
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 earlierAdding 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 pressuresinkage parameters described in Tire Models. The pressure at any point of the tiresoil interface is calculated based on the pressuresinkage relationship. The normal force is obtained by integrating the pressure along the tiresoil 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.
The Magic Formula Tire Model is based on the work by Pacejka et al1H.B. Pacejka, "The Tyre as a Vehicle Component", Proc. XXVI FISITA Congress, Prague, June 1623, 1996 (CDROM). 2E. Bakker, L. Nyborg, and H.B. Pacejka, "Tyre Modelling for Use in Vehicle Dynamics Studies," Society of Automotive Engineers, paper 870421, 1987. 3E. Bakker, H.B. Pacejka, and L. Lidner, "A New Tire Model with an Application in Vehicle Dynamics Studies," Society of Automotive Engineers, paper 890087, 1989. 4H.B. Pacejka and I.J.M. Besselink, "Magic Formula Tyre Model with Transient Properties," in F. Bohm and H.P. Willumeit, Eds., Proc. 2nd Int. Colloquium on Tyre Models for Vehicle Dynamic Analysis, Berlin. Lisse, The Netherlands: Swets & Zeitlinger, 1997.. 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.
Symbol  Name in Vortex  Value  Units 

a_{0}  a[0]  1.6929  none 
a_{1}  a[1]  55.2084  1/meganewton 
a_{2}  a[2]  1271.28  10^{3} 
a_{3}  a[3]  1601.8  newton/degree 
a_{4}  a[4]  6.4946  kilonewton 
a_{5}  a[5]  4.7966e3  1/degree 
a_{6}  a[6]  0.3875  1/kilonewton 
a_{7}  a[7]  1.0  none 
a_{8}  a[8]  4.5399e2  none 
a_{9}  a[9]  4.2832e3  degree/kilonewton 
a_{10}  a[10]  8.6536e2  degree 
a_{11}  a[11]    substitute by a_{111} x F_{z} + a_{112} 
a_{12}  a[12]  7.668  10^{3} 
a_{13}  a[13]  45.8764  newton 
a_{111}  a_111  7.9730  1/meganewton/degree 
a_{112}  a_112  0.2231  1/kilodegree 
b_{0}  b[0]  1.65  none 
b_{1}  b[1]  7.6118  1/meganewton 
b_{2}  b[2]  1122.6  10^{3} 
b_{3}  b[3]  7.36e3  1/meganewton 
b_{4}  b[4]  144.82  10^{3} 
b_{5}  b[5]  7.6614e2  1/kilonewton 
b_{6}  b[6]  3.86e3  1/kilonewton^{2} 
b_{7}  b[7]  8.5055e2  1/kilonewton 
b_{8}  b[8]  7.5719e2  none 
b_{9}  b[9]  2.3655e2  1/kilonewton 
b_{10}  b[10]  2.3655e2  none 
c_{0}  c[0]  2.2264  none 
c_{1}  c[1]  3.0428  meter/meganewton 
c_{2}  c[2]  9.2284  millimeter 
c_{3}  c[3]  0.500088  meter/meganewton/degree 
c_{4}  c[4]  5.56696  millimeter/degree 
c_{5}  c[5]  0.25964  1/kilonewton 
c_{6}  c[6]  1.29724e3  1/degree 
c_{7}  c[7]  0.358348  1/kilonewton^{2} 
c_{8}  c[8]  3.74476  1/kilonewton 
c_{9}  c[9]  15.1566  none 
c_{10}  c[10]  2.1156e3  1/degree 
c_{11}  c[11]  3.46e4  none 
c_{12}  c[12]  0.0  degree/kilonewton 
c_{13}  c[13]  0.0  degree 
c_{14}  c[14]  0.100695  meter/meganewton/degree 
c_{15}  c[15]  1.398  millimeter/degree 
c_{16}  c[16]  0.0  millimeter 
c_{17}  c[17]  0.0  newtonmeter 
The Vortex implementation of the Magic Formula is the following for every component:
The Composite Slip Tire Model is based on the work by Szostak et al5H.T. Szostak, "Analytical Modeling of Driver Response in Crash Avoidance Maneuvering Volume II: An Interactive Tire Model for Driver/Vehicle Simulation", National Highway Traffic Safety Administration Report DOTHS807271, 1988.. 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 tire6H.T. Szostak, "Analytical Modeling of Driver Response in Crash Avoidance Maneuvering Volume II: An Interactive Tire Model for Driver/Vehicle Simulation", National Highway Traffic Safety Administration Report DOTHS807271, 1988.:
μ_{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 Fiala7E. Fiala, "Seitenkrafte am rollenden Luftreifen," VDIZeitschrift 96, 973, 1954.. 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%.
Symbol  Name in Vortex  Value  Units  Definition 

C_{s}  Cs  14166.0  N  Tire longitudinal stiffness 
C_{α}  Calpha  5.156e4  N/radian  Tire lateral stiffness due to the slip angle 
μ_{0}  mu0  0.85  none  Tiretoroad coefficient of static friction 
μ_{1}  mu1  0.65  none  Tiretoroad coefficient of sliding friction occurring at 100% slip with pure sliding 
R_{2}  R2  0.0975  Meters  Tire carcass radius 
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:
The default value of the Coulomb tire models is as follow:
Name  Default Value  Unit  Meaning  Notes 

Damping Ratio  0.4  none  Scale applies on the critical damping  
Friction Model Longitudinal  Scaled Box Fast  N/A  Friction model  Can be: None, Box, Scaled Box, Scaled Box Fast 
Friction Coefficient Longitudinal  0.46  none  Coulomb coefficient of friction  Used in case of Scaled Box and Scaled Box Fast 
Friction Force Longitudinal  20000.0  Newton  Maximum friction  Used in case of Box 
Static Friction Force Longitudinal  1.2  none  Scaling factor applied on friction when at rest  
Slip Longitudinal  0.0  sec/kg  Inverse of drag coefficient  None zero value reduces the friction forces (then conflicting with the values predicted by the model) 
Friction Model Lateral  Scaled Box Fast  N/A  Friction model  Can be: None, Box, Scaled Box, Scaled Box Fast 
Friction Coefficient Lateral  0.46  none  Coulomb coefficient of friction  Used in case of Scaled Box and Scaled Box Fast 
Friction Force Lateral  20000  Newton  Maximum friction  Used in case of Box 
Static Friction Force Lateral  1.2  none  Scaling factor applied on friction when at rest  
Slip Lateral  0.0  sec/kg  Inverse of drag coefficient  None zero value reduces the friction forces (then conflicting with the values predicated by the model) 
The Soft Ground Tire Model as implemented in Vortex is mainly based on the work by Bekker8M.G. Bekker, Theory of Land Locomotion, The University of Michigan Press, 1956. 9M.G. Bekker, OffTheRoad Locomotion, The University of Michigan Press, 1960. 10M.G. Bekker, Introduction to TerrainVehicle Systems, The University of Michigan Press, 1969. and Wong11J.Y. Wong, Theory of Ground Vehicles, John Wiley, New York. 3rd Edition, 2001. 12J.Y. Wong, Terramechanics and OffRoad Vehicles, Elsevier Science Publishers, Amsterdam, the Netherlands, 1989.. The tire normal and shearing forces are computed by integrating normal pressure and shear stress over the entire tireterrain contact area. In order to find normal stress distribution under the wheel, an appropriate pressuresinkage relation should be used depending on the type of soil with which the wheel is interacting (in terramechanics literature, the pressuresinkage relation is the relation between average pressure under a rigid plate versus the penetration depth). Then, we need a model to relate these pressuresinkage relations to the pressure under the wheel. Various available models in the literature have been implemented in Vortex. In Vortex, the pressuresinkage model refers to the combination of a particular pressuresinkage relation and a model relating it to a rolling wheel. These include Bekker, BekkerWong, Reece, Muskeg, and Snow.
The shear stressshear displacement relationship is characterized by the exponential shearing equation, the hump shearing equation, or the Wong shearing equation. This is detailed in section Shear StressShear Strain Models.
The Bekker equation is the default pressuresinkage equation for Soft Ground Tire Models in Vortex, which is used to represent the pressuresinkage 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_{ϕ} are the pressuresinkage 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 (θ_{2}) is assumed 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 13J.Y. Wong, Theory of Ground Vehicles, John Wiley, New York. 3rd Edition, 2001., Table 2.3 p130.
An alternative way to represent normal stress distribution under a wheel is developed by Wong and Reece14J.Y. Wong and A.R. Reece, "Prediction of Rigid Wheel Performance Based on the Analysis of SoilWheel Stresses Part I: Performance of Driven Rigid Wheels," J. Terramechanics, 1967, vol. 4, 8198.. Contrary to the Bekker model, maximum normal stress distribution may not occur at the bottomdeadcentre of the wheel. They proposed the empirical relation below for estimating angle θ_{m} as the location of the point of the maximum radial stress:
where a_{0} and a_{1} are dimensionless constants, angle θ_{1} defines the location of the point corresponding to the beginning of wheelsoil contact, and i is the wheel slip ratio defined by:
Based on this model, normal stress distribution is obtained using the next relations:
where θ_{2} is the exit angle. θ_{2} is assumed to be zero in this model.
The default values for BekkerWong parameters are shown in the table below.
The pressuresinkage 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 pressuresinkage parameters, which are dimensionless, and n is the pressuresinkage exponent. In this model, θ_{m} is calculated similar to the BekkerWong model. Normal stress distribution under the wheel is then calculated using the relations below15J.Y. Wong and A.R. Reece, "Prediction of Rigid Wheel Performance Based on the Analysis of SoilWheel Stresses Part I: Performance of Driven Rigid Wheels," J. Terramechanics, 1967, vol. 4, 8198.:
In this model, θ_{2} can be nonzero. Following an approach explained in 16G. Ishigami, A. Miwa, K. Nagatani, and K. Yoshida, "Terramechanicsbased model for steering maneuver of planetary exploration rovers on loose soil," J. Field Robotics, 2007, vol. 24, 233250., θ_{2} is calculated by the relation below:
where λ is a dimensionless parameter that can have a value from zero to 1. The default values for Reece parameters are shown in the table below.
The Muskeg equation is used to represent the pressuresinkage characteristics of various types of organic terrain (muskeg):
where p is the pressure, z is the sinkage, k_{p} is a stiffness parameter for the peat, m_{m} is a strength parameter for the surface mat and D_{h} is a value (in meters) related to the contact area. D_{h} = 4 A/L where A and L are the area and the perimeter of the contact patch, respectively.
Normal stress distribution under the wheel is obtained using the following relation:
The default values (corresponding to Muskeg A) for Muskeg parameters are shown in the table below. Values for a sample of terrains are given in 17J.Y. Wong, Theory of Ground Vehicles, John Wiley, New York. 3rd Edition, 2001. Table 2.4 p133.
The Snow equation is used to represent the pressuresinkage characteristics of snow on top of ice layers:
where p is the pressure, z is the sinkage and D is the smaller dimension at the contact patch (in cm). Normal stress distribution under the wheel is obtained using the relation below:
The default values for Snow parameters are shown in the table below.
Symbol  Name in Vortex  Value  Units 

k_{p1}  kp1  16.3  kPa 
k_{p2}  kp2  0  kPa/cm 
k_{z1}  kz1  33  cm 
k_{z2}  kz2  0  cm^{2} 
Soft ground tire models can include the effect of soil hardening caused by wheels passing over an area multiple times. This is of primary importance for vehicles with multiple wheels or terrains that will be driven over multiple times, as each wheel changes the mechanical properties of the soil when it has passed over it.
By using this feature:
To include this effect in the simulation, the ground surface must be modeled with a Deformable Terrain or an Earthwork Zone. Three pressure sinkage relations are supported: Bekker, Wong, and Reece. (Muskeg and Snow do not model mineralbased soils, therefore they are not supported). Each of these has a parameter named Enable Soil Hardening that must be set.
A description of the concepts can be found in Wong^{11 12}, where a description of the effects of ground response to repetitive loading is given.
The next section gives an overview of the concept.
From experiments, it has been observed that when a small surface of ground is loaded for the first time, the pressure increases with the sinkage according to the pressure sinkage relations (gray line in the picture below). If the pressure is released (at penetration z_{u}), the soil surface rises up to b_{u} (because of its elasticity) then stops even if no load is applied anymore. The soil surface is then permanently deformed. This is called plastic deformation.
If a load is subsequently applied to the same area, the experimental observation shows that the pressure will increase approximately linearly with the sinkage (along the black line) until the sinkage reaches the point where the load was released the first time (z_{u}). If the load is increased further, the pressure continues to increase following original pressure sinkage relation curve (gray). In the picture, the elastic range of z is labeled e.
When the sinkage z is within the elastic range, the pressure takes the form:
where z ∈ [e]
k_{u} represents the rate of increase of pressure in the elastic range.
Again, following references ^{11 12}, it has been shown experimentally that the elastic stiffness k_{u} in the elastic range depends on the level of sinkage in the following form:
Where k_{0} and A_{u} are parameters that can be determined from experimental data.
In cases where k_{0} and A_{u} are not available, the following section gives a procedure to choose these parameters. They control the size of the plastic and the elastic portions of the soil in case of repetitive loading.
The parameters k_{0} and A_{u} are used to determine the plasticity and elasticity of the soil. When those parameters are known from experimental results, it is suggested to use them. For cases where these parameters cannot be found, this section gives a procedure to determine approximate values for k_{0} and A_{u} that will represent realistic soil behavior.
The idea behind this procedure is to choose the amount of elasticity at two values of z in the pressure sinkage diagram. This is represented by penetrations z_{1} and z_{2} and the length of elasticity represented by e_{1}and e_{2} in the next diagram.
Following the repetitive loading principle, at penetration z_{i}, the soil stiffness is given by:

for i = 1, 2  (1) 
On the other hand, k_{i} can be seen as the rate of change of pressure during a second pass:

(2) 
Where we have written k_{eq} as a constant for a given pressure sinkage relation:
Here, k_{c} and k_{ϕ} are the pressure sinkage parameters and b is taken as the width of the wheel (using Bekker’s representation in this example).
By choosing a length of elasticity at two different penetrations points, we can use (1) and (2) to form a system of two equations for the two unknowns k_{o} and A_{u}.
By solving for k0 and Au, we find:
Here, k_{eq} and n are from the current pressure sinkage relation, and z_{1}, z_{2}, e_{1} and e_{2} are userdesired penetration and stiffness lengths.
This technique to choose the parameters k_{0} and A_{u} gives approximate values as it does not consider the actual shape of the wheel.
The procedure has been captured in the following Python script:
#wheel width W = 0.4 # Wheel width in meters assert (W>0), "invalid wheel width {}".format(W) # pressure sinkage parameter (units depends on parametrization) n = 0.97 Kc = 5.5 Kphi = 2293 assert(n>0), 'invalid exponent in pressure parameter {}'.format(n) # desired penetration and elastic length at two points in meters z1 = 0.02 e1 = 0.01 z2 = 0.04 e2 = 0.0 #validation: assert(z1>0),'z1 must be strictly larger than 0 z1={}'.format(z1) assert(e1>0.0), 'e1 must be strictly larger than 0 e1={}'.format(e1) assert(z1<z2), 'z2 must be strictly larger than z1 z1={} z2={}'.format(z1,z2) assert(e2>0),'e2 must be strictly larger than 0 e2={}'.format(e2) Keq = ( Kc/W + Kphi ) # this example is for Bekker C1 = Keq*(z1**n) C2 = Keq*(z2**n) # results K0 = (C1*z2 / e1  C2*z1/e2)/(z2z1) Au = (C2/e2  C1/e1)/ (z2z1) print ( 'K0 = {} and Au = {} '.format( K0, Au) ) 
In soft ground tire models, the traction force is based on the shear stressshear strain characteristics of the soil. Soft ground tire models implement three different soil shearing properties. In each of them, they give the ratio of shear force to the maximum shear force, s_{max}, as a function of the shear displacement (noted j in future equations). The maximum shear force occurs at the point where the shear displacement produces a failure of the soil. This point is commonly characterized by the MohrCoulomb failure criterion given by:
The Exponential equation is the default shearing equation, which is used to represent the shear stressdisplacement characteristics of loose sand, saturated clay or fresh, dry snow. This model was proposed by Janosi and Hanamoto18Z. Janosi and B. Hanamoto, “The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils,” in Proceedings of the ISTVS 1st International Conference on Mechanics of SoilVehicle Systems, pp. 707–736, Edizioni Minerva Tecnica, Torino, Italy, 1961..
In this case, the relative shear force is given by:
The default values for Exponential Shearing parameters are shown in the table below.
The Hump equation has a distinct "hump" of shear stress with a subsequent decrease to a value approaching zero. It has been used to represent the shearing characteristics of some types of muskeg mats.
In this case, the relative shear force is given by:
The default values for Hump Shearing parameters are shown in the table below.
The Wong equation is similar to the Hump equation, but it allows a level of residual shear stress s_{r} rather than dropping to zero after the hump. It is useful in representing the shearing characteristics of some types of snow and loam, usually in a compacted state.
In this case, the relative shear force is given by:
Where K_{w} is the shear displacement (in cm) where the maximum shear stress appears and K_{r} is the residual shear stress.
The default values for Wong Shearing parameters are shown in the table below.
In Vortex soft ground tire models, the traction force is based on the shear stressshear strain characteristics as explained in the previous section. Moreover, the traction force includes the effect of the lugs. Therefore, the traction force is the combination of the internal shearing model (at the carcass) and the external shearing model (at the surface of the lug). The effect of the two models is averaged in proportion of the lugtocarcass ratio. The actual contribution from the carcass depends on the sinkage and the lug height. In case where the lug's height is zero, the two ratios are ignored, and 100% of the traction comes from the Internal Model.
The default values for lug parameters are shown in the table below.
The lateral force is modeled as an exponential function of slip angle integrated over the contact area between the tire and the terrain.
where c_{s} is given by:
In these equations, F_{l} is the lateral force, A is a user constant, W is the tire width, R is the wheel radius, s_{a} is the slip angle, and p(θ) is the pressure under the tire at θ which depends on the actual normal stress distribution. Please refer to the diagram in the Bekker model section above for θ notation. As this formulation gives no lateral force when lateral slip (also called the slip angle) is zero, the model provides a way to set a minimum value for c_{s}.
The default values for lateral force parameters are shown in the table below. Note that A=0 is enough to obtain no lateral force.
Symbol  Name  Value  Unit 

A  A  1.5  none 
B  B  0.2  1/rd 
Min Threshold Coeff  0.0  none 
Although the effect of the tire pressure is not included in any theoretical tire models, Vortex takes it into account in a few ways.
The tire pressure is used to compute the normal stiffness at the contact and this introduces some tire deflection. In case of soft ground (Wong and Bekker only), this tire deflection is taken into account. It affects the sinkage, which has two consequences. First, it affects the compaction resistive force (and torque), and second, it affects the shear stress which in turn affects the traction force.
The deflection of the tire caused by the tire pressure also generates some rolling resistance torque. This is discussed below in Rolling Resistance Tire Pressure Model.
The tire pressure is given in pascals and the default value is as follows:
Name  Value  Unit  Notes 

Tire Pressure  220627.2  Pascal  Equivalent to 32.9 PSI. 
The resistance in the tireterrain interaction is composed of two parts in the tire model. One part, which is referred to as compaction resistance, is related to the compaction of the terrain, and it is computed and implemented in the soft ground tire model. This resistance is zero in the other tire models.
The other part of the resistance is referred to as rolling resistance; it is due to the deformation of tires, and it is computed and implemented in the tire model base on different models available. Note that soft ground tire models include resistive torque and force caused by the soil compaction and that the effects of the rolling resistance models are added to it.
The rolling resistance is computed based on the rolling resistance coefficient. The rolling resistance coefficient represents the ratio of the rolling resistance to the tire normal force. Depending on the type of rolling resistance model that is selected, it can be a function of only forward wheel speed or it can also depend on tire pressure and normal force. The rolling resistance induces a resistive torque of the form:
Where c_{r} is the coefficient of rolling resistance, R is the wheel radius and N is the normal force on the wheel; c_{r} is unitless.
There are three different models:
In this model, the coefficient is a function of the linear speed of the wheel.
The default values for this coefficient model are:
Name  Value  Unit  

a_{0}  Base coefficient  0.04  none 
a_{1}  Speed factor  6e5  (s/m)^{n} 
n  Speed exponent  2.0  none 
In this model, the coefficient is a function of the linear speed of the wheel and the tire pressure.
Where p is the current tire pressure, p_{0} is the nominal tire pressure, F_{n} is the load, alpha is the tire pressure exponent, and β is the normal load exponent.
The default values for this coefficient model are:
Name  Value  Unit  

p  Nominal pressure  330000  Pascal 
α  Pressure exponent  0.4  none 
β  Load exponent  0.9  none 
a_{0}  Base coefficient  0.04  none 
a_{1}  Speed factor  6e5  (s/m) 
a_{2}  Speed factor second order  0  (s/m)^{2} 
In this model, the coefficient is given by a lookup table provided by the user as a CSV file containing two columns. In the first column, we must have linear speed of the wheel in meter per second (must appear in increasing order). The second column contains the rolling resistance coefficient.
The default table given in the resources folder is:
Speed (m/s)  Coefficient 

5.55  1.0 
19.44  1.0 
27.78  1.03 
33.33  1.1 
36.11  1.14 
38.89  1.2 
44.44  1.3 