# Steering Pull Model and Its Sensitivity Analysis
## 2 Tire Pulling Force
### Plyseter and Conicity Forces  
When a vehicle turns, it has one center of rotation, and thus each tire has its direction of travel. Each tire also has its direction of heading, on the other hand, and the angle difference between the direction of travel and heading is called'slip angle' [10]. Due to this slip angle, the tread sideways with respect to the tire and this yields the lateral force in the contact patch [11]. Accordingly, in vehicle dynamics, the lateral force of a tire is zero when its slip angle is zero. However, in many pneumatic tires, the lateral force is also caused by the structure of the tire such as the asymmetry of carcass and tread. This lateral force by the tire structures is called'residual cornering force' [12].  
Figure 1: Centripetal force due to road bank angle.  
Figure 2: Pulling distance.  
The residual cornering force of a tire results from the tire's plysteer and conicity. A typical radial tire structure is made with multiple layers of plies bonded together. If the tire rolls, the shape of the tire becomes flat at the contact patch, which causes the generation of lateral and longitudinal shear stresses in the contact area. These shear stresses cause coupled reaction forces which is called 'plysteer force'. This plysteer force is generated in typical radial tires under straight rolling conditions, and it is an inherent property which means the nonzero lateral force at zero slip angles [13]. The acting direction of a plysteer force depends on the rolling direction of the tire as shown in Figure 3a. Generally, a plysteer force is constant for a given tire design and has very few variations between samples but it rather varies with tire design.  
The conicity force of a tire is caused by manufacturing errors and can be explained by a rolling cone [11]. The acting direction of conicity force does not depends on the rolling direction of the tire as shown in Figure 3b. For the tires recently produced by modern tire manufacturing systems, the conicity force is known to be very small due to high precision manufacturing processes, and when two tires from the same manufacturing system are installed in a vehicle, they usually have the same direction and magnitude of conicity force. In addition, when installed in the opposite direction, they offset each other, which results in a counterbalance. For these reasons, the conicity force is neglected in this study.  
On the other hand, the plysteer forces of left and right tires have the same direction and therefore, they need to be considered in the tire pulling force model. There have been several theoretical approaches to predict the plysteer force of a tire, as mentioned in Chapter 1, but this study employed experimental values of a tire in order to build an accurate steering pull model. The plysteer force of a tire can be obtained by averaging the plysteer values which are measured in two opposite rolling directions. The plysteer force of a tire is (Equation (4))  
\[F_{y,ply}=\frac{F_{y,ply,cw}-F_{y,ply,cw}}{2} \tag{4}\]  
where, \(F_{y,ply}\) = plysteer force, \(F_{y,ply,cw}\) = measured plysteer force in clockwise direction, \(F_{y,ply}\) = measured plysteer force in counter clockwise direction.  
Figure 3: Acting direction of residual cornering force (**a**) ply-steer force (**b**) conicity force.