Library / Automotive and vehicle dynamics / VEH-19
Automotive and vehicle dynamics
Tyre force load sensitivity
An empirical fit for how the peak tyre force grows with the normal load.
- Ref
- VEH-19
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Fmax [kN]
Fref is the peak tyre force measured at a reference load Fz,ref, Fz the load now acting, and q an exponent fitted to tyre data (0 < q < 1). Because q is less than 1, doubling the load raises the peak force by less than a factor of two, so the friction coefficient Fmax/Fz falls as the load increases. It is an empirical local fit, valid for loads near the reference load.
An empirical fit for how the peak tyre force grows with the normal load. Because the exponent q is less than 1, the friction coefficient falls as load rises.
Calculator
- Friction coefficient at this load, Fmax / Fz0.941
- Friction coefficient at the reference load, Fref / Fz,ref1
- Loss of friction coefficient compared with the reference5.901 %
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.
- Raising the load from 4 to 6 kN (up 50 %) raises the peak force from 4 to 5.646 kN (up 41.1 %), so the friction coefficient falls from 1 to 0.941.
- This is why lateral load transfer costs grip. Moving 2 kN from the inner to the outer tyre (6 kN and 2 kN) gives a combined peak of 7.865 kN, less than the 8 kN of two tyres at 4 kN each, so a car with a large load transfer corners at lower grip.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Fref Peak force at the reference load | 0.5 to 12 kN | 0.7057 | 16.94 | 16.23 | +10% | +10 % |
| Fz,ref Reference normal load | 1 to 12 kN | 2.219 | 18.34 | 16.12 | +10% | −7.78 % |
| Fz Tyre normal load | 1 to 12 kN | 1.231 | 10.18 | 8.946 | +10% | +8.44 % |
| q Load sensitivity exponent (0 < q < 1) | 0.5 to 1 | 4.899 | 6 | 1.101 | +10% | +3.51 % |
Sweep of Tyre normal load
| Tyre normal load (kN) | Peak tyre force (kN) |
|---|---|
| 1 | 1.231 |
| 2.375 | 2.568 |
| 3.75 | 3.786 |
| 5.125 | 4.938 |
| 6.5 | 6.043 |
| 7.875 | 7.114 |
| 9.25 | 8.157 |
| 10.63 | 9.177 |
| 12 | 10.18 |
Sweep one variable
Variables and how each one contributes
Peak tyre force
What the equation gives you. Everything else feeds it.
Peak force at the reference load
The peak tyre force (longitudinal or lateral) at the reference load, in kN. Directly proportional: it scales the entire curve. Raising it by 10 % (to 4.4) moves the result by +10 %, from 5.646 to 6.211 kN.
Tyre normal load
The normal load now acting on the tyre, in kN. Raises the peak force, but by less than the load, because q is less than 1. This is the load sensitivity of the tyre. Raising it by 10 % (to 6.6) moves the result by +8.44 %, from 5.646 to 6.122 kN.
Reference normal load
The reference normal load at which Fref was measured, in kN. The same peak force reached at a higher reference load means the tyre gives less force per kN, so the result falls. Raising it by 10 % (to 4.4) moves the result by −7.78 %, from 5.646 to 5.207 kN.
Load sensitivity exponent (0 < q < 1)
The load sensitivity exponent, fitted from tyre data: 1 means grip is proportional to load, and smaller values mean stronger load sensitivity. Typical road tyres are 0.8 to 0.9. With the load above the reference, a higher exponent gives more grip. At q = 1 the friction coefficient would be constant. Raising it by 10 % (to 0.935) moves the result by +3.51 %, from 5.646 to 5.844 kN.
Worked example with the values above
- Load ratio: Fz / Fz,ref = 6 / 4 = 1.5.
- Raised to the power q: 1.5^0.85 = 1.411.
- Fmax = Fref × 1.411 = 4 × 1.411 = 5.646 kN.
Assumptions and limits
- An empirical power-law fit with a constant exponent, valid for loads near the reference load.
- Constant tyre temperature, pressure, camber and surface, and the exponent q between 0 and 1.
Common mistakes
- Assuming grip is proportional to load (q = 1), which hides the effect of load transfer.
- Extrapolating the fit to loads far from the reference load.
Statistics and uncertainty
The exponent q is fitted from tyre test data and is known to about ±0.05, which changes the peak force by a few per cent at a load ratio of 1.5. The fit is local: far from the reference load (below about 0.3 or above about 2 times) it can be seriously wrong, since real tyres saturate or fall off differently. Use measured tyre maps when the load range is wide.
Where it is used
- Explaining why lateral and longitudinal load transfer reduce the total grip of an axle.
- Tuning the front-to-rear roll stiffness balance for handling.
Maths you need
- Powers and ratios: x^q for 0 < q < 1
- Friction coefficient: peak force divided by load