The vehicle dynamics models i.e. the complex interactions between the tires and the road are the factors that determine the motion of the vehicle. We don’t delve too heavily into the mathematical details here. Rather we’ll focus more on the intuition of what these forces do. Out of the many forces in a dynamic model, they essentially boil down to lateral and longitudinal forces acting on the vehicle. Think of the two forces i.e. longitudinal & lateral force moving the vehicle. Longitudinal force is the force pushing the vehicle forward or backward and lateral force is a force pushing the vehicle side to side. Tires play a huge role here. Forces on the tires is the sum of the longitudinal and lateral force on the vehicle.
The slip angle can be interpreted as the angle between the velocity vector of the wheel and the orientation of the wheel. You’re probably familiar with the sound of a tire squealing in a parking lot. That’s a result of having a slip angle. The slip angle can be expressed as the arc tangent of the wheel longitudinal velocity divided by the wheel lateral velocity. The vehicle actually turns due to the force generated by the slip angle. Otherwise, inertia would carry the vehicle off the road. Different tires have different characteristics though. Racing tires generate much more force from the same slip angle than conventional tires.
The ratio between the speed of the vehicle wheel and the expected longitudinal velocity is called slip ratio. This means, there is always some slipping when the vehicle is moving on roads. So, there’s also a slipping motion in addition to the tire’s natural rolling motion,. As the slip angle is required to generate lateral force, the longitudinal force is generated by slip ratio. In either case, all of this begins when the contact patch of the tire meets the road.
Race car drivers have a fantastic mental model of the graph and can usually capture optimal longitudinal or lateral force for the situation. The slip angle is the only one component used to calculate the lateral & longitudinal forces of tires. There is a lot of literature around tire dynamics. We could have an entire course just on tire forces. The most popular Magic Tire Formula model.