Library / Automotive and vehicle dynamics / VEH-27
Automotive and vehicle dynamics
Yaw rate gain
Steady-state yaw rate produced per unit of steer angle.
- Ref
- VEH-27
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- r/δ [1/s]
U is the forward speed, L the wheelbase and Ku the understeer gradient in rad per g. The gain r/δ is the steady-state yaw rate (rad/s) per unit steer angle (rad), which also equals the yaw rate in °/s for 1° of steer. For Ku = 0 it equals U/L. If Ku is negative, the denominator becomes zero at the critical speed √(L g / |Ku|), beyond which the linear model predicts instability.
Steady-state yaw rate produced per unit of steer angle. For an understeering vehicle the gain rises with speed to a maximum and then falls, while for oversteer it grows without limit towards a critical speed.
Calculator
- Yaw rate for 1° of steer angle5.377 °/s
- Gain of a neutral-steer vehicle, U / L7.407 1/s
- Characteristic speed, √(L g / Ku), if understeering32.55 m/s
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 20 m/s, 1° of steering produces 5.377° per second of yaw rate. A neutral-steer vehicle would give 7.407°/s, so the understeer reduces the response by 27.4 %.
- For an understeering vehicle the gain rises with speed to a maximum at the characteristic speed 32.55 m/s, where it equals 6.028 s⁻¹, and then falls. Beyond the characteristic speed, more speed makes the vehicle less responsive.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ku Understeer gradient | −0.01 to 0.1 rad per g | 2.951 | 8.725 | 5.774 | +0.01 rad per g | −9.88 % |
| U Speed | 3 to 60 m/s | 1.102 | 6.026 | 4.924 | +10% | +4.01 % |
| L Wheelbase | 2.2 to 3.5 m | 4.425 | 6.212 | 1.787 | +10% | −6.77 % |
Sweep of Speed
| Speed (m/s) | Yaw rate gain (1/s) |
|---|---|
| 3 | 1.102 |
| 10.13 | 3.419 |
| 17.25 | 4.988 |
| 24.38 | 5.784 |
| 31.5 | 6.024 |
| 38.63 | 5.941 |
| 45.75 | 5.695 |
| 52.88 | 5.382 |
| 60 | 5.053 |
Sweep one variable
Variables and how each one contributes
Yaw rate gain
What the equation gives you. Everything else feeds it.
Speed
The forward speed, in m/s. Raises the gain at low speed, in near proportion, but the understeer term in the denominator flattens it as speed rises. Raising it by 10 % (to 22) moves the result by +4.01 %, from 5.377 to 5.593 1/s.
Wheelbase
The wheelbase, in m. Lowers the gain, since a longer wheelbase makes the vehicle turn more slowly for the same steering. Raising it by 10 % (to 2.97) moves the result by −6.77 %, from 5.377 to 5.013 1/s.
Understeer gradient
The understeer gradient, in rad per g. Lowers the gain at higher speeds, since an understeering vehicle turns less per degree of steering as it goes faster. Raising it by 0.01 (to 0.035) moves the result by −9.88 %, from 5.377 to 4.846 1/s.
Worked example with the values above
- Understeer term: Ku U² / g = 0.025 × 20² / 9.81 = 1.019 m.
- Denominator: L + KuU²/g = 2.7 + 1.019 = 3.719 m.
- r/δ = U / 3.719 = 20 / 3.719 = 5.377 s⁻¹.
Assumptions and limits
- Steady-state cornering with small steering and slip angles and a linear tyre model.
- A constant speed and a constant understeer gradient. Transient effects such as yaw inertia and tyre lag are excluded.
Common mistakes
- Using the formula for a transient manoeuvre. It is a steady-state result.
- Taking a very large gain as a feature. For Ku < 0 the denominator approaches zero at the critical speed, which signals instability.
Statistics and uncertainty
The gain depends on Ku through the denominator, so at high speed the uncertainty of the understeer gradient (±30 %) dominates: at 30 m/s a 30 % error in Ku changes the gain by about 12 %. At low speed the gain is nearly U/L and known to a few per cent. Real yaw response is also affected by transients (a delay of 0.1 to 0.3 s), which this steady-state result does not include.
Where it is used
- Judging how responsive a vehicle feels to steering at different speeds.
- Finding the characteristic or critical speed of a vehicle from Ku.
Maths you need
- Understeer gradient: handling balance
- Yaw rate: rate of turn about a vertical axis