Reply to "Rolling Tire Characteristics"

I am not very good at drawing, it is easier for me to show it as calculations and arguments, because I don't know how to show the compression of the tyre in a drawing. Here is a sketch roughly to scale. I have magnified the area around the contact point by 20 times so that you can see the difference between the 3 cases. It is only a range of 0.9% between the No-SLIP CASE and the NEARLY ZERO FRICTION CASE - these are shown by dashed lines when not touching the ground but I cant calculate exactly the shape assumed there. The MOTOR BIKE CASE lies between these two extremes and is shown by the heavy line in the magnified region. The 0.9% change in the contact length does change your angle alpha at the wheel axle, but the effect of the change on the geometry is negligible and in particular it does not affect the axle ground height significantly or the rate of feeding uncompressed tyre into the contact area.


I am not very familiar with the literature. When I searched 11 years ago in libraries, I found very little. Searching is so much easier now since so much more is found by Google (in fact I dont think I used Google then which was just in its infancy as a University project). What a change in 11 years. Google now turns up numerous text books as well as a lot of research papers. I cant say whether anyone has treated the rolling bicycle wheel. The purpose of nearly everything I have seen so far is treating the problems of how to accelerate, brake, or corner a car, or a motorbike. There are many papers which I have not looked at on steel wheels rolling on steel rails, which I have assumed would not be applicable. I am not surprised we have not so far found someone treating something equivalent to the measuring bicycle wheel. We may well have to go back to the era of Whipple's 1899 paper to find a treatment of the simple rolling-no slipping case.

Now I have a better idea what to look for I will spend some time in the book shops in Cambridge on Sunday. I will also search the library indexes for Oxford. But these are rather long term projects. It would be really helpful if anyone can lay their hands on a good treatment of the bike wheel.

You ask:In order for R=H, the length of the contact region needs to be shorter than what I have shown in Figure 1. How does this happen?

It is shorter because the tyre does not slip and its longitudinal compression is constant everywhere along the length of the contact patch. For the motorbike formula to be true, the friction with the road is not able to provide the necessary compression of the tyre near the end of the contact patch so the tyre slips and has a lower average value of compression, this means that the contact region is in fact longer than the no slip case, which has a 0.6% smaller value for your angle alpha. There have to be changes to the tyre geometry outside the contact patch which enables this to happen without changing H. As I see it the whole non contacting part of the tyre changes slightly so that it is no longer an arc of a circle.

This is what I am finding difficult to calculate and then draw. We would be able to do it with a full finite element model of the tyre, but I do not have access to any finite element software. Is there free software available? Even then I should think such a model would take a week or so even in an expert practitioner's hands. I would rather rely on my experimental data, and it would be great if someone else could get measurements to compare.