40 Electric and Hybrid Vehicles overall behavior of the vehicle. The traction properties at the tire–road interface are fundamental in evaluating the dynamic behavior of a vehicle. The study of tire mechanics helps understand the mechanism of force transmission between the axle and the road. The subject is complex, and sev- eral theories have been proposed to provide the understanding of the process involved [1–3]; a basic overview is presented in this book. 2.7.1 s lip The friction forces that are fundamental to vehicle traction depend on the difference between the tire rolling speed and its linear speed of travel.
The tire rolling speed is related to the wheel angular speed and is given by v r tire w = ω h wh where w wh is the wheel speed and r wh is the driven wheel radius. The wheel speed of travel is equivalent to the vehicle translatory speed v . Vehicle linear velocity v and the tire speed v tire differ in magnitude and direction as shown in Figure 2.17. The angle between the tire velocity and the vehicle velocity is known as slip angle α , shown in Figure 2.17 with exaggeration. The difference in speed itself is generated due to the tire properties and the interaction of forces at the tire–road interface.
The tire forces do not act through a point but are distributed over the contact patch area of the tire with the road surface. Furthermore, the forces are not uniform along the contact patch in either longitudinal or lateral directions. The vehicle traction force F TR works in the longitudinal direction. Longitudinal and lateral forces will be discussed further later; first, let us discuss the vehicle slip in the longitudinal direction. The tire treads gradually enter the contact patch of a moving vehicle when traction torque is applied to the driven wheel.
The tire treads in front end of the contact patch undergo high levels of compression compared to the rear end due to the traction torque applied at the axle. The rolling of the wheels and the traction torque also causes a shift in instantaneous center of rotation. The side- wall of the tire simultaneously goes through shear deformation. The distance that the tire travels when subject to propulsion torque will be less than that in a free rolling tire due to the compression. This results in a difference between the wheel angular speed and vehicle speed, which is known as longitudinal slip or simply slip .
Slip is defined as the ratio of linear vehicle velocity and the spin velocity of the wheel. Mathematically, slip s of a vehicle is given as v s = − 1 (2.24) ω wh r wh where v is the vehicle velocity. Since vehicle velocity v is smaller than wheel angular velocity converted to linear velocity ω wh r wh , the slip is a positive number between 0 and 1.0 for propulsion. Slip is often expressed as a percentage. v tire v F TR F lateral FIGURE 2.17 Speed and forces in the tire–road contact patch area.