glossary

Explanation

The Magic Formula is an empirical model that describes the relationship between tire forces/moments and the slip (either longitudinal or lateral) experienced by the tire. It’s called the “Magic Formula” because of its ability to accurately fit experimental data across a wide range of conditions, despite its relatively simple mathematical structure.

The general form of the Magic Formula for longitudinal force $F_x$ can be expressed as:

$$F_x=D\sin \left(C\arctan \left(B\kappa -E\left(B\kappa -\arctan (B\kappa )\right)\right)\right)$$

where:

• $\kappa$ is the Longitudinal Slip Ratio.
• $B$, $C$, $D$, and $E$ are fitting coefficients, often referred to as stiffness, shape, peak, and curvature factors, respectively.

For lateral force $F_y$, the formula is similar but uses the Slip Angle $\alpha$ as the input variable:

$$F_y=D\sin \left(C\arctan \left(B\alpha -E\left(B\alpha -\arctan (B\alpha )\right)\right)\right)$$

Version 6.2

Differences Between the Latest Magic Formula (MF 6.2) and the Original Magic Formula

Parameterization

• Original Magic Formula: Used a simpler set of parameters $B$, $C$, $D$, and $E$ primarily for fitting longitudinal and lateral forces based on slip ratio and slip angle.
• MF 6.2: Includes more detailed and refined parameters that consider additional factors such as load dependency, camber angle, and combined slip conditions. Parameters are often functions of normal load $F_z$, camber angle $\gamma$, and other conditions.

Handling Combined Slip

• Original Magic Formula: Primarily focused on separate handling of longitudinal and lateral slip with limited ability to accurately model combined slip conditions.
• MF 6.2: Better handles combined slip (where both longitudinal and lateral slips occur simultaneously) with refined equations that incorporate interactions between different slip components.

Inclusion of Camber Angle

• Original Magic Formula: Did not explicitly include camber angle effects in the equations.
• MF 6.2: Explicitly includes camber angle $\gamma$ in the equations, allowing for more accurate modeling of its effects on tire forces and moments.

Dynamic and Transient Behavior

• Original Magic Formula: Focused on steady-state behavior with limited capability to model dynamic and transient responses.
• MF 6.2: Incorporates parameters and terms to better capture dynamic behavior and transient responses, including tire relaxation length and other dynamic effects.

Load Sensitivity

• Original Magic Formula: Parameters were generally static and did not account for variations in normal load.
• MF 6.2: Parameters are load-dependent, allowing for more accurate modeling of how tire forces and moments change with varying normal loads.

Equations

Longitudinal Force Equation

Original Magic Formula:

$$F_x=D\sin \left(C\arctan \left(B\kappa -E\left(B\kappa -\arctan (B\kappa )\right)\right)\right)$$

MF 6.2:

$$F_x=D_x\sin \left(C_x\arctan \left(B_x\kappa -E_x\left(B_x\kappa -\arctan (B_x\kappa )\right)\right)\right)$$

where $D_x$, $C_x$, $B_x$, and $E_x$ are functions of normal load $F_z$, camber angle $\gamma$, and other conditions.

Lateral Force Equation

Original Magic Formula:

$$F_y=D\sin \left(C\arctan \left(B\alpha -E\left(B\alpha -\arctan (B\alpha )\right)\right)\right)$$

MF 6.2:

$$F_y=D_y\sin \left(C_y\arctan \left(B_y(\alpha +S_H)-E_y\left(B_y(\alpha +S_H)-\arctan (B_y(\alpha +S_H))\right)\right)\right)+S_V$$

where $D_y$, $C_y$, $B_y$, $E_y$, $S_H$ (horizontal shift), and $S_V$ (vertical shift) are functions of normal load $F_z$, camber angle $\gamma$, and combined slip effects.

Aligning Moment Equation

Original Magic Formula:

$$M_z=D\sin \left(C\arctan \left(B\alpha -E\left(B\alpha -\arctan (B\alpha )\right)\right)\right)$$

MF 6.2:

$$M_z=D_z\sin \left(C_z\arctan \left(B_z(\alpha +S_H)-E_z\left(B_z(\alpha +S_H)-\arctan (B_z(\alpha +S_H))\right)\right)\right)+S_{MZ}$$

where $D_z$, $C_z$, $B_z$, $E_z$, and $S_{MZ}$ (moment shift) are functions of normal load $F_z$, camber angle $\gamma$, and combined slip effects.

$S$ Parameters in MF 6.2

In the Magic Formula 6.2 (MF 6.2) tire model, the $S$ parameters are introduced to provide additional flexibility and accuracy in modeling the tire’s behavior. These parameters serve to fine-tune the model’s equations, providing horizontal and vertical shifts to better fit empirical data. These adjustments help to account for offsets and ensure that the modeled forces and moments accurately reflect the behavior observed in tire testing. The key $S$ parameters and their roles are as follow.

1. $S_H$ (Horizontal Shift):

   • Function: Adjusts the slip variable (angle or ratio) to account for shifts in the peak force/moment.
   • Impact: Helps to align the peak force/moment with the actual data, improving accuracy.

2. $S_V$ (Vertical Shift):

   • Function: Adjusts the output force or moment vertically.
   • Impact: Compensates for baseline offsets in the force/moment data, ensuring the model’s output matches the observed values.

3. $S_{MZ}$ (Moment Shift):

   • Function: Adjusts the aligning moment vertically.
   • Impact: Ensures the modeled aligning moment aligns with the empirical data by correcting baseline shifts.

Summary

The progression from the original Magic Formula to version 6.2 represents significant advancements in tire modeling, incorporating more detailed parameterization, handling of combined slip conditions, inclusion of camber angle effects, better dynamic and transient behavior representation, and load sensitivity. These enhancements make MF 6.2 a more robust and comprehensive model for accurately predicting tire forces and moments across a wide range of operating conditions.