 108)/108 = 1.6 (to 3 figures it’s 1.55, but it was xmas and my 18 in lumped distance might be a little low, so I rounded up.)

Yes, it’s way, way more important to take weight off near the outside of the wheel/tire. Since the RKE varies as the square of radius of the mass from from the center, taking 1 pound off the tire tread (say at a radius of 10 in.) is about equal to taking 25 pounds off the hub (say at a radius of about 2 in.) (10/2) squared = 25!!!!!!

I didn’t want to do a bunch of detailed calculations figuring out the inertia of the wheel and tire. For my examples, I just assumed everything was lumped 1/2 in beyond the rim, at a diameter of 18 in. I admit this could be off, but as my favorite college professor said “the most important calculation an engineer does is the first one on the napkin. You don’t want 3 significant figures. you want the right order of magnitude and 1 significant number. You want to know if the answer is .06 or .6 or 6 or 60 or 6 million.” That’s what I’m after here.

I’ll look up the spreadsheet and try to figure out how it was done.

I’m not trying to figure out how to calculate the 0-60 time from scratch; that’s more complicated. That time depends on launch techique, how much power the engine has at different rpms, the gearing, how long it takes to shift, air drag, etc., etc., and the weights and inertias. I’m saying that given all the other stuff as equal (garfield’s stock runs), what can you expect just by reducing the weight and rotational inertia. And by a quick and easy calculation.

I don’t know what car/wheels/tires SCC was using. For now, for the MCS, I stand by the 1.6 and the .2.

looking forward to your results.

(I wrote the posts out in Word. Copying them into this editor loses some punctuation and the fraction symbols, so I’m going to correct some of those in my original post, indicated by [ ] )

[flyboy2160]

dave,

i'll explain what the spreadsheet is doing later, but at first glance it looks like the author perhaps made a mistake in calculating the tire alone inertias.

i can't find what RA amd RB are, so i'm not 100% sure, but if they are the inner and outer tire radii, the formulas in cells D24 and D28 are wrong in that the author squared both radii and added the results instead of subtracting the inner from the outer. this will overstate the tire inertia by almost a factor of 2.

[Garfield]

I hope you guys have jobs that deal with this kind of thing, or you've got way too much free time

Hey, I just make the numbers, you guys figure out the rest

[DaveNagy]

Give us a little more time, Garfield, and we'll be able to deduce your very existence from first principles!

Actually, I don't understand any of this very well. That's why I was bugging Flyboy to explain how come such different "weight equivalencies" kept popping up.

I see what Flyboy means about those suspicious formulas in cells D24 and D28. I know even less about spreadsheets than I know about most things, but lemme see if I can figure out what those variables coorespond to. (I won't even attempt to work out what the formula actually does.)

Ah ha! If you double click on the the C24 cell, the "real" formula is revealed as:

0.5 * t2M *(w2Radius^2 + t2Radius^2)

This is supposed to equal the rotational inertia of the tire alone. Excel's nice color-coding system leads me to believe that:

t2M = the second tire's mass, in 'slugs'.
w2Radius = the second wheel's radius, in feet.
t2Radius = the second tire's radius, in feet.

So yeah, since the wheel's radius is the same as the tire's "inner" radius, this does seem wacky.

I'll try changing the formula to what Flyboy suggested and see what happens... Good, that seemed to reduce the calculated "torque difference" between the heavy wheels and the light wheels by about 30%. That drops the weight ratio that we've been discussing into the 2.6-3.2 range, depending on how the weight distibution is modeled. Still above what Flyboy is betting on, but quite a bit closer.

Oh, but I've been modeling lighter tires than Garfield's. Lemme change that. Nah, I also was assuming a lighter car. When I plug in the same weights as Flyboy used, I get a ratio up around 4:1 again. Sigh.

I still think it has something to do with the rate of acceleration. (Not speed, acceleration.) Are we sure that rotational inertia scales the same way that linear inertia does? (I'm just making up terms now!)

-Dave

PS. My car's delivery was delayed. Again. But I'm assured that I'll get it tomorrow. (I've heard that before!)

[DaveNagy]

Oh, while I'm here: What would you guys recommend as a test methodology when I get my car? I want to compare acceleration between the heavy stock wheels, and the lighter aftermarket wheels. Things to keep in mind:

• The car won't be broken in, and won't be until after the heavy wheels are long gone.
• I've already sold the heavy wheels/tires, so wheelspin is a no no.

I'm thinking I'll leave the DSC on to keep me honest, and try to release the clutch pretty quickly, even if that lugs the engine a tad on launch. That way I'll get a fairly long 1st-gear "pull" where I've got my foot to the floor (don't tell MINI!) and the clutch fully out. This won't be a fast launch, but it should be pretty reproducible. (I'll "trim off" all the data at the very beginning of the run, so it'll sorta be like a "street start".)

I hope to end up with "before and after" 7MPH-to-35MPH pulls that vary only due to changes in tire/wheel weight and, perhaps, temperature. I'll try to do the runs at similar ambient temperatures, but what do I do about controlling the intercooler temp? Should I do both runs after letting the car cool off for a while? Or should I do multiple runs back to back and only compare the "hottest" runs? What's the best way to cool the intercooler off? Let it sit, or drive down the road at a brisk but steady speed?

-Dave

[BrantV]

flyboy,

Nice post.

Although your calculations can't account for gained traction on non perfectly smooth surfaces in the real world. A potentially larger gain from wheel weight loss. The lower the weight, the faster the suspension reacts, and the more time the tire spends in contact with the ground. Less wheel hop should give you faster acceleration

Brant

[Wynn]

Flyboy,

Thanks for your fine assessment. I think that for MOST of us-- those who aren't racers and guys with precise testing equipment-- the conclusion we can draw comfortably is that the weight of our wheels and tires will not have a perceptible impact on our accelleration. Choosing MINI's smaller wheels, relative to the heavier 17" S-spokes, should have about as much impact as eliminating the sunroof (a difference of 70 pounds, apparently).

I had been concerned about the weight of the 17s before your report, and now feel comfortable knowing that I won't notice the difference. 0.2 seconds is statistically meaningful, but practically meaningless to drivers on the street.

Cool!

Thanks,
wynn

[Garfield]

Keep in mind that loosing unsprung weight is not about 0-60 numbers, it's really about handling. THAT is where you'll notice the bigger difference, if you care about handling. Hitting small bumps around turns is inevitable. With heavier wheels, your suspension cannot react nearly as fast, and therefore you loose grip or skid much sooner.

Brian

[dpayne1]

Not to mention the improvements in braking............

[flyboy2160]

his [correct] answer is 1.5 to 2.0

http://www.miataforum.com/ubb/ultima...c;f=3;t=006390

after looking more closely at the s lites and finding out that the sidewalls of the runflats aren't as thick as i thought, my suspicion that my 18" diameter was a little low was well founded. my guess now is that the wheel tire mass is lumped closer to 20". so the chassis equivalent number is closer to the 2.0, say 1.8.

my point again, though, is that the ballpark number is in this range, NOT 4 or 6 or 10!!!!

glad you guys found this useful and fun!

[tomstork]

Great postings; very entertaining--and educational. What I take from all of this is:
1) You'd have to have a *lot* lighter tires and/or rims to notice much difference in the real world (FWIW, most of the car magazines would not consider 0.2 0-60 difference to be "significant."
2) You'll probably improve your times a lot more by practicing your launching technique and finding sticky tires--regardless of their weight, than by going for the lightest tire/rim combination.

My bottom line? I guess I have a couple of them:
- I can't imagine anything much harder on a car--or worse for the lifespan of the drivetrain--than running a lot of all-out 0-60 tests.
- I'm a lot more interested in having strong wheels than light ones (hopefully both can be good-looking), especially if I'm driving on real-world city streets.

YMMV, of course--I'm really not trying to pick a fight. Thanks again for the work, Fly Boy!
- T.