Library / Automotive and vehicle dynamics / VEH-21
Automotive and vehicle dynamics
Tyre longitudinal slip ratio
Signed slip of a rolling tyre: how much faster (driving, positive) or slower (braking, negative) the wheel surface moves than the wheel centre.
- Ref
- VEH-21
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- κ
Re ω is the speed of the tyre surface (effective rolling radius times wheel angular speed), and Ux is the speed of the wheel centre along the wheel plane. The slip ratio is their difference divided by the larger of the two speeds, so it stays between −1 and +1. It is positive when driving (the surface moves faster than the vehicle) and negative when braking, and −1 corresponds to a locked wheel and +1 to a wheel spinning on the spot. This is one convention: use the matching sign and normalisation in the tyre model.
Signed slip of a rolling tyre: how much faster (driving, positive) or slower (braking, negative) the wheel surface moves than the wheel centre. The denominator keeps the value bounded between −1 and 1.
Calculator
- Slip in per cent4.762 %
- Tyre surface speed, Re ω21 m/s
- Slip speed, Re ω − Ux1 m/s
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 wheel surface speed of 21 m/s against a vehicle speed of 20 m/s is a slip ratio of 0.04762 (4.76 %), a light driving slip. Tyre force normally peaks at about 8 to 15 % slip and then falls.
- The limits are the locked wheel (ω = 0, which gives κ = −1) and the spinning wheel on a stationary vehicle (κ = +1). The denominator keeps the ratio finite at both extremes, unlike the simple (slip speed)/(vehicle speed) form.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ω Wheel angular speed | 10 to 150 rad/s | −0.85 | 0.5556 | 1.406 | +10% | +182 % |
| Ux Wheel-centre speed along the wheel plane | 5 to 50 m/s | −0.58 | 0.7619 | 1.342 | +10% | −195 % |
| Re Effective rolling radius | 0.2 to 0.5 m | −0.3 | 0.4286 | 0.7286 | +10% | +182 % |
Sweep of Wheel angular speed
| Wheel angular speed (rad/s) | Slip ratio |
|---|---|
| 10 | −0.85 |
| 27.5 | −0.5875 |
| 45 | −0.325 |
| 62.5 | −0.0625 |
| 80 | 0.1667 |
| 97.5 | 0.3162 |
| 115 | 0.4203 |
| 132.5 | 0.4969 |
| 150 | 0.5556 |
Sweep one variable
Variables and how each one contributes
Slip ratio
What the equation gives you. Everything else feeds it.
Effective rolling radius
The effective rolling radius of the tyre, in m. Scales the surface speed Reω, and so moves the slip ratio. A radius error of 1 % shifts κ by about 0.01. Raising it by 10 % (to 0.33) moves the result by +182 %, from 0.04762 to 0.1342.
Wheel angular speed
The angular speed of the wheel, in rad/s, from a wheel-speed sensor. Raises the surface speed. Since the default slip is small, a 10 % rise in ω moves κ a great deal. Raising it by 10 % (to 77) moves the result by +182 %, from 0.04762 to 0.1342.
Wheel-centre speed along the wheel plane
The speed of the wheel centre along the direction the wheel points, in m/s (the vehicle speed over the ground for a straight run). A higher wheel-centre speed with the same wheel speed means the tyre is slipping less (or braking more). Raising it by 10 % (to 22) moves the result by −195 %, from 0.04762 to −0.04545.
Worked example with the values above
- Surface speed: Re ω = 0.3 × 70 = 21 m/s, against Ux = 20 m/s.
- Slip speed: Re ω − Ux = 1 m/s. Denominator: the larger of the two speeds = 21 m/s.
- κ = 1 / 21 = 0.04762 (4.762 % driving).
Assumptions and limits
- Straight rolling with a fixed effective radius, and wheel-centre speed measured along the wheel plane.
- A bounded slip definition with the larger speed as the denominator. Matching tyre-model coefficients must use the same convention.
Common mistakes
- Using the unloaded radius in place of the effective rolling radius, which gives a false slip at zero true slip.
- Mixing the sign convention (positive in driving) with a tyre model that uses the opposite one.
Statistics and uncertainty
κ is a small difference of two nearly equal speeds, so it is very sensitive to speed errors: with a 1 % error in the effective radius, κ changes by about 0.01, which at a typical working slip of 0.05 to 0.1 is a 10 to 20 % error in the slip ratio. The wheel-centre speed is also hard to measure on a real vehicle, which is why ABS and traction controllers estimate it.
Where it is used
- ABS and traction control design, and any tyre model driven by longitudinal slip.
- Analysing wheelspin and wheel lock in test data.
Maths you need
- Rolling without slip: speed = R ω
- Ratios: a normalised difference