Tyre-road friction and tyre slip from: https://www.tut.fi/ms/muo/tyreschool/moduulit/moduuli_10/hypertext/3/3_1.html#3_1_3
Let us consider the truck-tyre in detail.
When braking the tyre in stationary conditions, the normal force acts just in front of the wheel centre. Within the contact area a shear stress arises that increases until the adhesion limit is reached and sliding occurs, after which it decreases broadly proportional (Coulomb friction assumed) with the locally occurring normal stress.
The speed of the tyre relative to the wheel centre first increases. When the adhesion limit is reached, a sliding speed arises locally, resulting in a nonzero average speed Vslip of the rubber within the contact area. The ratio of this speed and the wheel speed in percent is designated the longitudinal slip sx = - k:
If a side force operates on the tyre, a lateral deformation appears in the tyre belt and its tread. Points on the running surface experience the belt deformation before they make contact with the road at which point the tyre first attempts to maintain contact with the road surface.
Figure 1 - Braking the tyre
This corresponds to a gradually increasing shear stress in lateral direction. Once the adhesion limits are reached, the rubber will start to slide relative to the road with a lateral motion, perpendicular to the wheel plane. The asymmetry in the distribution of shear stress causes the resulting force not to grip exactly in the middle of the contact area just under the wheel centre. Rather there is a pneumatic trail which, in combination with the side force, produces an aligning torque which tries to push the tyre in the direction of the wheel speed. The tangent of the slipangle between wheelplane and wheel speed, denoted as side slip -sy :
sy = tan(a)
in conjunction with the wheel load and the camber angle, are decisive for the side force and the aligning torque.
Figure 2 - Cornering of a tyre
The relationships between the position of the tyre, in terms of the slipvalues, and tyreresponse in terms of longitudinal and lateral force, pneumatic trail and aligning torque are of essential importance in studying vehicle behaviour. Without a good description of these tyrecharacteristics, such kind of research �s impossible.
Figure 3 - A tyre under combined slip conditions
Besides the properties described in pure slip conditions, one is also interested in situations of combined slip which are pertinent to braking when cornering. The maximum shear force between tyre and road surface is given by the existent coefficient of friction multiplied by the wheel load. The implication for braking in a bend is that the possible maximum brake force relative to a situation of straight line braking will be reduced. One has therefore sacrificed braking potential which is indicated by the friction ellipse, in which realistic combinations of brake or drive force and side force are shown separately.
The above discussion finally leads to relationships between:
- Side force versus lateral slip
- Pneumatic trail versus lateral slip
- Aligning torque versus lateral slip
- Brake or drive force versus longitudinal slip
under pure slip conditions (only lateral or longitudinal slip). In case of combined slip, the side force also depends on longitudinal slip, etc. Typical examples of these pure slip characteristics are shown in figure 4. One observes a strong nonlinear behaviour for larger slip. These relationships are of essential importance in studying vehicle behaviour. Without a good description of these tyre characteristics, such kind of research is impossible. The slope of the side force Fy vs. slip angle a near a = 0 (the cornering stiffness) is the determining parameter in the linear basic handling and stability theory of automobiles, as we shall see later.
Figure 4 - Some typical tyre characteristics
Under combined slip conditions, typical plots of Fx (brake or drive force) versus Fy (cornering force) are shown below for fixed values of slip angle a (taken from DELFT-TYRE). For small values of a, the side force almost vanishes. As aincreases, the side force Fy becomes apparent at the cost of a lower maximum value of the longitudinal force Fx.
Figure 5 - Combined slip, Fx versus Fy
It is important to note that tyre shear forces depend on tyre load. This dependence is usually nonlinear, where for increasing tyre load, the absolute slope of the tyre force versus tyre load reduces. This is the reason why during cornering, the average lateral tyre force per axle reduces due to force redistribution from inner to outer wheel. Consequently, as we shall see later, the steering performance of the vehicle is changed which might even lead to yaw-instability (oversteer conditions).
An example is shown in figure 6 with Fy depicted vs. load Fz for three different slip angle. One observes the decreasing absolute slope of these curves, meaning that under load transfer of for example 1500 N (being the increase, decrease of the tyre load at outer and inner tyre, respectively) and with an axle slip angle of 0.08 rad., the total lateral force is reduced. In other words, the cornering stiffness is reduced due to this load transfer.
Figure 6 - Load sensitivity lateral force
The illustration is for passenger cars, for which a restricted load variation is apparent. This is different for truck tyres where large variations in payload and thus large variations in tyre load will occur. This is illustrated in figure 7 where the normalized cornering stiffnesses (cornering stiffness coefficient: tyre cornering stiffness divided by the tyre load) for typical truck and passenger car tyres are depicted vs, tyre load. One oberves a decreasing trend for both passenger and truck tyres, with a much smaller sensitivity for truck tyres compared to passenger tyres.
Figure 7 - Load sensitivity passenger car and truck tyres
We close this section with some remarks concerning factors that influence tyre characteristic curves. The variations of the longitudinal force coefficient (defined as Fx/Fz) and longitudinal force versus longitudinal slip of two truck radial tyres are depicted in figure 8 and figure 9. The peak value is the maximum that can be reached without wheel locking, while the slide value is obtained during the locked wheel. Figure 10 presents some typical ranges of values of the longitudinal force coefficient obtained on a dry concrete surface for bias and radial tyres with two type of tread patterns design.
The dependencies of the longitudinal force coefficient to the load and speed are presented in figure 11 and figure 12.
Figure 8 - Longitudinal force coefficient vs. slip
Figure 9 - Braking force vs. slip of a truck tyre [3.1]
Figure 10 - Longitudinal force coefficient on dry road [2.1]
Figure 11 - Effect of load (radial tyre) [2.1]
Figure 12 - Effect of speed (radial tyre) [2.1]
Next, we consider the lateral force in more detail. This force is a function not only of simple friction but also of the size, design, construction and operating condition (i.e. load and inflation pressure).
The influences of the tyre construction and loading condition to lateral force coefficient (defined as Fy/Fz) are presented in figure 13 and figure 14.
It must be noticed that radial truck tyres are more responsive than bias truck tyres and the low profile radial truck tyre has a more constant lateral force coefficient through the load change (important in suspension design).
Combining braking and cornering by adding braking force to a tyre which rolls with slip angle results the friction ellipse concept. This ellipse envelops all the plotted curves of lateral forces versus longitudinal forces for different slip angles, as discussed above. Figure 15 and figure 16 show some combined forces for two truck tyres (bias and radial) while figure 17 shows the influence of slip angle on braking forces.
Figure 15 - Braking and cornering (bias tyre) [2.1]
Figure 16 - Braking and cornering (radial tyre) [3.1]
Figure 17 - Influence of slip angle on braking forces