Library / Automotive and vehicle dynamics / VEH-44
Automotive and vehicle dynamics
Transient tyre force after a slip-angle step
Lateral force building up as a tyre rolls a distance s after a step in slip angle, with relaxation length ℓ.
- Ref
- VEH-44
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Fy [N]
After a step change in slip angle the lateral force does not appear instantly: the tyre must roll a distance of the order of its relaxation length ℓ before the contact patch deflects. The first-order model F(s) = Fss + (F₀ − Fss)e^(−s/ℓ) describes this build-up against rolling distance s, so the lag in time is ℓ/U and falls as the speed rises.
Lateral force building up as a tyre rolls a distance s after a step in slip angle, with relaxation length ℓ. A first-order build-up from the initial force F₀ to the steady target F_ss, with a constant target and constant relaxation length.
Calculator
- Fraction of the step completed, 1 − e^(−s/ℓ)86.47 %
- Gap still to close, Fy − Fss−406 N
- Distance to complete 95 % of the step, 3ℓ1.5 m
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.
- After 1 m of rolling the force has reached 86.5 % of the step, 2,594 N out of 3,000 N.
- At 2 relaxation lengths the force is nearly there; at 3ℓ = 1.5 m it would be at 95 %. At 20 m/s that delay is 75 ms.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Fss Steady-state target force | −20,000 to 20,000 N | −17,290 | 17,290 | 34,590 | +800 N | +26.7 % |
| F0 Initial force | −20,000 to 20,000 N | −112.7 | 5,301 | 5,413 | +800 N | +4.17 % |
| s Rolling distance since the step | 0 to 20 m | 0 | 3,000 | 3,000 | +10% | +2.84 % |
| ℓ Relaxation length | 0.05 to 5 m | 543.8 | 3,000 | 2,456 | +10% | −3.12 % |
Sweep of Rolling distance since the step
| Rolling distance since the step (m) | Lateral force (N) |
|---|---|
| 0 | 0 |
| 2.5 | 2,980 |
| 5 | 3,000 |
| 7.5 | 3,000 |
| 10 | 3,000 |
| 12.5 | 3,000 |
| 15 | 3,000 |
| 17.5 | 3,000 |
| 20 | 3,000 |
Sweep one variable
Variables and how each one contributes
Lateral force
What the equation gives you. Everything else feeds it.
Steady-state target force
The steady-state force the slip angle finally gives, in N. Raises the final force and the build-up in proportion. Raising it by 10 % (to 3,300) moves the result by +10 %, from 2,594 to 2,853 N.
Initial force
The force at the moment of the step, in N. A higher starting force reduces the build-up that remains. Raising it by 10 % (to 0) moves the result by +0 %, from 2,594 to 2,594 N.
Rolling distance since the step
The rolling distance since the step, in m. More distance brings the force closer to the steady value; the gap shrinks as e^(−s/ℓ). Raising it by 10 % (to 1.1) moves the result by +2.84 %, from 2,594 to 2,668 N.
Relaxation length
The relaxation length, in m. A longer relaxation length slows the build-up, so the force lags the steering more. Raising it by 10 % (to 0.55) moves the result by −3.12 %, from 2,594 to 2,513 N.
Worked example with the values above
- Decay factor: e^(−s/ℓ) = e^(−1/0.5) = 0.13534.
- Gap at the start: F₀ − Fss = 0 − 3,000 = −3,000 N.
- Fy = 3,000 + (−3,000) × 0.13534 = 2,594 N.
Assumptions and limits
- A rolling tyre on a flat road with a point contact, small angles where a linearised form is used, and constant properties during the event.
- Velocities resolved in the body or tyre axes named on the page, with the sign convention stated there.
Common mistakes
- Mixing the SAE and ISO sign conventions for slip angle and lateral force.
- Entering steer angle in radians when degrees are expected, or the reverse.
- Using the relaxation-length formula for a step that is not small enough to stay in the linear tyre range.
Statistics and uncertainty
Tyre forces are known only to about 5 % to 15 % in practice, and slip angle and slip ratio are small differences of large velocities, so small velocity errors give large relative errors in slip.
Where it is used
- Computing tyre slip in handling simulations from measured vehicle motion.
- Estimating steering feel and wheel-speed response in control design.
Maths you need
- Rigid-body velocity: v_wheel = v_cg + ω × r
- Slip angle: the angle between a tyre’s heading and its velocity
- Relaxation length: the rolling distance over which tyre force builds up