Library / Automotive and vehicle dynamics / VEH-23
Automotive and vehicle dynamics
Kinematic turning radius
Turning radius of the rear axle midpoint for a front steer angle δ at low speed, with no tyre slip.
- Ref
- VEH-23
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- R [m]
L is the wheelbase and δ the steer angle of the front wheels, taken as a single equivalent angle (the kinematic bicycle model). R is the radius of the circle followed by the midpoint of the rear axle. The model assumes no tyre slip, which holds at walking pace and low lateral acceleration. At higher speed the tyres must slip to generate the cornering force, and the real radius becomes larger or smaller depending on the understeer.
Turning radius of the rear axle midpoint for a front steer angle δ at low speed, with no tyre slip. This is the geometric (Ackermann) turn that every real vehicle approaches as speed falls.
Calculator
- Path curvature, 1 / R0.01941 1/m
- Turning circle diameter, 2R103 m
- Lateral acceleration at 10 m/s, U² / R1.941 m/s²
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 2.7 m wheelbase and 3° of steering give a 51.52 m radius. For small angles tan δ ≈ δ, so R ≈ L / δ: halving the angle doubles the radius (103.1 m at 1.5°).
- The same equation gives the tightest turn: at 30° of steering the radius is 4.677 m. At 10 m/s (36 km/h) that radius would need 21.38 m/s² (2.18 g) of lateral acceleration, which no tyre can supply, so the kinematic radius applies at low speed only.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| δ Front steer angle | 0.5 to 30 ° | 4.677 | 309.4 | 304.7 | +10% | −9.11 % |
| L Wheelbase | 2 to 6 m | 38.16 | 114.5 | 76.32 | +10% | +10 % |
Sweep of Front steer angle
| Front steer angle (°) | Turning radius (m) |
|---|---|
| 0.5 | 309.4 |
| 4.188 | 36.88 |
| 7.875 | 19.52 |
| 11.56 | 13.2 |
| 15.25 | 9.903 |
| 18.94 | 7.869 |
| 22.63 | 6.478 |
| 26.31 | 5.46 |
| 30 | 4.677 |
Sweep one variable
Variables and how each one contributes
Turning radius
What the equation gives you. Everything else feeds it.
Wheelbase
The wheelbase, in m: the distance between the front and rear axles. Directly proportional: a longer vehicle needs a larger circle for the same steering angle. Raising it by 10 % (to 2.97) moves the result by +10 %, from 51.52 to 56.67 m.
Front steer angle
The steer angle of the front wheels, in degrees, positive for a left turn. The radius is roughly inversely proportional to the angle for small angles, so steering a little more makes a much tighter turn at the small end. Raising it by 10 % (to 3.3) moves the result by −9.11 %, from 51.52 to 46.83 m.
Worked example with the values above
- Convert the angle: δ = 3° = 0.05236 rad, and tan δ = 0.05241.
- Divide the wheelbase by it: R = L / tan δ = 2.7 / 0.05241.
- R = 51.52 m.
Assumptions and limits
- The kinematic bicycle model: no tyre slip, a single equivalent front steer angle and a rigid vehicle.
- The radius is that of the rear axle midpoint, at low speed and small lateral acceleration.
Common mistakes
- Using the steering wheel angle rather than the road wheel angle. They differ by the steering ratio, typically 12:1 to 18:1.
- Applying the result at speed. Tyre slip changes the path.
Statistics and uncertainty
The kinematic radius is accurate only at low lateral acceleration, below about 0.1 g, where tyre slip is small. At higher speed the real radius differs from this by the understeer term, by 10 to 50 % at typical corner speeds, so this formula is a geometric starting point, not a prediction of the path. The wheelbase is known to 0.1 %; the equivalent steer angle (taken from the steering wheel) is the uncertain input.
Where it is used
- Turning circles and manoeuvring in confined spaces.
- The geometric part of the steady-state steering equation.
Maths you need
- Trigonometry: tan of an angle in a right-angled triangle
- Circular motion: path radius and curvature