Library / Automotive and vehicle dynamics / VEH-48
Automotive and vehicle dynamics
Steady CG sideslip in a linear bicycle model
Body sideslip angle at the centre of gravity in a steady turn of signed path radius R at speed U, for a linear bicycle model with rear cornering stiffness Cr.
- Ref
- VEH-48
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- β [rad]
In a steady turn of radius R at speed U, a linear bicycle model with rear cornering stiffness Cr has the rear axle force m·U²·a/(L·R) = Cr·αr. The rear-axle velocity direction then gives the body sideslip β ≈ b/R + αr·(sign), which reduces to β ≈ b/R − m a U²/(L Cr R). At low speed the vehicle sideslip is b/R, set by geometry alone, and it falls and crosses zero as speed rises.
Body sideslip angle at the centre of gravity in a steady turn of signed path radius R at speed U, for a linear bicycle model with rear cornering stiffness Cr. Small angles, no rear steer. Positive β means the velocity points left of the heading.
Calculator
- Kinematic (geometric) term, b/R0.036 rad
- Dynamic term, m a U²/(L Cr R)0.045 rad
- Speed at which the sideslip is zero, √(b L Cr/(m a))13.42 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.
- The sideslip is −0.009 rad (−0.516°), so the vehicle nose points inside its path at this speed.
- The sideslip is zero at 13.42 m/s; the default speed is above that.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Vehicle mass | 100 to 50,000 kg | −1.464 | 0.033 | 1.497 | +10% | +50 % |
| U Forward speed | 0 to 60 m/s | −0.684 | 0.036 | 0.72 | +10% | +105 % |
| Cr Rear axle cornering stiffness | 5,000 to 1,000,000 N/rad | −0.504 | 0.0333 | 0.5373 | +10% | −45.5 % |
| b CG to rear axle | 0.2 to 5 m | −0.09243 | 0.07823 | 0.1707 | +10% | −68.3 % |
| R Path radius (signed) | 5 to 1,000 m | −0.09 | −4.5e-4 | 0.08955 | +10% | −9.09 % |
| a CG to front axle | 0.2 to 5 m | −0.04672 | 0.02475 | 0.07147 | +10% | +28.8 % |
Sweep of Forward speed
| Forward speed (m/s) | CG sideslip angle (rad) |
|---|---|
| 0 | 0.036 |
| 7.5 | 0.02475 |
| 15 | −0.009 |
| 22.5 | −0.06525 |
| 30 | −0.144 |
| 37.5 | −0.2452 |
| 45 | −0.369 |
| 52.5 | −0.5152 |
| 60 | −0.684 |
Sweep one variable
Variables and how each one contributes
CG sideslip angle
What the equation gives you. Everything else feeds it.
CG to rear axle
The distance from the centre of gravity to the rear axle, in m. Raises the low-speed sideslip b/R in direct proportion. Raising it by 10 % (to 1.98) moves the result by −68.3 %, from −0.009 to −0.002853 rad.
Path radius (signed)
The path radius, in m. A bigger radius lowers both terms, so the sideslip magnitude falls in inverse proportion. Raising it by 10 % (to 55) moves the result by −9.09 %, from −0.009 to −0.008182 rad.
Vehicle mass
The vehicle mass, in kg. Raises the dynamic term and drives the sideslip more negative. Raising it by 10 % (to 1,650) moves the result by +50 %, from −0.009 to −0.0135 rad.
CG to front axle
The distance from the centre of gravity to the front axle, in m. Raises the dynamic term, since a larger share of the lateral force is at the rear. Raising it by 10 % (to 1.32) moves the result by +28.8 %, from −0.009 to −0.0116 rad.
Forward speed
The forward speed, in m/s. Raises the dynamic term with U², so the sideslip changes sign above a particular speed. Raising it by 10 % (to 16.5) moves the result by +105 %, from −0.009 to −0.01845 rad.
Rear axle cornering stiffness
The rear-axle cornering stiffness, in N/rad. A stiffer rear axle needs less slip angle and so lowers the dynamic term. Raising it by 10 % (to 66,000) moves the result by −45.5 %, from −0.009 to −0.004909 rad.
Worked example with the values above
- Geometric term: b/R = 1.8 / 50 = 0.036 rad.
- Dynamic term: m a U²/(L Cr R) = 1,500 × 1.2 × 225 / (3 × 60,000 × 50) = 0.045 rad.
- β = 0.036 − 0.045 = −0.009 rad.
Assumptions and limits
- A rigid vehicle on a flat road in steady or slowly varying motion, small angles and a linear single-track (bicycle) model where stated.
- Level road, no aerodynamic yaw moment and no roll effects on the axle forces unless the page says so.
Common mistakes
- Entering the path radius without its sign.
- Using the front cornering stiffness in place of the rear one.
- Applying the small-angle bicycle model at high lateral acceleration, where tyre saturation breaks it.
Statistics and uncertainty
Yaw inertia and cornering stiffness are usually known to 10 % to 20 %, and steady sideslip is the small difference of two similar terms, so it is more sensitive to errors than either term alone.
Where it is used
- Splitting cornering force between axles and checking handling balance.
- Yaw-control and torque-vectoring design, and banked test-track layout.
Maths you need
- Newton’s second law: F = ma, for force and for moment
- Single-track model: two tyres lumped as one per axle
- Centripetal acceleration: a = U²/R