Introduction to the problem
I first tried to think about how rolling tyres work when I was trying to understand why pneumatic tyres give a smaller constant on rougher roads compared to a smooth road, whereas a solid tyre gives a larger constant on rougher roads. My work was described in 4 papers which appeared in MN in 1998. They can be dowloaded here. The paper relevant to the present discussion is number 3, in which I tried to model the rolling pneumatic tyre. I derived a formula which enables one to calculate the effective rolling radius from the tyre radius and the axle ground separation. The formula depended on the assumption that there was no slipping between the road and the contact patch. I here call this THE NO SLIP CASE.
I did not succeed in finding the cause of the variation in calibration constant with roughness, but I did eliminate one possibility. I concluded:
quote:Surface roughness effects probably arise in the region near the point of first contact between the wheel and the ground where they affect the amount of initial circumferential compression of the tyre. I speculate that there are three possible causes:
1) tyre deformation extending beyond the point of first contact,
2) road height irregularities modifying the geometry of initial contact,
3) varying skidding at the point of first contact.
After reconsidering the analysis which appears on page 5 on Paper 3, entitled "A Deformed Rolling Tyre:Effective Radius", I think the derivation there still stands. I would now clarify conclusion 3 and say that a cause of calibration variation may include 3) varying skidding in the region very near the point of first contact.
Prompted by Mark Neal I have now examined two other possible cases:
1. The case given by the MOTOR BIKE EQUATION for which Mark has pointed out a geometrical/trigonometrical derivation. This case must some involve slipping over part of the contact patch. The Motor Bike equation also enables one to calculate the effective rolling radius from the wheel radius and the height of the axle above the ground.
2. The case when there is NEARLY ZERO FRICTION CASE. In this case the tyre circumference does not change length, and the effective rolling radius in equal to the tyre radius.
THE NO SLIP CASE
In this case I find that the effective rolling radius is equal to the axle/ ground separation. When the tyre meets the ground at an angle at the and of the contact patch it is slightly compressed and the friction forces hold it at a constant value of strain as it passes under the axle. If the strain were to vary then the tyre would need to slip, contrary to the assumption I made for this case.
MOTOR BIKE EQUATION CASE
In this case when the tyre meets the ground it does not stick, there is a region for which it slips and where it is compressed by the forces provided by the coefficient of sliding friction. As the piece of tyre moves through the contact patch towards the axle it encounters an area where it stops sliding, the static friction is then sufficient to provide the necessary forces. In this case the effective rolling radius is intermediate between the NO SLIP CASE and the NEARLY ZERO FRICTION CASE
NEARLY ZERO FRICTION CASE
In this case the tyre slips everywhere throughout the contact patch. We assume just sufficient sliding friction to ensure that the wheel turns. Since the longitudinal circumference of the tyre is not significantly changed the rolling radius equals the tyre radius.
Conclusion of the work
In the second post of this thread I posted the results of an experiment to measure the axle ground separation. When I considered the variation of constant I find between riding and pushing the bike, I found that for my pneumatic tyre the riding calibration fits THE NO SLIP CASE, and excluded the MOTOR BIKE EQUATION CASE and the NEARLY ZERO FRICTION CASE.