45 Vehicle Mechanics Typical friction coefficients for both longitudinal and lateral forces as a function of slip for a fixed slip angle are shown in Figure 2.23. The traction force limit in the contact patch increases almost linearly with slip at the beginning as the majority of tire tread elements are worked most effectively without significant slip. The traction force limit reaches its peak at around 20% of the slip. The relationship between traction effort limit and slip becomes nonlinear beyond the peak; when the slip reaches 100%, the vehicle is sliding only without any linear velocity. On a dry pavement, the traction force limit peak is about 80%–90% of the vertical load.
Peak friction coefficient µ p defines this limit. The friction coefficient when the slip reaches 100% is known as sliding coefficient µ s . The friction coefficients vary depending on the road condition, although the shape of the curves remains the same. The nominal peak and sliding friction coeffi- cients for several ground conditions are given in Table 2.1. The peak and sliding friction coefficients of an icy road will be much smaller than that of a dry pavement meaning the wheel will slip much more easily for the same traction torque from the propulsion system.
In wheel slippage conditions, the tire–road interface cannot develop the traction force due to the lower friction force limits even though there is traction torque at the wheels. The location of where the peak will appear depends on the slip angle. For smaller slip angles, the peak appears at smaller slip values. Traction control system (TCS) is incorporated in today’s vehicles for controlling both longitu- dinal and lateral forces. The longitudinal control algorithm limits the traction force to prevent slip- page. The lateral control is achieved by keeping the yaw motion to zero. This can be achieved, for example, by controlling the steering angle.
The independent slippage prevention on each wheel of a four-wheel-drive vehicle greatly improves the lateral stability. The dynamics of traction systems are highly nonlinear and time-variant; consequently, the design of model-based TCS is also nonlinear and time-varying. The major difficulty for designing such systems is the real-time estimation of friction coefficient versus slip characteristics for different tires and road surfaces. μ Coefficient of Friction 75 100 25 50 Slip s (%) FIGURE 2.23 Longitudinal and lateral traction force coefficient variation.
TABLE 2.1 Nominal Longitudinal Peak and Sliding Friction Coefficients Surface Peak Coefficient, μ p Sliding Coefficient, μ s Asphalt and concrete (dry) Concrete (wet) Asphalt (wet) Gravel 0.75 0.7 0.5 0.55 Snow 0.2 0.15 Ice 0.1 0.07 0.9 0.8 0.6 0.6 μ(s) x μ p μ μ(s) s z