Magic Formula - Pacejka 1989 Tire Model
A good approximation can be obtained for the longitudinal force, the lateral force and the the aligning moment as the function of the longitudinal slip and the slip angle by using the same empirical equation called the magic formula. The early version of this formula proposed by Pacejka in [1] is often referred to as the PAC89 tire model. It has fewer parameters than later models, but still quite accurate for pure slip conditions. Therefore it is still used today when simplicity and computational efficiency are more important than high precision. The main components of the model are described below, along with implementation details specific to the Vortex platform. This implementation has been validated against published data.
Coordinate systems
The original PAC89 magic formula model formulation uses the SAE coordinate system where the x-axis points forward, the z-axis points down and the y-axis points to the right. In Vortex the TYDEX Wheel Axis (ISO) convention is used with opposite y and z axis direction. This is shown in the figure below which also indicates the considered positive direction for the forces, and moment acting on the wheel as well as the contact point velocity components. Note that the lateral/cornering force and the aligning moment have opposite signs. Also, in case of the Magic Formula a negative angular velocity is needed to produce a positive forward velocity, and the normal reaction force Fz is considered positive when it points upwards (opposite to the local z axis of the contact coordinate system). These differences have to be accounted for in the cornering force, aligning moment and slip angle calculations.
Magic Formula equation
Parameter | Description | Typical Tire Characteristic |
|---|---|---|
x, X | Slip or slip angle |  |
Y | Output quantity | |
B | Stiffness factor | |
C | Shape factor | |
D | Peak factor | |
E | Curvature factor | |
Sh | Horizontal shift | |
Sv | Vertical shift |
Model Inputs
Simulated quantities
Input Variable | Name in Vortex | Default Value | Units | Description |
|---|---|---|---|---|
|
| kN | Resultant contact normal force acting on the tire | |
- | - | degrees | Camber angle |
Secondary coefficients of the lateral force (see Appending A.2 in [2])
Input Variable | Name in Vortex | Default Value | Units | Description |
|---|---|---|---|---|
A[0] | 1.6929 | - |
| |
A[1] | -55.2084 | 10-3/kN |
| |
A[2] | 1271.28 | 10-3 |
| |
A[3] | 1601.8 | N/degree |
| |
A[4] | 6.4946 | kN |
| |
[5] | 4.7966e-3 | 1/degree |
| |
| [6] | -0.3875 | 1/kN |
|
| [7] | 1 | - |
|
| [8] | -4.5399e-2 | - |
|
[9] | 4.2832e-3 | degree/kN |
| |
[10] | 8.6536e-2 | degree |
| |
| [11] | 0 | 10-3/degree | This coefficient is often substituted as to obtain a better description of the camber thrust [2]. In Vortex, this is the default behaviour. To fall back to Pacejka’s original definition one needs to set and . |
[12] | 7.668 | 10-3 |
| |
[13] | 45.8764 | N |
| |
| a1 | -7.973 | 10-6/(N-deg.) |
|
| a2 | -0.2231 | 10-3/degree |
|
Secondary coefficients of the longitudinal force (see Appending A.2 in [2])
Input Variable | Name in Vortex | Default Value | Units | Description |
|---|---|---|---|---|
B[0] | 1.65 | - |
| |
B[1] | -7.6118 | 10-3/kN |
| |
B[2] | 1122.6 | 10-3 |
| |
B[3] | -7.36e-03 | 10-3/kN |
| |
B[4] | 144.82 | 10-3 |
| |
b[5] | -7.6614e-02 | 1/kN |
| |
b[6] | -3.86e-03 | 1/(kN)2 |
| |
b[7] | 8.5055e-02 | 1/kN |
| |
b[8] | 7.5719e-02 | - |
| |
b[9] | 2.3655e-02 | 1/kN |
| |
b[10] | 2.3655e-02 | - |
|
Secondary coefficients of the aligning moment (see Appending A.2 in [2])
Input Variable | Name in Vortex | Default Value | Units | Description |
|---|---|---|---|---|
c[0] | 2.2264 | - |
| |
| c[1] | -3.0428 | mm/kN |
|
| c[2] | -9.2284 | mm |
|
c[3] | 0.500088 | mm/(kN-deg.) |
| |
c[4] | -5.56696 | mm/degree |
| |
c[5] | -0.25964 | 1/kN |
| |
c[6] | -1.30E-03 | 1/degree |
| |
c[7] | -0.35835 | 1/(kN)2 |
| |
c[8] | 3.74476 | 1/kN |
| |
c[9] | -15.1566 | - |
| |
c[10] | 2.12E-03 | 1/degree |
| |
c[11] | 3.46E-04 | - |
| |
c[12] | 0 | degree/kN |
| |
c[13] | 0 | degree |
| |
c[14] | 0.100695 | mm/(kN-deg.) |
| |
c[15] | -1.398 | mm/degree |
| |
c[16] | 0 | mm |
| |
c[17] | 0 | N-m |
|
Slip
Calculation | Equation |
|---|---|
Longitudinal Slip |
where is the forward velocity of the wheel Note: In [3] the longitudinal slip is defined as with . Point S is thought to be attached to the rotating wheel body in a distance equal to the effective radius from the wheel’s center along the contact normal. In the Vortex implementation of this tire model, the effective radius is approximated by the radius of the unloaded tire. The longitudinal slip is negative for braking and positive for accelerating conditions. At low forward velocities its value can become large. The slip is -1 for a completely locked wheel, but can also become smaller in the special case when the wheel rotates backwards while it is sliding forward. The value of the slip is not limited to the range (-1,+1) range. In Vortex, to handle the transition from zero to a finite forward velocity, the longitudinal slip is set to when . |
Slip Angle | when Note: This definition follows opposite sign convention for the lateral velocity compared to reference [1]. A positive vy is needed for having a positive slip angle when the wheel moves forward. |
Longitudinal / Traction Force
Calculation | Equation |
|---|---|
Longitudinal / Traction Force | input variable: output variable: Note: The magic formula has mixed units. The longitudinal slip is expressed in percentages, the normal force Fz in the coefficients below is in kilonewtons, but the output force is in Newtons. |
Peak Factor | where is the peak longitudinal friction coefficient |
Shape Factor | |
Stiffness Factor | where the product gives the slope of the curve at |
Curvature Factor | |
Horizontal Shift | |
Vertical Shift |
Lateral / Cornering Force
Calculation | Equation |
|---|---|
Lateral / Cornering Force (Pure Slip) | input variable: output variable: where the value is negated to match the sign convention (TYDEX Wheel Axis) used in Vortex. Note: The magic formula has mixed units. The slip angle is expressed in degrees, the normal force Fz in the coefficients below is in kilonewtons, but the output force is in Newtons. |
Lateral / Cornering Force (Combined Slip) | To consider the limiting effect of the longitudinal force Fx on the lateral force Fy(Fx) the following elliptical approximation [2] is considered: with where for pure longitudinal slip and for pure lateral slip |
Peak Factor | where is the peak lateral friction coefficient |
Shape Factor | |
Stiffness Factor | where the product gives the slope of the curve at |
Curvature Factor | |
Horizontal Shift | |
Vertical Shift | with |
Aligning Moment
Calculation | Equation |
|---|---|
Aligning Moment | input variable: output variable: where the value is negated to match the sign convention (TYDEX Wheel Axis) used in Vortex. Note: The magic formula has mixed units. The slip angle is expressed in degrees, the normal force Fz in the coefficients below is in kilonewtons, but the output torque is in Newton-meters. |
Peak Factor | |
Shape Factor | |
Stiffness Factor | where the product gives the slope of the curve at |
Curvature Factor | |
Horizontal Shift | |
Vertical Shift |
|
References
[1] E. 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.
[2] G. Genta, Motor Vehicle Dynamics: Modeling and Simulation, World Scientific, Singapore, 2003 (first published in 1997).
[3] H.B. Pacejka and E. Bakker, The Magic Formula Tyre Model, Vehicle System Dynamics, Vol. 21, Supplement 1, pp. 1-18, 1992.