41 Vehicle Mechanics The vehicle velocity is then given by v r = − ω wh wh (1 s ) During vehicle braking, the tire is subject to similar compression due to the applied braking torque, and a difference in vehicle speed and wheel angular speed occurs enabling the vehicle stop. The slip during braking is given as ω r s = − 1 . wh wh (2.25) v 2.7.2 T racTion f orce aT T ire –r oad i nTerface The rotary motion of the axles is converted to linear motion of the vehicle at the tire–road inter- face to enable vehicle motion.
When a torque is applied at a driven wheel through the axle of a vehicle, friction forces proportional to the traction torque are developed at the tire–road interface. The forces on the tire are not localized at a point as mentioned previously but are the resultant of the normal and shear stresses exerted by each element of the tire tread distributed in the contact patch. The stress under the tire is distributed non-uniformly in the x- and z -directions along the two-dimensional contact patch at the tire–road interface; this two-dimensional stress distribution gives rise to both longitudinal and lateral forces.
The longitudinal and lateral forces can be obtained by integrating the shear stress in the contact patch over the contact area. The longitudinal force is the traction force responsible for the forward velocity of the vehicle. We will focus our discussion on the longitudinal force; this force has been referred to as traction force F TR earlier in the chapter. The forces on a wheel and the stress distribution at the tire–road interface are shown in Figure 2.18. The stress distribution and the resultant forces are shown in relation with the contact patch area at the tire–road interface.
The traction torque T TR working around the axis of rotation of the wheel results in the wheel angular velocity ω wh ; this traction torque is primarily responsible for the distributed pressure or stress at the tire–road interface. The longitudinal stress distribution F gyT y wh T TR F Roll F TR x z Contact patch F gyT Normal Stress Distance x Longitudinal Stress F TR Distance x FIGURE 2.18 Forces on the wheel and stress distribution at the tire–road interface.