Originally Posted by AZE92M3
I think I'm going to have to respectfully disagree with your first point. A 1 inch larger diameter rim that is overall 1lb heavier with 3lbs less wrapped around the outside, rear wheel example, should rotate much easier. The rim will play a much lesser roll than what's wrapped around the outer edge of it because that extra 3lbs is at the farthest point. The 1lb is only an extra .5inch farther from center. The 3lbs of tire is much farther from center and is 3 times heavier than that 1lb of rim. Plus the overall weight of the rim & tire is 2lbs less than the 19.
If you look at the wheel & tire from the center out this is what you get. Take the 20" setup for example. The overall diamter is 26.73 or a radius of 13.365 inches. Zero to 9.5 inches from center they are the same(19 inch diameter). Then the next .5 inch is 1lb heavier. Then the next 3.365 inches is 3lbs lighter than the 19" setup.
I'm not so sure about your second point except for the fact that each car is engineered from the factory to each specific sidewall. Otherwise your argument wouldn't hold any water for the Porsche 911 GT3 or the new Corvette ZR-1. Both of which use tire diameter or sidewall depths which violate what you say. Thinner sidewalls allow the suspension engineers to more accurately dial in a suspension because there is less tire variable to compensate for.
Fair enough. I will respectfully disagree back, but this time with a wealth of evidence to make my case.
Have a look at the table below. I have made a nice spreadsheet (and even attached it for the real nerds) to calculate wheel moment of inertia (among other things). As you can see in your particular cases colums D vs. E and F vs. G the larger wheel (despite not
being quite apples to apples in terms of widths) uniformly have MUCH larger moments of inertia. The moments are larger because of the effect of the wheel, not the tire. The larger wheels have a much higher concentration of mass at a large distance away from the center of rotation. I can't quite follow/decipher your "reasoning" above but it is not correct and does not arrive at realistic numbers, trends nor conclusions. Your insight here is simply flawed.
I should add some key assumptions in my calculations (again for all the real nerds that may scrutinize the details):
1. I assumed the cross section of the wheel is a squared "U" on its side, i.e. with no spokes but solid. This is a fair assumption as you can make that section of a uniform thickness that approximates tapered spokes as are present in most real wheels.
2. I also assumed that a tire is a squared U (rotated 90 degrees clockwise to mate with the wheel).
3. I used the "thin shell" volume approximation for the barrel parts of the wheel and tire for volume calculations.
4. The thicknesses of the various sections are reasonable educated guesses, but as you can see they give very reasonable values of component weights, in many places properly predicting which wheel/tire is heavier/lighter compared to your data. As well you can tweak these values around quite a bit and the overall conclusion about I total will not change.
5. Don't worry about the weight prediction being spot on - it really does not matter much. The moment of inertia is the key parameter here and it depends much more on simply predicting where the wheel is in space and where the tire is in space and having those densities close.
6. I used a reasonable value of .02 lb/in^3 density for a tire (note almost 5 times less than aluminum as per my original point!). This value was determined by computing the volume of my simplified U shaped tire and dividing a known tire weight by this.
As far as my second point in my previous post, I'll disagree again. Your counter example doesn't prove what is or is not good for the M3. See comments from footie above. It isn't that you can't make a car work with less sidewall than the very particular M3 18" tire. It just takes engineering the entire suspension and suspension mounting elastomers and other minutia AROUND the particular choice of tire. And, as I said, for the M3 the 18 is ideal.
Once again (for the E9X M3)....
18s for performance
20s for looks
19s for a decent compromise
And of course the driver will be an even bigger factor than these differences.