Mark:
I can't calculate the precise deformed shape so I cant draw it with certainty. However, by measurement and calculation I do know certain features of it so I could try illustrating these. I really need to illustrate the compression in the tread and for the cases where there is slipping, show this. The only way I can think of doing this is accompanying the diagrams for each of my 3 cases with graphs. This will take time to do, but if it really helps I will do it. However, I cant help feeling that people should be convinced by my experimental result even if I have not drawn a picture:
For the benefit of all I reiterate the experimental steps and the key argument here:
(First, note I have just corrected a mistake I made in rearranging the formula in the second post of this thread. I originally got the formula upside down I had n=0.87. Now the corrected version reads n=(R-H)/(R-Ro), we get n= 3.57/3.1 = 1.15. This is encouragingly close to 1 which is predicted if my thin membrane theory applies. In fact the difference between 1.15 and 1 could easily be accounted for by ..... This mistake does not affect my conclusions.)
1. I measured the calibration constant walking the bike. From this I calculated the effective rolling radius of the tyre when not loaded. This I used as the value for R in the above formula.
2. I measured the calibration constant riding the bike on the same calibration course. From this I calculated the effective rolling radius of the tyre when loaded with my weight. This I used as the value for Ro in the above formula.
3. In the static experiment described in post 2 of this thread, I directly measured how much closer the axle got to the ground when loaded. I did this by measuring the rim to ground distance when I put my weight on the bike. This gave me the value of R-H, H being the height of the axle above the ground when loaded. When there is no load, the axle height will be R.
4. I put the measured quantities into the equation and found that n was 1.17 (+/- 0.2 to allow for my measurement errors). So the true value of n may be anywhere between 0.97 and 1.37, most probably it is very slightly larger than 1, but is certainly not as high as 3 which apparently applies for motor bikes.
5. Ten years ago I developed a simple theory based on NO SLIPPING of the tyre over the ground. This theory results in n=1, so can explain the observed results.
It does indeed turn out that the effective rolling radius is very close to the height of the axle to the ground. If it differs at all it is less than 1 mm larger than the axle height.
It is quite easy for measurers to repeat for themselves step 1 and 2 above. Step 3 is harder to do accurately. I did it with the help of an assistant to photograph the compression of the tyre with my weight bearing on it. I would be interested to hear if others manage to do this. We could then find out how many tyres are similar to mine and according to my theory do not slip over the area of the contact patch (or at least only slip over a small part of the contact patch area.)