Modelling Tyres

The model which is probably most widely known is the Pacejka "Magic Formula" fit to empirical data, since Hans Pacejka started his research into tyres in the 1960s.

The MF was updated at various times:

- 1989 original formulation

- 1994 updated formulation

- 1996 updated formulation (78 coefficients)

- PAC2002 (89 coefficients): adds contact-mass transient capabilities, tyre spin and parking which make it suitable also for low and zero speed applications, ABS braking and changing friction properties.

- PAC2002 with inflation pressure (99 coefficients)

- 5.2 (90 coefficients): models camber also in longitudinal direction

- The PAC2006 (97 coefficients) is one of the most recent models

Lateral Force estimation from sensors and algorithms

TyreFrictionEstimationForChassisControl

Lateral Force Load Sensitivity

Experimental data shows that as load is increased on the tyre, its "efficiency" is reduced. During cornering, the average lateral force per axle reduces due to force redistribution from inner to outer wheel.

Adherence under different road conditions

The road surface conditions, temperature and contamination are important for the adherence values. The influence of load and vehicle speed on wet roads are shown below.

When the water layer thickness and vehicle speed increase the hydroplaning phenomenon appears. So, the tyre is lifting from the road surface due to the pressure created by water under tyre. The tyre behavior between the state in which hydroplaning begin and the full (total) hydroplaning state, is commonly described with the "three-zone concept" elaborated by Dr. V.E. Gough in 1959. As can be seen in figure 4, zone A (unbroken water film) is in forward region of the contact area where the hydrodynamic pressure lifts and completely separates the tread from the road. In this region the water film remains unbroken and the local coefficient of friction is substantially zero. Zone B (transition zone) is the region where a progressive breakdown of the water is dissipated but where a thin film layer of water still remains. In this region the effective coefficient of friction varies widely. Zone C ("dry" contact) is the region where the lubrication film has been substantially removed and the road frictional forces can be generated.

While on dry and wet road surfaces the maximum lateral forces appear in slip angle domains above 10deg, correspondingly smaller maximum lateral forces are attained already at slip angle of 5deg to 7deg on snow-packed road surfaces due to the lower coefficient of friction. For braking forces the maximum on a dry asphalt road appears in the range of 20% slip, on wet road are already evident at 15 % slip and on snow-packed roads at 8% already. As the friction between tyre and road decreases, the load dependence decreases also.

Tyre Force vs Slip Angle in Commercially Available Software

Several commercial software tools are able to model tyres with reasonable precision, such as CarSim:

Tyre Force and Moment Plots

Any tyre is able to produce forces in the longitudinal (acceleration-braking) and lateral (cornering) planes and in a combination of the two. Typically it's maximum F_Lat vs F_long envelope of operation forms an ellipse rather than a circle, because the force in one plane may exceed that in its perpendicular plane. Thus one of the useful data plots is the "Ellipse Diagram". Shown here below is an example from the very good resources at WhiteSmoke.

There is also another way to plot data, with slip angle and slip ratio as horizontal axes, and force in the vertical axis. This is a little bit like taking a F vs Slip line and revolving it around the origin: it makes a plot which can be called "Black Hole Diagram". Shown here below is again an example from WhiteSmoke.

Self aligning torque data gathered on a stability rig is generally only showing the effect of pneumatic trail. The plot can be in 3 dimensions (Vertical force - Slip Angle - Self Aligning Torque), or in two dimensions (Gough Plot). It is possible to extract more lateral force from a tyre after the SAT has peaked but it becomes somewhat harder to make out when the tyre will begin to lose grip. It takes considerable driver training to adjust to this tyre characteristic.

The plot below is an example from the Vehicle Dynamics department of Florence University (Italy): passenger car tyre (Fz = 1000 (1000) 7000, α = 0.04 (0.04) 0.2 rad, tp = 0.01 (0.01) 0.05 ).

Tyres in Racing simulators and games

The following games use reasonably realistic tyre models

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