Library / Automotive and vehicle dynamics / VEH-20
Automotive and vehicle dynamics
Linear tyre lateral force
Lateral force of a tyre at small slip angle: proportional to the slip angle with cornering stiffness Cα as the constant.
- Ref
- VEH-20
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Fy [kN]
Cα is the cornering stiffness of the tyre (kN per radian of slip angle) and α the slip angle: the angle between the direction the wheel points and the direction it actually moves. A positive slip angle means the velocity points to the left of the wheel heading, so the tyre force on the vehicle points to the right: Fy is negative. The linear law applies near zero slip, before the tyre saturates. Entered values of Cα are positive and the slip angle is in degrees.
Lateral force of a tyre at small slip angle: proportional to the slip angle with cornering stiffness Cα as the constant. The minus sign shows that the force opposes the sideways drift.
Calculator
- Slip angle in radians0.03491 rad
- Lateral force in newtons−2,094 N
- Cornering stiffness per degree1.047 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.
- A tyre with Cα = 60 kN/rad at a 2° slip angle gives 2.094 kN of lateral force, opposing the sideways motion. Doubling the slip angle to 4° would double it in this linear law, but a real tyre starts to saturate at 4° to 8°.
- The slope of the real tyre curve at the origin is the cornering stiffness, so this formula is the first step of the Magic Formula. The two agree at small angles and diverge as the tyre approaches its peak.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| α Slip angle | −8 to 8 ° | −8.378 | 8.378 | 16.76 | +0.32 ° | +16 % |
| C_α Cornering stiffness | 10 to 200 kN/rad | −6.981 | −0.3491 | 6.632 | +10% | +10 % |
Sweep of Slip angle
| Slip angle (°) | Lateral force (kN) |
|---|---|
| −8 | 8.378 |
| −6 | 6.283 |
| −4 | 4.189 |
| −2 | 2.094 |
| 0 | 0 |
| 2 | −2.094 |
| 4 | −4.189 |
| 6 | −6.283 |
| 8 | −8.378 |
Sweep one variable
Variables and how each one contributes
Lateral force
What the equation gives you. Everything else feeds it.
Cornering stiffness
The cornering stiffness of one tyre, in kN/rad: about 40 to 100 for a car tyre, rising with load and falling with a worn or soft tyre. Directly proportional: a stiffer tyre gives more lateral force for the same slip angle. Raising it by 10 % (to 66) moves the result by +10 %, from −2.094 to −2.304 kN.
Slip angle
The slip angle, in degrees: the angle between the wheel heading and its direction of travel. Positive when the velocity points to the left of the heading. Directly proportional in this linear law, but only up to a few degrees, where the tyre starts to saturate. Raising it by 10 % (to 2.2) moves the result by +10 %, from −2.094 to −2.304 kN.
Worked example with the values above
- Convert the slip angle to radians: α = 2 × π/180 = 0.03491 rad.
- Cornering stiffness: Cα = 60 kN/rad.
- Fy = −Cα α = −60 × 0.03491 = −2.094 kN (the sign shows it opposes the drift).
Assumptions and limits
- Small slip angles, in the linear range of the tyre (typically below about 4° for a road tyre).
- Steady rolling at constant load and speed, with no longitudinal slip or camber thrust.
Common mistakes
- Entering the slip angle in radians when the input expects degrees.
- Using the linear law near the limit of grip, where the force is much less than this formula gives.
Statistics and uncertainty
The cornering stiffness is not a fixed property of the tyre. It changes by 20 % or more with load, inflation pressure, temperature, speed and tread wear, and each tyre of a set differs a little. For a vehicle model, treat Cα as known to ±15 %, which gives the same relative error in Fy at small slip.
Where it is used
- Bicycle-model handling analysis, and stability calculations at small steering angles.
- Reading cornering stiffness from tyre test data or estimating the slip angle needed for a given lateral force.
Maths you need
- Angles: degrees and radians
- Linear relationships: force proportional to a slip angle