Library / Automotive and vehicle dynamics / VEH-22
Automotive and vehicle dynamics
Magic Formula tyre force
The generic pure-slip Magic Formula curve for tyre force against slip.
- Ref
- VEH-22
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- F [kN]
X is the slip variable: the slip ratio κ for a longitudinal force or the slip angle in radians for a lateral force. B, C, D and E are fitted to tyre test data: B is the stiffness factor, C the shape factor, D the peak force and E the curvature factor, which controls how sharp the peak is and how the force falls after it. This is the simplified zero-shift form without combined-slip coupling or any explicit dependence on load, camber and temperature. The sign convention must match the slip variable used.
The generic pure-slip Magic Formula curve for tyre force against slip. B sets the initial slope, C the shape, D the peak and E the curvature near and beyond the peak. This is the simplified zero-shift form.
Calculator
- Normalised force, F / D0.9812
- Initial slope, B C D79.2 kN per unit slip
- Force from the linear law, B C D X7.92 kN
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With these parameters the curve peaks at about X = 0.132 with a force of 4 kN (100 % of D), and the default X = 0.1 gives 3.925 kN. For a longitudinal force, a peak at 10 to 15 % slip is typical.
- Beyond the peak the force falls: at X = 0.4 it is 3.317 kN. Far past the peak it tends to D sin(Cπ/2) = 2.09 kN. This is why a locked or spinning wheel gives less force than a rolling one, and why ABS works.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| D Peak force | 0.5 to 10 kN | 0.4906 | 9.812 | 9.321 | +10% | +10 % |
| X Slip variable (slip ratio, or slip angle in rad) | 0 to 0.4 | 0 | 4 | 4 | +10% | +1.12 % |
| B Stiffness factor | 3 to 25 | 1.837 | 4 | 2.163 | +10% | +1.12 % |
| C Shape factor | 1 to 2.5 | 2.963 | 4 | 1.036 | +10% | +1.75 % |
| E Curvature factor | −1 to 1 | 3.709 | 4 | 0.2909 | +0.04 | −0.196 % |
Sweep of Slip variable (slip ratio, or slip angle in rad)
| Slip variable (slip ratio, or slip angle in rad) | Tyre force (kN) |
|---|---|
| 0 | 0 |
| 0.05 | 3.057 |
| 0.1 | 3.925 |
| 0.15 | 3.984 |
| 0.2 | 3.859 |
| 0.25 | 3.704 |
| 0.3 | 3.557 |
| 0.35 | 3.428 |
| 0.4 | 3.317 |
Sweep one variable
Variables and how each one contributes
Tyre force
What the equation gives you. Everything else feeds it.
Stiffness factor
The stiffness factor, which sets the initial slope of the curve (the slope at the origin is B C D). Typical values are 8 to 14 for longitudinal and 8 to 12 for lateral force. Steepens the initial rise and moves the peak to a smaller slip. At the default slip it matters a good deal, because the tyre is already near its peak. Raising it by 10 % (to 13.2) moves the result by +1.12 %, from 3.925 to 3.969 kN.
Shape factor
The shape factor, which sets the overall form of the curve: about 1.3 to 1.9 for road tyres. Changes the height and fall-off of the curve. A larger C pushes the force further down after the peak. Raising it by 10 % (to 1.815) moves the result by +1.75 %, from 3.925 to 3.994 kN.
Peak force
The peak force that the tyre can produce, in kN (usually μ Fz for the tyre load). Directly proportional, since D scales the whole curve. Raising it by 10 % (to 4.4) moves the result by +10 %, from 3.925 to 4.317 kN.
Curvature factor
The curvature factor, which controls how sharp the peak is. Values from 0 to 1 give a rounded peak, negative values a sharper one. Moves the peak to a higher slip and changes the fall-off. Its effect is small before the peak and larger after it. Raising it by 0.1 (to 0.4) moves the result by −0.512 %, from 3.925 to 3.905 kN.
Slip variable (slip ratio, or slip angle in rad)
The slip variable: the slip ratio (dimensionless) or the slip angle in radians. Positive values give a positive force. Pushes the tyre along its force curve. Below the peak a rise in slip raises the force, but beyond the peak it lowers it. Raising it by 10 % (to 0.11) moves the result by +1.12 %, from 3.925 to 3.969 kN.
Worked example with the values above
- Scaled slip: B X = 12 × 0.1 = 1.2. Inner term: B X − E (B X − tan⁻¹(B X)) = 1.2 − 0.3 × (1.2 − 0.8761) = 1.103.
- Shape: C tan⁻¹(1.103) = 1.65 × 0.8343 = 1.377 rad, and sin(1.377) = 0.9812.
- F = D × 0.9812 = 4 × 0.9812 = 3.925 kN.
Assumptions and limits
- Pure slip (longitudinal or lateral, not both), with the simplified zero-shift form and no horizontal or vertical offset.
- Parameters fitted at one load, pressure, camber and temperature.
Common mistakes
- Using the same parameters for longitudinal and lateral force. Each direction has its own set.
- Entering the slip angle in degrees when the formula expects radians.
Statistics and uncertainty
The four parameters are fitted to measured data and are strongly correlated, so a fit that matches the data well can still give quite different parameter sets. Typical fit errors on force are 3 to 5 % within the range of the test. The parameters also vary with load, camber, pressure and temperature, which this simplified form does not include; use it with a full tyre model when load varies.
Where it is used
- Fitting measured tyre data to give a curve for vehicle simulation.
- Illustrating the peak and fall-off of grip with slip for ABS and traction control.
Maths you need
- Trigonometric functions: sin and inverse tan in radians
- Curve fitting: parameters chosen to match measured data