Tire Models Technical Notes
The tire models provided in the Vortex® toolkit are based on real scientific research. This page is meant to provide more technical background information than is found in the Adding Tire Models section.

When the hard ground tire model is used, the stiffness at the contact patch is deduced from the tire pressure. Tire pressure can be set to infinity in cases where no compliance in the contact is desired.
When a soft ground tire model is used, the stiffness at the contact patch is computed based on the pressure-sinkage parameters described in Tire Models. The pressure at any point of the tire-soil interface is calculated based on the pressure-sinkage relationship. The normal force is obtained by integrating the pressure along the tire-soil interface. The contribution of shear stress is also considered in computing the normal force. The tire stiffness based on the tire pressure is combined to the stiffness from the soft ground model. This contribution vanishes if the tire pressure is set to infinity.

The following section describes the hard grounds tire models. Each model is used to compute the longitudinal and lateral friction force (the traction and the cornering, respectively), and the alignment moment (which is along the normal vector at the contact). The stiffness at the contact patch is computed based on the tire pressure, and the rolling resistance depends on the actual rolling resistance model in use.

Note The following information relates only to the 1997 Magic Formula model. For the Pacejka Magic Formula 2002 model, make sure you are using the correct data before using it. The explanation for the 2002 model is beyond the scope of this documentation but is readily available online.
The Magic Formula Tire Model is based on the work by Pacejka et al 1, 2, 3, 4. It uses mathematical functions that relate the lateral force as a function of lateral slip, the longitudinal force as a function of longitudinal slip, and the aligning moment as a function of lateral slip.
The following table shows the default values of the magic formula.

The Composite Slip Tire Model is based on the work by Szostak et al 6 . This model produces tire forces taking into account the interaction of longitudinal and lateral forces from small through saturation. Tire longitudinal and lateral forces are computed from composite force which is a function of composite slip. The composite slip is computed from slip angle, longitudinal slip, normal load, tire patch length, friction coefficient, longitudinal and lateral stiffness coefficients, etc.
The next table shows the default value of the composite slip tire model.

Example of values for three types of tire 6 :

norm should be 0.85 for normal road conditions, 0.3 for wet road conditions, and 0.1 for icy road conditions.

The Fiala tire model is based on the work by Fiala 7  . This model uses the parameters that are directly related to the physical properties of the tire to determine tire longitudinal forces, lateral forces, and aligning moments. Note that the carcass radius is used only in the case of the alignment moment.
According to that theory, in case where there is no lateral slip, the maximum traction force would be:

where N is the load applied on the wheel.
The slip value where half of the maximum traction happens is called the critical slip and is represented by:

A typical value for the critical slip is around 15%.
The following table shows the default values of the Fiala tire model in Vortex. They are based on a wheel load of 5000 newtons and critical slip of 15%.

The Coulomb friction model is not like the other hard ground tire models. It has been introduced in Vortex to give the possibility of using the classic friction models of Vortex in the same way as the other tire models (see VxMaterial for more details). Note that the Coulomb tire model has been stripped of a few properties that were not relevant for tire interaction with a hard ground. For instance:
Stiffness/compliance has been removed since tire pressure models it (note that the damping has been kept as a scaling factor on the critical damping).
Restitution has been removed since this feature is more likely to be used for pure elastic contact interaction.
Friction around the three axes have been removed: Around the primary axis of the contact, friction is not needed. Around the secondary axis of the contact, the rolling resistance model (if set) gives a more realistic result. Finally, around the normal axis, the friction is implicitly added by the lateral and the longitudinal friction of the two contacts at the tire patch.
The default value of the Coulomb tire models is as follow:

The Soft Ground Tire Model as implemented in Vortex is mainly based on the work by Bekker 9, 10, 11 and Wong 12, 13 . The tire normal and shearing forces are computed by integrating normal pressure and shear stress over the entire tire-terrain contact area. In order to find normal stress distribution under the wheel, an appropriate pressure-sinkage relation should be used depending on the type of soil with which the wheel is interacting (in terramechanics literature, the pressure-sinkage relation is the relation between average pressure under a rigid plate versus the penetration depth). Then, we need a model to relate these pressure-sinkage relations to the pressure under the wheel. Various available models in the literature have been implemented in Vortex. In Vortex, the pressure-sinkage model refers to the combination of a particular pressure-sinkage relation and a model relating it to a rolling wheel. These include Bekker, Bekker-Wong, Reece, Muskeg, and Snow.
The shear stress-shear displacement relationship is characterized by the exponential shearing equation, the hump shearing equation, or the Wong shearing equation. This is detailed in section Shear Stress-Shear Strain Models.

The Bekker equation is the default pressure-sinkage equation for Soft Ground Tire Models in Vortex, which is used to represent the pressure-sinkage characteristics of homogeneous terrain, such as sand and clay. It is given by:

where p is the pressure, b is the smaller dimension of the contact patch, z is the sinkage and n, k_c, k are the pressure-sinkage parameters. In this model, the normal stress distribution under the wheel is obtained by the relation below:

The following is a schematic of rigid wheel and soil contact.

The exit angle θ₂ is assumed to be zero in this model. The default values (corresponding to sandy loam) for Bekker parameters are shown in the table below.
Values for a sample of terrains are given in 12 .

An alternative way to represent normal stress distribution under a wheel is developed by Wong and Reece 14. Contrary to the Bekker model, maximum normal stress distribution may not occur at the bottom-dead-centre of the wheel. They proposed the empirical relation below for estimating θ_m as the location of the point of the maximum radial stress:

where a₀ and a₁ are dimensionless constants, θ₁ defines the location of the point corresponding to the beginning of wheel-soil contact, and i is the wheel slip ratio defined by:

Based on this model, normal stress distribution is obtained using the next relations:

where θ₂ is the exit angle. θ₂ is assumed to be zero in this model.

The default values for Bekker-Wong parameters are shown in the table below.

The pressure-sinkage relationship is characterized differently in the Reece model (compared to the Bekker's parameters):

where c is soil cohesion, γ_s is weight density or specific weight of terrain, z is the wheel sinkage, and b is the wheel width or the smaller dimension of the contact patch. k'_c and k'_φ are the new pressure-sinkage parameters, which are dimension-less, and n is the pressure-sinkage exponent. In this model, θ_m is calculated similar to the Bekker-Wong model. Normal stress distribution under the wheel is then calculated using the relations below 14:

In this model, θ₂ can be non-zero. Following an approach explained in 15 , θ₂ is calculated by the relation below:

where λ is a dimensionless parameter that can have a value from 0 to 1. The default values for Reece parameters are shown in the table below.