More philosophical thoughts about rolling resistance--
Let's assume that it is true that the rolling resistance is proportional to the amount of flexing going on in a tire. One measure of the amount of flexing would be the angle formed between the flat contact patch and the tangent to the tire tread face just outside the patch. Let's call this angle "alpha." (If you want to draw a picture, make a circle, then draw a flat spot on it for the contact patch. Alpha is the angle between the original circle and the flat (chord) line, or it's half the angle you get by drawing lines from the ends of the patch to the center of the circle.)
So, then it should be true (our assumption) that the rolling resistance is proportional to alpha:
RR = F alpha,
where F is the "flexing constant" of the tire.
As mentioned before, the load supported by the tire is equal to the tire pressure times the area of the contact patch:
L = P A
This time, we need to further describe the contact patch-- the area of the patch is equal to its length times the tread width, well roughly so:
A = W C
Where W is the width of the contact patch and C is the length, the chord across the circle. Now, I want to invoke a little geometry. The formula for the length of the contact patch is:
C = 2R sin(alpha), or C = 2R alpha
for small angles, that is for a small contact patch. Putting all this stuff together gives a relationship between the load L and the tire pressure, tread width, tire radius R, and alpha:
L = P W 2R alpha
Way up at the top we had a formula for the rolling resistance and now we have a formula for the load. Dividing one by the other gives a formula for the rolling resistance coefficient:
Crr = RR /L
Substituting,
Crr = F /(P W 2R)
Alpha was in both equations and cancelled out, that's cool. This is an interesting formula, but we have to be very mindful of F, the flexing constant. This "constant" depends on the construction of the tire. There is experimental evidence that it Might be true that F is proportional to the tire width, which is just about the same as W. This means that a tire's rolling resistance coefficient Might not depend on how wide the tire is. (But, remember, the aero drag DOES depend on tire width.) The only other new variable to talk about is the tire radius, R, or the tire diameter, 2R. Now, we have an expanded very interesting relationship for rolling resistance and tire properties:
Rolling resistance is proportional to one over pressure, and one over tire diameter.
Provided that we are talking about tires of identical construction except for the tire diameter.
Nothing has been said here about "dynamics", how the tire and vehicle behave when the car is accelerating, turning, and hitting bumps. (We have lots of those in Pleasant Grove, Utah.)
When choosing tires, dynamics considerations can affect choices too.
Ernie Rogers