Library / Automotive and vehicle dynamics / VEH-34
Automotive and vehicle dynamics
Cornering speed limit with downforce
The highest speed an ideal point-mass vehicle can hold on a corner of radius R when all tyre grip is used laterally and aerodynamic downforce adds to the tyre load.
- Ref
- VEH-34
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Umax [m/s]
The corner is taken as a point mass at constant speed U round a radius R, with the friction coefficient μ constant and all the grip used laterally. The required centripetal force m U²/R must equal μ times the normal load, which is the weight m g plus the downforce ½ρCLAU². Solving for U gives the equation shown. The denominator must be positive: if the downforce term grows faster than the centripetal term, this simple model gives no finite limit (the car would corner at any speed) and tyre load sensitivity and the aerodynamic limits take over.
The highest speed an ideal point-mass vehicle can hold on a corner of radius R when all tyre grip is used laterally and aerodynamic downforce adds to the tyre load. Without downforce it reduces to U² = μgR.
Calculator
- Speed without downforce, √(μ g R)38.36 m/s
- Maximum speed in km/h156.9 km/h
- Lateral acceleration, U² / R1.935 g
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.
- Without downforce this car could take the 100 m corner at 38.36 m/s; with the downforce it can take it at 43.57 m/s, 5.214 m/s faster, at a lateral acceleration of 1.94 g.
- Downforce helps most in fast corners, because it grows with the square of the speed: on a 150 m corner the gain is 10.74 m/s (57.72 m/s against 46.98 m/s), against 5.214 m/s on the 100 m corner.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Corner radius | 30 to 500 m | 21.76 | 634.2 | 612.4 | +10% | +6.44 % |
| A Reference area | 0.3 to 3 m² | 40.09 | 97.04 | 56.96 | +10% | +1.48 % |
| μ Tyre friction coefficient | 0.8 to 2.5 | 29.86 | 62.64 | 32.78 | +10% | +6.44 % |
| CL Downforce coefficient | 0.5 to 5 | 39.25 | 51.72 | 12.47 | +10% | +1.48 % |
| m Vehicle mass | 500 to 2,000 kg | 40.21 | 47.95 | 7.738 | +10% | −1.29 % |
| ρ Air density | 0.8 to 1.3 kg/m³ | 41.61 | 44.11 | 2.504 | +10% | +1.48 % |
Sweep of Downforce coefficient
| Downforce coefficient | Maximum cornering speed (m/s) |
|---|---|
| 0.5 | 39.25 |
| 1.063 | 40.34 |
| 1.625 | 41.52 |
| 2.188 | 42.8 |
| 2.75 | 44.22 |
| 3.313 | 45.79 |
| 3.875 | 47.53 |
| 4.438 | 49.5 |
| 5 | 51.72 |
Sweep one variable
Variables and how each one contributes
Maximum cornering speed
What the equation gives you. Everything else feeds it.
Tyre friction coefficient
The tyre friction coefficient used laterally: 1 to 1.2 for road tyres and 1.5 or more for racing tyres. Raises the speed, with a bit less than a square root effect because the downforce term in the denominator also grows. Raising it by 10 % (to 1.65) moves the result by +6.44 %, from 43.57 to 46.38 m/s.
Vehicle mass
The vehicle mass, in kg. Lowers the speed slightly: a heavier car gets less benefit from its downforce, because the weight dominates the tyre load. Raising it by 10 % (to 880) moves the result by −1.29 %, from 43.57 to 43.01 m/s.
Corner radius
The radius of the corner, in m. Raises the speed. Without downforce U grows with the square root of R; with downforce the effect is stronger. Raising it by 10 % (to 110) moves the result by +6.44 %, from 43.57 to 46.38 m/s.
Air density
The air density, in kg/m³. Raises the downforce and so the cornering speed. Thinner air at altitude reduces it. Raising it by 10 % (to 1.32) moves the result by +1.48 %, from 43.57 to 44.22 m/s.
Downforce coefficient
The downforce coefficient (positive means downwards). Raises the speed: more downforce means more grip. Raising it by 10 % (to 2.75) moves the result by +1.48 %, from 43.57 to 44.22 m/s.
Reference area
The reference area of the downforce coefficient, in m². Raises the speed in the same way as the coefficient. Raising it by 10 % (to 0.88) moves the result by +1.48 %, from 43.57 to 44.22 m/s.
Worked example with the values above
- Denominator: m/R − ½μρCLA = 8 − 1.8 = 6.2 kg/m. Numerator: μ m g = 11,770 N.
- U² = 11,770 / 6.2 = 1,899 m²/s².
- Umax = √1,899 = 43.57 m/s (156.9 km/h).
Assumptions and limits
- A point mass with constant μ and constant aerodynamic coefficients, on a flat track in still air, using all the grip laterally.
- The tyre load sensitivity is ignored, so the real speed is lower; the denominator must be positive.
Common mistakes
- Using a friction coefficient greater than the tyres can sustain at the higher downforce load. Load sensitivity lowers μ as the load rises.
- Using the result when the denominator is zero or negative.
Statistics and uncertainty
The answer comes from the friction coefficient, which is known to ±5 to 10 % even for racing tyres and varies with temperature and the rubber on the track. The downforce term is known to ±5 %. Close to the point where the denominator reaches zero the speed becomes very large and very sensitive to the aerodynamic terms, so a result more than about twice the no-downforce speed should be treated with suspicion.
Where it is used
- Race car lap simulations and judging what speed downforce is worth in a given corner.
- Setting wing levels for a circuit.
Maths you need
- Circular motion: centripetal force m U² / R
- Friction: F ≤ μ N