Library / Automotive and vehicle dynamics / VEH-42
Automotive and vehicle dynamics
Tyre slip angle from local velocity
Slip angle of a rolling tyre from the forward and left velocity components ut and vt of the wheel centre in tyre axes.
- Ref
- VEH-42
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- α [rad]
The slip angle is the angle between a tyre’s heading and its velocity direction. With the lateral velocity vt positive to the left, α = arctan(−vt/ut): a wheel drifting to the left of its heading has negative vt and positive α under this sign convention, which gives positive lateral force for positive cornering stiffness. The equation needs ut > 0 and fails when the wheel is not rolling forward.
Slip angle of a rolling tyre from the forward and left velocity components ut and vt of the wheel centre in tyre axes. This sign convention gives a positive lateral force for positive slip angle and positive cornering stiffness. It needs forward rolling, ut > 0.
Calculator
- Slip angle in degrees5.711 °
- Wheel-centre speed, √(ut² + vt²)10.05 m/s
- Tangent of the slip angle, −vt/ut0.1
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.
- The wheel slips at 0.09967 rad (5.71°), well inside the linear tyre range of a few degrees.
- For small angles α ≈ −vt/ut = 0.1, so the arctan changes the answer by 0.331 % here.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| vt Lateral velocity component at the wheel centre | −20 to 20 m/s | −1.107 | 1.107 | 2.214 | +0.8 m/s | −79.9 % |
| ut Forward velocity component at the wheel centre | 0.1 to 100 m/s | 0.01 | 1.471 | 1.461 | +10% | −9.04 % |
Sweep of Lateral velocity component at the wheel centre
| Lateral velocity component at the wheel centre (m/s) | Slip angle (rad) |
|---|---|
| −20 | 1.107 |
| −15 | 0.9828 |
| −10 | 0.7854 |
| −5 | 0.4636 |
| 0 | 0 |
| 5 | −0.4636 |
| 10 | −0.7854 |
| 15 | −0.9828 |
| 20 | −1.107 |
Sweep one variable
Variables and how each one contributes
Slip angle
What the equation gives you. Everything else feeds it.
Forward velocity component at the wheel centre
The forward velocity component at the wheel centre, in m/s. A higher forward speed gives a smaller slip angle for the same sideways drift. Raising it by 10 % (to 11) moves the result by −9.04 %, from 0.09967 to 0.09066 rad.
Lateral velocity component at the wheel centre
The lateral velocity component at the wheel centre, in m/s, positive to the left. More leftward drift makes the slip angle more negative; more rightward drift makes it more positive. Raising it by 10 % (to −1.1) moves the result by +9.92 %, from 0.09967 to 0.1096 rad.
Worked example with the values above
- Ratio: −vt/ut = −(−1) / 10 = 0.1.
- α = arctan(0.1) = 0.099669 rad.
- In degrees: 5.7106°.
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