Library / Automotive and vehicle dynamics / VEH-24
Automotive and vehicle dynamics
Ackermann outer wheel steer angle
For perfect Ackermann geometry the inner and outer front wheels must steer to different angles so that all wheels roll about one centre.
- Ref
- VEH-24
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- δo [°]
For all four wheels to roll about one point, the inner and outer front wheels must steer to different angles that satisfy cot δo − cot δi = tf/L. Here L is the wheelbase, tf the front track width, and δi and δo the inner and outer steer-angle magnitudes in degrees. The result is the outer angle for a given inner angle under perfect Ackermann geometry. Real linkages approximate it, often with 50 to 100 % of the full Ackermann effect.
For perfect Ackermann geometry the inner and outer front wheels must steer to different angles so that all wheels roll about one centre. This gives the outer wheel angle from the inner wheel angle.
Calculator
- Difference, δi − δo3.333 °
- Mean of the two steer angles18.33 °
- Radius of the rear axle midpoint, L cot δi + tf/28.218 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.
- With the inner wheel at 20° the outer wheel steers 16.67°, 3.333° less. The inner wheel runs on the smaller circle, so it must turn more.
- The difference is almost nothing for small angles: at an inner angle of 5° the outer is 4.755°. That is why cars on the road, which mostly steer through a few degrees, work well with partial Ackermann, and why racing cars use little or none to account for slip angles.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| δi Inner wheel steer angle | 2 to 40 ° | 1.959 | 29.27 | 27.31 | +10% | +1.39 ° |
| L Wheelbase | 2 to 6 m | 15.74 | 18.35 | 2.612 | +10% | +0.258 ° |
| tf Front track width | 1.2 to 2.2 m | 15.68 | 17.4 | 1.715 | +10% | −0.275 ° |
Sweep of Inner wheel steer angle
| Inner wheel steer angle (°) | Outer wheel steer angle (°) |
|---|---|
| 2 | 1.959 |
| 6.75 | 6.311 |
| 11.5 | 10.29 |
| 16.25 | 13.96 |
| 21 | 17.37 |
| 25.75 | 20.56 |
| 30.5 | 23.59 |
| 35.25 | 26.48 |
| 40 | 29.27 |
Sweep one variable
Variables and how each one contributes
Outer wheel steer angle
What the equation gives you. Everything else feeds it.
Wheelbase
The wheelbase, in m. A longer wheelbase makes the required angle difference smaller (the track-to-wheelbase ratio is what matters). Raising it by 10 % (to 2.97) moves the result by +1.55 %, from 16.67 to 16.93 °.
Front track width
The front track width, in m: the distance between the centres of the front tyre contacts. A wider track needs a larger difference between the inner and outer angles. Raising it by 10 % (to 1.76) moves the result by −1.65 %, from 16.67 to 16.39 °.
Inner wheel steer angle
The steer angle of the inner front wheel, in degrees. Raises the outer angle, but by less. The gap widens as the steering angle grows. Raising it by 10 % (to 22) moves the result by +8.32 %, from 16.67 to 18.05 °.
Worked example with the values above
- Inner wheel: cot δi = 1 / tan(20°) = 2.747. Track term: tf / L = 1.6 / 2.7 = 0.5926.
- Outer wheel: cot δo = 2.747 + 0.5926 = 3.34, so tan δo = 0.2994.
- δo = tan⁻¹(0.2994) = 16.67°.
Assumptions and limits
- Rolling without slip and a rigid vehicle, with all four wheels turning about a common centre on the rear axle line.
- The track width is that between the steering pivots or tyre contacts, equal at left and right.
Common mistakes
- Swapping the inner and outer angles. The inner wheel always has the larger steering angle.
- Using the rear track width in place of the front track.
Statistics and uncertainty
The geometry is exact for rolling without slip. At speed the tyres run at slip angles, which are larger on the more heavily loaded outer tyre, so the best steering geometry is not full Ackermann and typically has 50 % or less, or even anti-Ackermann for race cars. Linkage errors of 0.5° are common, so compare with measurements before drawing conclusions about the angle difference.
Where it is used
- Steering linkage design and tyre-wear checks.
- Understanding why the inner wheel turns more than the outer on every vehicle.
Maths you need
- Cotangent: cot δ = 1 / tan δ
- Rolling without slip: all wheels turn about one centre