Library / Automotive and vehicle dynamics / VEH-41
Automotive and vehicle dynamics
Wheel-centre velocity in tyre coordinates
Forward component of the wheel-centre velocity in the steered tyre frame.
- Ref
- VEH-41
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- ut [m/s]
Transport theorem for a rigid body: the velocity of a wheel at (x, y) is the centre-of-gravity velocity plus ω × r, which in body axes is (u − r·y, v + r·x). Rotating through the steer angle δ gives the tyre-frame forward component ut = (u − r y)cos δ + (v + r x)sin δ. The tyre frame lateral component then follows as vt = −(u − r y)sin δ + (v + r x)cos δ.
Forward component of the wheel-centre velocity in the steered tyre frame. Body velocities u and v at the centre of gravity are carried to the wheel at (x, y) using the yaw rate r, then rotated by the steer angle δ. The sideways component is in the derived rows. Positive y and steer are to the left.
Calculator
- Lateral velocity in the tyre frame, −(u − r y)sin δ + (v + r x)cos δ1 m/s
- Wheel-centre speed9.552 m/s
- Slip angle of this wheel−6.009 °
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 centre moves forward at 9.5 m/s in its own axes, slightly below the 10 m/s of the centre of gravity because the wheel is on the inside of a left turn.
- The yaw rate changes the forward speed by r·y = 0.5 m/s here, so wheel-speed sensors on opposite sides read differently in a turn.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| u Forward speed of the CG | 0 to 100 m/s | −0.5 | 99.5 | 100 | +10% | +10.5 % |
| r Yaw rate | −2 to 2 rad/s | 8 | 12 | 4 | +0.08 rad/s | −0.842 % |
| δ Steer angle | −40 to 40 ° | 6.635 | 9.552 | 2.917 | +1.6 ° | +0.255 % |
| y Wheel position left of the CG | −1.5 to 1.5 m | 9.25 | 10.75 | 1.5 | +0.06 m | −0.316 % |
| v Lateral speed of the CG | −10 to 10 m/s | 9.5 | 9.5 | 0 | +0.4 m/s | 0 % |
| x Wheel position ahead of the CG | −3 to 3 m | 9.5 | 9.5 | 0 | +0.12 m | 0 % |
Sweep of Steer angle
| Steer angle (°) | Forward velocity in the tyre frame (m/s) |
|---|---|
| −40 | 6.635 |
| −30 | 7.727 |
| −20 | 8.585 |
| −10 | 9.182 |
| 0 | 9.5 |
| 10 | 9.529 |
| 20 | 9.269 |
| 30 | 8.727 |
| 40 | 7.92 |
Sweep one variable
Variables and how each one contributes
Forward velocity in the tyre frame
What the equation gives you. Everything else feeds it.
Forward speed of the CG
The forward speed of the centre of gravity, in m/s. Raises the tyre forward speed almost one for one. Raising it by 10 % (to 11) moves the result by +10.5 %, from 9.5 to 10.5 m/s.
Lateral speed of the CG
The lateral speed of the centre of gravity, in m/s, positive to the left. Adds to the forward tyre speed in proportion to sin δ, so it matters only when the wheel is steered. Raising it by 10 % (to 0) moves the result by +0 %, from 9.5 to 9.5 m/s.
Yaw rate
The yaw rate, in rad/s, positive for a left turn. Raises the lateral wheel speed in proportion to x, and changes the forward wheel speed in proportion to y. Raising it by 10 % (to 0.55) moves the result by −0.526 %, from 9.5 to 9.45 m/s.
Wheel position ahead of the CG
The wheel position ahead of the centre of gravity, in m. A wheel further forward gets a larger lateral speed from the yaw rate. Raising it by 10 % (to 2.2) moves the result by +0 %, from 9.5 to 9.5 m/s.
Wheel position left of the CG
The wheel position to the left of the centre of gravity, in m. A wheel further left sees a lower forward speed in a left turn. Raising it by 10 % (to 1.1) moves the result by −0.526 %, from 9.5 to 9.45 m/s.
Steer angle
The steer angle of the wheel, in degrees, positive to the left. Rotates the tyre axes, shifting speed from the forward to the lateral component. Raising it by 10 % (to 0) moves the result by +0 %, from 9.5 to 9.5 m/s.
Worked example with the values above
- Body-axes velocity at the wheel: (u − r y, v + r x) = (9.5, 1) m/s.
- Rotation: cos δ = 1, sin δ = 0.
- ut = 9.5 × 1 + 1 × 0 = 9.5 m/s.
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