Library / Automotive and vehicle dynamics / VEH-26
Automotive and vehicle dynamics
Steady-state steering angle
The front steer angle needed to hold a steady turn of radius R at speed U: the geometric angle plus an understeer term that grows with lateral acceleration.
- Ref
- VEH-26
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- δ [°]
L is the wheelbase, R the radius of the path followed by the rear axle midpoint, U the speed and Ku the understeer gradient in rad per g. The first term L/R is the geometric (kinematic) steer angle and the second is the extra angle needed to compensate for tyre slip, which grows with the lateral acceleration ay = U²/R. The result is the equivalent front wheel angle in degrees (not the steering wheel angle). For negative Ku the vehicle oversteers and the required angle falls with speed.
The front steer angle needed to hold a steady turn of radius R at speed U: the geometric angle plus an understeer term that grows with lateral acceleration.
Calculator
- Geometric (Ackermann) angle, L / R3.094 °
- Extra angle for understeer, Ku ay / g1.314 °
- Lateral acceleration, U² / R0.4587 g
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.
- At 15 m/s round a 50 m corner the geometric angle is 3.094° and the understeer adds 1.314°, 29.8 % of the total. The lateral acceleration is 0.459 g.
- Because the understeer term grows with the square of the speed, the steering angle increases as the vehicle goes faster round the same corner: at 20 m/s it is 5.43° (0.815 g).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Turn radius | 20 to 300 m | 0.7347 | 11.02 | 10.29 | +10% | −0.401 ° |
| U Speed | 3 to 40 m/s | 3.147 | 12.44 | 9.292 | +10% | +0.276 ° |
| Ku Understeer gradient | −0.03 to 0.15 rad per g | 2.305 | 7.036 | 4.731 | +0.01 rad per g | +0.263 ° |
| L Wheelbase | 2.2 to 3.5 m | 3.835 | 5.325 | 1.49 | +10% | +0.309 ° |
Sweep of Speed
| Speed (m/s) | Steer angle (°) |
|---|---|
| 3 | 3.147 |
| 7.625 | 3.434 |
| 12.25 | 3.97 |
| 16.88 | 4.757 |
| 21.5 | 5.794 |
| 26.13 | 7.08 |
| 30.75 | 8.617 |
| 35.38 | 10.4 |
| 40 | 12.44 |
Sweep one variable
Variables and how each one contributes
Steer angle
What the equation gives you. Everything else feeds it.
Wheelbase
The wheelbase, in m. Raises the geometric part in direct proportion. Raising it by 10 % (to 2.97) moves the result by +7.02 %, from 4.408 to 4.717 °.
Turn radius
The radius of the path, in m. Tighter corners (smaller R) need more steering and give more lateral acceleration. Lowers both terms, so the angle falls as the corner opens out. Raising it by 10 % (to 55) moves the result by −9.09 %, from 4.408 to 4.007 °.
Speed
The speed round the corner, in m/s. Affects only the understeer term, which grows with the square of the speed, so fast cornering needs extra steering. Raising it by 10 % (to 16.5) moves the result by +6.26 %, from 4.408 to 4.684 °.
Understeer gradient
The understeer gradient, in rad per g: positive for understeer, negative for oversteer, zero for neutral steer. Typical cars are 0.02 to 0.05. Affects only the understeer term and is directly proportional to it. A neutral vehicle needs only the geometric angle. Raising it by 0.01 (to 0.06) moves the result by +5.96 %, from 4.408 to 4.671 °.
Worked example with the values above
- Geometric angle: L / R = 2.7 / 50 = 0.054 rad. Lateral acceleration: U² / (R g) = 15² / (50 × 9.81) = 0.4587 g.
- Understeer term: Ku × ay/g = 0.05 × 0.4587 = 0.02294 rad.
- δ = 0.054 + 0.02294 = 0.07694 rad = 4.408°.
Assumptions and limits
- Steady-state cornering at constant speed and radius, with small angles and a linear understeer gradient.
- A flat road, and no significant aerodynamic or suspension effects on the handling balance.
Common mistakes
- Reading the road wheel angle as the steering wheel angle. Multiply by the steering ratio for the steering wheel.
- Mixing the gradient in degrees per g with radians per g.
Statistics and uncertainty
The geometric term is exact, so the uncertainty comes from the understeer term. Ku is known to ±30 % or worse from calculation alone, and the term is a sizeable fraction of the total, so the steering angle is uncertain to about 10 % at moderate speed and more at high lateral acceleration, where the linear model also stops working. Measured steering-wheel angle against lateral acceleration is the best check.
Where it is used
- Predicting the steering effort and angle needed in a corner.
- Calculating Ku from test-track data of steer angle against lateral acceleration.
Maths you need
- Understeer gradient: handling balance of the vehicle
- Circular motion: a = U² / R