Didn't really understand what you wanted to know X_ouch. Here's a bit of physics from Doug Muir, who has a grasp of these things:
Unfortunately, doing accurate physical modeling of pneumatic tires is a very difficult task. To begin with, even for the tires on your street car (to say nothing of the tires found in professional racing series) the equations are very stiff, which means you will either need a small timestep, an advanced integrator, or maybe both.
A simple model of tire forces is what's known as the brush model. As the tire rotates, a new piece of rubber comes into contact with the ground at the front of the contact patch. As that piece of rubber continues to rotate, frictional forces will deflect it from the circular path around the axis of rotation if there is a slip angle (for now we'll concentrate on side slip which gives rise to cornering forces, but a similar analysis applies to longitudinal slip, or spin, which gives rise to acceleration and braking forces). The amount of deflection for a given piece of rubber will be given by its distance from the front of the contact patch and the tangent of the slip angle. Now if we imagine that there is a spring connecting this piece of rubber to the point corresponding to it's natural path around the axis of rotation, we have a force of magnitude k*x*tan(slip) where k is the spring constant and x is the distance from the front of the contact patch. Integrating this force over each piece of rubber in the contact patch gives us the total force generated by that tire. The basic integral will look something like:
_ x = l _ y = w
F =_ / _/ k * x * tan(slip)
x = 0 y = 0
That is the double integral from x to l (where l is the length of the contact patch) and from y to w (where w is the width of the contact patch) over k * x * tan(slip). Noting that the expression we're integrating doesn't depend on y, we can rewrite it as:
_ x = l
F = _/ k * w * x * tan(slip)
x = 0
Which reduces to
F = 0.5 * k * w * l * tan(slip);
Now if that's all we did, we would get cornering forces (assuming our simulator was stable, as I mentioned, for real tires k*w is a large quantity). However, this model lets the cornering force grow purely as a function of slip angle. Going back to our model though, the deflection is caused by friction with the ground -- when the deflection force on that piece of rubber exceeds the frictional force, it will no longer continue to deflect with the tangent of the slip angle. The point at which it will no longer deflect is: k * w * d * tan(slip) = u * N / l
where d is the distance from the front of the contact patch where the rubber will start to slip, u is the coefficient of friction between the rubber and the road, and N is the normal load on the tire. Assuming that the coefficient of friction is constant, our new equation for the tire force is: F = 0.5 * k * w * l * tan(slip) when (d >= l)
F = 0.5 * k * w * d * tan(slip) + (l - d) * u * N / l when (d < l) In the real world, the coefficient of friction will depend on many factors, including the temperature of the rubber, the normal load on the tire, the rolling speed of the tire, the sliding speed of the rubber, and of course the composition of the rubber and of the surface it's in contact with. Modern racing tires can have coefficients of friction well over 1.0.
Anyway, this is a simple model of how a tire works, useful mainly to demonstrate how complex it can be to do an accurate simulation of vehicle dynamics. Excellent references for this type of modeling are:
Tires, Suspension and Handling
John C. Dixon
ISBN: 15609-18314
Race Car Vehicle Dynamics
William F. Milliken
Douglas L. Milliken
ISBN: 1-156091-526-9
(sorry about the misshapen integral signs.-BT)
