Library / Automotive and vehicle dynamics / VEH-25
Automotive and vehicle dynamics
Understeer gradient
The steering handling balance of a vehicle: the weight carried by each axle divided by its cornering stiffness.
- Ref
- VEH-25
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Ku [rad per g]
Wf and Wr are the static loads on the front and rear axles, found from the mass m, the wheelbase L and the distance b from the centre of gravity to the rear axle (a = L − b is the distance to the front axle). Cf and Cr are the cornering stiffnesses of the whole front and rear axles. Ku is in radians per unit of lateral acceleration in g. A positive value means understeer: the front tyres need more slip angle than the rear. A negative value means oversteer and zero is neutral steer.
The steering handling balance of a vehicle: the weight carried by each axle divided by its cornering stiffness. A positive value means understeer, a negative value oversteer, and zero is neutral steer.
Calculator
- Understeer gradient in degrees per g1.226 °/g
- Front axle static load, Wf7.63 kN
- Rear axle static load, Wr7.085 kN
- Characteristic speed, √(L g / Ku), if understeering35.18 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 the defaults Ku = 0.0214 rad per g (1.23° per g): a mild understeer, which is the usual design target. Each 1 g of lateral acceleration needs 1.23° more steering than the geometric angle.
- Moving the centre of gravity back by 40 cm (b from 1.4 to 1.0 m) gives Ku = −0.00684 rad per g, which is oversteer. Understeer is a matter of where the weight sits and how stiff the tyres on each axle are.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad per g) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Cf Front axle cornering stiffness | 50 to 300 kN/rad | −0.01186 | 0.1153 | 0.1272 | +10% | −24.9 % |
| Cr Rear axle cornering stiffness | 50 to 300 kN/rad | −0.08301 | 0.03508 | 0.1181 | +10% | +15.8 % |
| m Vehicle mass | 500 to 5,000 kg | 0.007134 | 0.07134 | 0.06421 | +10% | +10 % |
| b Distance from the centre of gravity to the rear axle | 0.9 to 1.8 m | −0.0139 | 0.04965 | 0.06355 | +10% | +46.2 % |
| L Wheelbase | 2.2 to 3.5 m | −0.001191 | 0.04387 | 0.04506 | +10% | −42 % |
Sweep of Distance from the centre of gravity to the rear axle
| Distance from the centre of gravity to the rear axle (m) | Understeer gradient (rad per g) |
|---|---|
| 0.9 | −0.0139 |
| 1.012 | −0.005957 |
| 1.125 | 0.001986 |
| 1.238 | 0.009929 |
| 1.35 | 0.01787 |
| 1.462 | 0.02582 |
| 1.575 | 0.03376 |
| 1.688 | 0.0417 |
| 1.8 | 0.04965 |
Sweep one variable
Variables and how each one contributes
Understeer gradient
What the equation gives you. Everything else feeds it.
Vehicle mass
The vehicle mass, in kg. The axle loads scale with mass, but so does the tyre stiffness need. With fixed stiffnesses a heavier vehicle understeers more (or oversteers less). Raising it by 10 % (to 1,650) moves the result by +10 %, from 0.0214 to 0.02354 rad per g.
Wheelbase
The wheelbase, in m. Changes the static load split for a fixed b, and so the balance. Raising it by 10 % (to 2.97) moves the result by −42 %, from 0.0214 to 0.01242 rad per g.
Distance from the centre of gravity to the rear axle
The distance from the centre of gravity to the rear axle, in m. A larger b puts more weight on the front axle. Shifts weight to the front axle, which increases understeer. It is the strongest lever here. Raising it by 10 % (to 1.54) moves the result by +46.2 %, from 0.0214 to 0.03129 rad per g.
Front axle cornering stiffness
The cornering stiffness of the whole front axle (both tyres), in kN/rad. A stiffer front axle needs less slip angle for the same load, which lowers the understeer. Raising it by 10 % (to 143) moves the result by −24.9 %, from 0.0214 to 0.01607 rad per g.
Rear axle cornering stiffness
The cornering stiffness of the whole rear axle (both tyres), in kN/rad. A stiffer rear axle needs less slip angle, which raises the understeer. Raising it by 10 % (to 209) moves the result by +15.8 %, from 0.0214 to 0.02479 rad per g.
Worked example with the values above
- Axle loads: Wf = m g b / L = 1,500 × 9.81 × 1.4 / 2.7 = 7,630 N, and Wr = m g (L − b) / L = 7,085 N.
- Slip per axle: Wf / Cf = 7,630 / 130,000 = 0.05869 rad, and Wr / Cr = 7,085 / 190,000 = 0.03729 rad.
- Ku = 0.05869 − 0.03729 = 0.0214 rad per g (1.226° per g).
Assumptions and limits
- Linear tyres and small slip angles, with the cornering stiffness of each axle constant and no suspension or steering compliance effects.
- Steady-state cornering on a flat road, and axle loads equal to the static loads, ignoring weight transfer and aerodynamics.
Common mistakes
- Using the stiffness of one tyre in place of the whole axle.
- Swapping a and b: Wf depends on the distance from the centre of gravity to the rear axle, b.
Statistics and uncertainty
Ku is a small difference of two nearly equal terms, so it is very sensitive to the inputs. At the defaults a 10 % error in the rear stiffness Cr changes Ku by about 16 to 20 %, and the sign can flip for a vehicle close to neutral steer. Cornering stiffness varies with load, pressure and temperature, so the calculated Ku is best compared with a measured steering-wheel angle against lateral acceleration plot.
Where it is used
- Predicting handling balance from axle weights and tyre stiffness.
- Tuning understeer with tyre pressure, tyre choice or weight distribution.
Maths you need
- Static axle loads: moments about the axles
- Cornering stiffness: lateral force per unit slip angle