The Pacejka / Magic Formula (MF) equations were conceived to fit the data gathered from experimental tests with real tires. The equations can then predict the behavior of each specific real tire with great precision. Real Pacejka/MF data sets are heavily protected intellectual property of the tire manufacturers. However, when used in video games where gameplay is important, the Pacejka curves can be fine tuned in a variety of ways in order to achieve the desired result.

This guide explains how each parameter of the **Pacejka ’94 specification** affects the resulting curve, as well as the typical range for the values.

## Longitudinal Force

Pacejka ’94 Longitudinal Force parameters:

Parameter | Role | Units | Typical range | Sample |
---|---|---|---|---|

b0 | Shape factor | 1.4 .. 1.8 | 1.5 | |

b1 | Load influence on longitudinal friction coefficient (*1000) | 1/kN | -80 .. +80 | 0 |

b2 | Longitudinal friction coefficient (*1000) | 900 .. 1700 | 1100 | |

b3 | Curvature factor of stiffness/load | N/%/kN^2 | -20 .. +20 | 0 |

b4 | Change of stiffness with slip | N/% | 100 .. 500 | 300 |

b5 | Change of progressivity of stiffness/load | 1/kN | -1 .. +1 | 0 |

b6 | Curvature change with load^2 | -0.1 .. +0.1 | 0 | |

b7 | Curvature change with load | -1 .. +1 | 0 | |

b8 | Curvature factor | -20 .. +1 | -2 | |

b9 | Load influence on horizontal shift | %/kN | -1 .. +1 | 0 |

b10 | Horizontal shift | % | -5 .. +5 | 0 |

b11 | Vertical shift | N | -100 .. +100 | 0 |

b12 | Vertical shift at load = 0 | N | -10 .. +10 | 0 |

b13 | Curvature shift | -1 .. +1 | 0 |

#### Pacejka ’94 longitudinal formula

Coefficient | Name | Parameters | Formula |
---|---|---|---|

C | Shape factor | b0 | C = b0 |

D | Peak factor | b1, b2 | D = Fz · (b1·Fz + b2) |

BCD | Stiffness | b3, b4, b5 | BCD = (b3·Fz^{2} + b4·Fz) · e^{(-b5·Fz)} |

B | Stiffness factor | BCD, C, D | B = BCD / (C·D) |

E | Curvature factor | b6, b7, b8, b13 | E = (b6·Fz^{2} + b7·Fz + B8) · (1 – b13·sign(slip+H)) |

H | Horizontal shift | b9, b10 | H = b9·Fz + b10 |

V | Vertical shift | b11, b12 | V = b11·Fz + b12 |

Bx1 | (composite) |
Bx1 = B · (slip + H) |

**F = D · sin(C · arctan(Bx1 – E · (Bx1 – arctan(Bx1)))) + V**

where:

- F = longitudinal force in N (newtons)
- Fz = vertical force in kN (kilonewtons)
- slip = slip ratio in percentage (0..100)

#### Initial values for setting up the longitudinal curve

Reference load | 4000 |

Maximum load | 13000 |

b0 | 1.5 |

b2 | 1100 |

b4 | 300 |

b8 | -2 |

all other | 0 |

#### Load-independent parameters

Load don’t have influence on these. If all other parameters are 0 then the curve keeps itself invariant with respect to the vertical load.

The first four (**b0, b2, b4, b8**) are the most relevant parameters that define the curve’s shape.

Parameter | Typical range | Description | Related |
---|---|---|---|

b0 |
1.4 .. 1.8 | General shape of the curve. Defines the amount of falloff after the peak.The Pacejka model defines b0 = 1.65 for the longitudinal force. |
C |

b2 |
900 .. 1700 | Friction coefficient at the peak (vertical coordinate) *1000. | D |

b4 |
100 .. 500 | Peak’s horizontal position specified as “ascent rate”. | BCD |

b8 |
-20 .. +1 | Curvature at the peak. The more negative = more “sharp”. Has influence on the falloff afterwards. | E |

b10 | -5 .. +5 | Curve’s horizontal shift | Sh |

b11 | -100 .. +100 | Curve’s vertical shift | Sv |

b13 | -1 .. +1 | Adjustment of the curvature at the peak. Similar to b8 | E |

#### Load-dependent parameters

The desired curve’s shape must be configured at the reference load. Typically the related parameters (load independent) must also be tweaked in order to conserve the shape.

These load-dependent parameters must be verified along all the load range, from minimum to maximum, in order to ensure their coherency. The coherency limit of each parameter can be located at the maximum load. The exception is b12, which must be verified at the minimum load.

Parameter | Typical range | Description | Related |
---|---|---|---|

b1 | -80 .. +80 | Change of the friction coefficient at the peak. Positive = more friction with more load. Negative = less friction with more load. |
D, b2 |

b3 | -20 .. +20 | Change of the peak’s horizontal position. Positive = increases ascent rate with load (moves to the left). Negative = decreases ascent rate with load (moves to the right). |
BCD, b4 |

b5 | -1 .. +1 | Lineal change of the peak’s horizontal position. Similar to b3 but more lineal and with reverse effect positive-negative. Positive = decreases ascent rate with load. Negative = increases ascent rate with load. |
BCD, b4 |

b6 | -0.1 .. +0.1 | Quadratic change of the curvature at the peak. Positive = more flat with load. Negative = sharper with load. |
E, b8 |

b7 | -1 .. +1 | Change of the curvature at the peak. Same as b6 but more lineal. Positive = more flat with load. Negative = sharper with load. |
E, b8 |

b9 | -1 .. +1 | Change of the horizontal shift. Positive = shifts to the left with more load. Negative = shifts to the right with more load. |
Sh, b10 |

b12 | -10 .. +10 | Vertical shift when approaching zero load. Must be verified for coherency at the configured minimum load. |
Sv, b11 |

## Lateral Force

Pacejka ’94 Lateral Force parameters:

Parameter | Role | Units | Typical range | Sample |
---|---|---|---|---|

a0 | Shape factor | 1.2 .. 18 | 1.4 | |

a1 | Load influence on lateral friction coefficient (*1000) | 1/kN | -80 .. +80 | 0 |

a2 | Lateral friction coefficient (*1000) | 900 .. 1700 | 1100 | |

a3 | Change of stiffness with slip | N/deg | 500 .. 2000 | 1100 |

a4 | Change of progressivity of stiffness / load | 1/kN | 0 .. 50 | 10 |

a5 | Camber influence on stiffness | %/deg/100 | -0.1 .. +0.1 | 0 |

a6 | Curvature change with load | -2 .. +2 | 0 | |

a7 | Curvature factor | -20 .. +1 | -2 | |

a8 | Load influence on horizontal shift | deg/kN | -1 .. +1 | 0 |

a9 | Horizontal shift at load = 0 and camber = 0 | deg | -1 .. +1 | 0 |

a10 | Camber influence on horizontal shift | deg/deg | -0.1 .. +0.1 | 0 |

a11 | Vertical shift | N | -200 .. +200 | 0 |

a12 | Vertical shift at load = 0 | N | -10 .. +10 | 0 |

a13 | Camber influence on vertical shift, load dependent | N/deg/kN | -10 .. +10 | 0 |

a14 | Camber influence on vertical shift | N/deg | -15 .. +15 | 0 |

a15 | Camber influence on lateral friction coefficient | 1/deg | -0.01 .. +0.01 | 0 |

a16 | Curvature change with camber | -0.1 .. +0.1 | 0 | |

a17 | Curvature shift | -1 .. +1 | 0 |

#### Pacejka ’94 lateral formula

Coefficient | Name | Parameters | Formula |
---|---|---|---|

C | Shape factor | a0 | C = a0 |

D | Peak factor | a1, a2, a15 | D = Fz · (a1·Fz + a2) · (1 – a15·γ^{2}) |

BCD | Stiffness | a3, a4, a5 | BCD = a3 · sin(atan(Fz / a4) · 2) · (1 – a5·|γ|) |

B | Stiffness factor | BCD, C, D | B = BCD / (C·D) |

E | Curvature factor | a6, a7, a16, a17 | E = (a6·Fz + a7) · (1 – (a16·γ + a17)·sign(slip+H)) |

H | Horizontal shift | a8, a9, a10 | H = a8·Fz + a9 + a10·γ |

V | Vertical shift | a11, a12, a13, a14 | V = a11·Fz + a12 + (a13·Fz + a14)·γ·Fz |

Bx1 | (composite) |
Bx1 = B · (slip + H) |

**F = D · sin(C · arctan(Bx1 – E · (Bx1 – arctan(Bx1)))) + V**

where:

- F = lateral force in N (newtons)
- Fz = vertical force in kN (kilonewtons)
- slip = slip angle in degrees
- γ = camber angle in degrees

#### Initial values for setting up the lateral curve

Reference load | 4000 |

Maximum load | 13000 |

Camber | 0 |

a0 | 1.4 |

a2 | 1100 |

a3 | 1100 |

a4 | 10 |

a7 | -2 |

all other | 0 |

#### Curve’s shape and horizontal behavior with load

The curve’s shape itself always depend on load at the horizontal axis according to the parameters a3 and a4.

Parameter | Typical range | Description | Related |
---|---|---|---|

a0 |
1.2 .. 1.8 | General shape of the curve. Defines the amount of falloff after the peak.The Pacejka model defines a0 = 1.3 for the lateral force. |
C |

a2 |
900 .. 1700 | Friction coefficient at the peak (vertical coordinate) *1000. | D |

a3* |
500 .. 2000 | Peak’s horizontal position at the reference load, specified as “ascent rate”. | BCD |

a4* |
0 .. 50 | Change of the peak’s horizontal position with load. Smaller value = bigger change with load. | BCD |

a7 |
-20 .. +1 | Curvature at the peak. The more negative = more “sharp”. Has influence on the falloff afterwards. | E |

a9 | -1 .. +1 | Curve’s horizontal shift | Sh |

a11 | -200 .. +200 | Curve’s vertical shift | Sv |

a17 | -1 .. +1 | Adjustment of the curvature at the peak. Similar to a7. | E |

* Configure the horizontal behavior with load

#### Load-dependent parameters

The desired curve’s shape must be configured at the reference load. Typically the related parameters (load independent) must also be tweaked in order to keep the shape.

These load-dependent parameters must be verified along all the load range, from minimum to maximum, in order to ensure their coherency. The coherency limit of each parameter can be located at the maximum load. The exception is a12, which must be verified at the minimum load.

Parameter | Typical range | Description | Related |
---|---|---|---|

a1 | -80 .. +80 | Change of the friction coefficient at the peak. Positive = more friction with more load. Negative = less friction with more load. |
D, a2 |

a6 | -2 .. +2 | Change of the curvature at the peak. Positive = more flat with load. Negative = sharper with load. |
E, a7 |

a8 | -1 .. +1 | Change of the horizontal shift. Positive = shifts to the left with more load. Negative = shifts to the right with more load. |
Sh, a9 |

a12 | -10 .. +10 | Vertical shift when approaching zero load. Must be verified for coherency at the configured minimum load. |
Sv, a11 |

#### Camber-dependent parameters

They must ve verified along all the camber range in order to ensure their coherency. The coherency limit of each parameter can be located at the limits of the camber range. These parameters don’t have effect if camber is 0.

Parameter | Typical range | Description | Related |
---|---|---|---|

a5 | -0.1 .. +0.1 | Change of the peak’s horizontal position. Positive = decreases ascent rate with camber (moves to the right). Negative = increases ascent rate with load (moves to the left). |
BCD, a3 |

a10 | -0.1 .. +0.1 | Change of the horizontal shift. Same sign as camber = shifts to the left. Opposite sign as camber = shifts to the right. |
Sh, a9 |

a13 | -10 .. +10 | Change of the vertical shift according to camber and load. Same sign as camber = shifts upwards. Opposite sign as camber = shifts downwards. The more load the more camber effect. |
Sv, a11 |

a14 | -15 .. +15 | Change of the vertical shift. Same sign as camber = shifts upwards. Opposite sign as camber = shifts downwards. |
Sv, a11 |

a15 | -0.01 .. +0.01 | Change of the friction coefficient at the peak. Positive = less friction with camber. Negative = more friction with camber. |
D, a2 |

a16 | -0.1 .. +0.1 | Change of the curvature at the peak. Same same sign as camber = more flat. Opposite sign as camber = sharper. |
E, a7 |

## Simplified Magic Formula with constant coefficients

This simplification uses the four dimensionless B, C, D, E coefficients only. Numerical values are based on empirical tire data. The coefficients do not depend on load, camber, or further parameters. Source

Coefficient | Name | Typical range | Typical values for longitudinal forces | |||
---|---|---|---|---|---|---|

Dry tarmac | Wet tarmac | Snow | Ice | |||

B | Stiffness | 4 .. 12 | 10 | 12 | 5 | 4 |

C* | Shape | 1 .. 2 | 1.9 | 2.3 | 2 | 2 |

D | Peak | 0.1 .. 1.9 | 1 | 0.82 | 0.3 | 0.1 |

E | Curvature | -10 .. 1 | 0.97 | 1 | 1 | 1 |

**F = Fz · D · sin(C · arctan(B·slip – E · (B·slip – arctan(B·slip))))**

where:

- F = tire force in N (newtons)
- Fz = vertical force in N (newtons)
- slip = slip ratio in percentage (0..100, for longitudinal force) or slip angle in degrees (for lateral force)

* The Pacekja model specifies the shape as C=1.65 for the longitudinal force and C=1.3 for the lateral force.

Hi,

I am currently doing my Mechanical Engineering Honours degree and I am investigating the Magic Formula with respect to lateral forces and pure slip.

How can I determine the parameters when the lateral force vs slip angle is given?

The Magic Formula is an empirical method. The parameters are somewhat iterated so the curve fits the real data. Here is an example:

http://white-smoke.wikifoundry.com/page/Tyre+curve+fitting+and+validation

Hi Edy,

have you got any example about tractor tires?