When we drive a car through a curve, all wheels must turn around the same instant center of rotation, allowing better control and minimizing tire wear. Each front wheel must therefore follow a path with a different radius, being obviously larger for the outer wheel.
The instant center of rotation is the point around which a body rotates in an specific moment. The line connecting the instant center of rotation and any point of the body must be perpendicular to the velocity vector in that point.
According to Ackermann, the angle formed by the outer wheel with the extension of the rear axle (α) must be smaller than that formed by the inner wheel and the rear axle (β).
If a vehicle is designed not taking into account the Ackermann steering rules and the two front wheels turn with the same angle, they will not have the same instant center of rotation. This will lead to stability issues and excessive tire wear.
If the Ackermann principle is applied, the inner wheel will turn with a bigger angle, so that the instant center of rotation is the same for all wheels.
To achieve this effect, the steering links will form a specific angle with the longitudinal axis, as described in the image below.
Ackermann is calculated using the following formula:
Track width/Wheelbase= cotg(α) – cotg(β)
From where we can obtain:
Ackermann = arctan ( Wheelbase/ (Wheelbase/tan(β) – Track width) )
Percentage = 100 · ( α / Ackermann )
100% Ackerman means that the extensions of the connecting rods merge in the center of the rear axle. If the percentage is >100, they will merge in front of the rear axle; and behind it if the percentage is <100.
In the real world we have to take into account the tire deformation, known as slip angle which is the difference between the turning angle and the angle that the contact patch actually acquires due to the forces exerted on it.
A Pure Ackermann steering is no longer used in modern cars. The explanation to this is that, because of the lateral mass transfer, the wheels in the inner side of the curve (where the load is lower) require less slip angle to reach their limit of adhesion.
To get the desired effects on the geometry, toe concept must be used. When talking about toe in, we mean that the wheels will have convergence towards the center of the paths they are following, that is, the inner wheel will try to describe a slightly larger circumference and the outer wheel a slightly smaller circumference. With this geometry the slip angle of the inner wheel is reduced, while the slip angle of the outer wheel is increased.
When we have toe out, the inner wheel will try to describe a smaller circumference than the one it’s following while the outer one a larger circumference. With this geometry the slip angle of the inner wheel increases, reducing that of the outer wheel.