Library / Automotive and vehicle dynamics / VEH-18
Automotive and vehicle dynamics
Friction ellipse remaining longitudinal force
The longitudinal tyre force still available when part of the tyre grip is already used by a lateral force.
- Ref
- VEH-18
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Fx,max [kN]
Fz is the normal load on the tyre, Fy the lateral force already being used, and μx, μy the longitudinal and lateral friction coefficients. The friction ellipse says that (Fx/μxFz)² + (Fy/μyFz)² ≤ 1; it is a circle when the two coefficients are equal. This solves it for the largest longitudinal force Fx that can still be added. It is an approximate model, calibrated for the load, temperature, pressure and surface. If the lateral force exceeds μyFz, no longitudinal force is possible.
The longitudinal tyre force still available when part of the tyre grip is already used by a lateral force. It is the friction ellipse (a circle when μₓ = μᵧ) solved for the remaining capacity.
Calculator
- Peak longitudinal force with no lateral force, μx Fz4 kN
- Fraction of the lateral capacity in use75 %
- Fraction of the longitudinal capacity left66.14 %
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 4 kN tyre with μ = 1 has a 4 kN force circle. Using 3 kN of lateral force leaves 2.646 kN for braking or driving, not 1 kN as simple subtraction would suggest: forces combine as a vector, not as a sum.
- The remaining force collapses near the limit: at 3.9 kN lateral the tyre has only 0.8888 kN left, and at 3.6 kN it has 1.744 kN. This is why braking hard in a corner can overload the tyre and cause a slide.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kN) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Fz Tyre normal load | 1 to 10 kN | 0.781 | 9.539 | 8.758 | +10% | +21.7 % |
| Fy Lateral tyre force in use | 0 to 8 kN | 0 | 4 | 4 | +10% | −14.6 % |
| μx Longitudinal friction coefficient | 0.3 to 1.8 | 0.7937 | 4.762 | 3.969 | +10% | +10 % |
| μy Lateral friction coefficient | 0.3 to 1.8 | 0 | 3.636 | 3.636 | +10% | +10.6 % |
Sweep of Lateral tyre force in use
| Lateral tyre force in use (kN) | Remaining longitudinal force (kN) |
|---|---|
| 0 | 4 |
| 1 | 3.873 |
| 2 | 3.464 |
| 3 | 2.646 |
| 4 | 0 |
| 5 | — |
| 6 | — |
| 7 | — |
| 8 | — |
Sweep one variable
Variables and how each one contributes
Remaining longitudinal force
What the equation gives you. Everything else feeds it.
Tyre normal load
The normal load on the tyre, in kN. Raises both force limits in proportion. With Fy fixed, more load means less of the lateral capacity is used, so the remaining longitudinal force rises faster than the load. Raising it by 10 % (to 4.4) moves the result by +21.7 %, from 2.646 to 3.219 kN.
Lateral tyre force in use
The lateral force already being used, in kN, taken as positive. Lowers the remaining capacity, slowly at first and then sharply as Fy approaches the lateral limit. Raising it by 10 % (to 3.3) moves the result by −14.6 %, from 2.646 to 2.261 kN.
Longitudinal friction coefficient
The peak longitudinal friction coefficient of the tyre on this surface: about 1 for a road tyre on dry tarmac. Directly proportional: it scales the whole longitudinal capacity. Raising it by 10 % (to 1.1) moves the result by +10 %, from 2.646 to 2.91 kN.
Lateral friction coefficient
The peak lateral friction coefficient of the tyre on this surface: about 1 for a road tyre on dry tarmac. A higher lateral friction coefficient means a given Fy uses less of the capacity, so more longitudinal force remains. Raising it by 10 % (to 1.1) moves the result by +10.6 %, from 2.646 to 2.926 kN.
Worked example with the values above
- Fraction of lateral capacity in use: Fy / (μyFz) = 3 / (1 × 4) = 0.75.
- Square root term: √(1 − 0.75²) = √0.4375 = 0.6614.
- Fx,max = μx Fz × 0.6614 = 4 × 0.6614 = 2.646 kN.
Assumptions and limits
- An elliptical (or circular) combined-slip limit, which is a simplification of real tyres, with constant friction coefficients.
- The tyre normal load, temperature and surface are fixed, and the forces are the peak values the tyre can give.
Common mistakes
- Subtracting the lateral force from the peak force instead of combining them as a vector.
- Entering a lateral force above the limit μyFz, where no longitudinal capacity remains.
Statistics and uncertainty
The friction coefficients are the dominant uncertainty and are known to ±10 % or worse for a given surface. Near the limit the square root makes the answer extremely sensitive: when 90 % of the lateral capacity is used, a 5 % error in Fy or μy changes the remaining longitudinal force by roughly 25 %. Treat results near the limit as indicating a risk, not an exact prediction.
Where it is used
- Working out how much braking or acceleration is possible while cornering (trail braking).
- Explaining why traction control and ABS cut power or brake force in a corner.
Maths you need
- Pythagoras and ellipses: x²/a² + y²/b² = 1
- Friction: F ≤ μ N