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Automotive and vehicle dynamics
Yaw acceleration from axle forces
Yaw acceleration of a vehicle from the front and rear axle lateral forces, the distances a and b from the centre of gravity to the axles, any separate applied yaw moment Mz and the yaw inertia Izz.
- Ref
- VEH-47
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- ṙ [rad/s²]
The yaw equation of motion is Izz·ṙ = a·Fyf − b·Fyr + Mz: the front axle force acts at a lever arm a ahead of the centre of gravity and rotates the vehicle to the left, while the rear force at b behind rotates it back. A separate applied yaw moment Mz, from torque vectoring or differential braking, adds directly.
Yaw acceleration of a vehicle from the front and rear axle lateral forces, the distances a and b from the centre of gravity to the axles, any separate applied yaw moment Mz and the yaw inertia Izz. Forces are resolved in body axes; roll and yaw products of inertia are ignored.
Calculator
- Front-axle moment, a Fyf4,200 N m
- Rear-axle moment, b Fyr3,600 N m
- Net yaw moment, a Fyf − b Fyr + Mz600 N 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.
- The net yaw moment of 600 N m gives 0.24 rad/s², which is 13.8°/s².
- The front axle contributes 4,200 N m and the rear axle takes back 3,600 N m, so the result is the small difference of two large terms and is sensitive to each force.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Fyr Rear-axle lateral force | −30,000 to 30,000 N | −19.92 | 23.28 | 43.2 | +1200 N | −360 % |
| Fyf Front-axle lateral force | −30,000 to 30,000 N | −15.84 | 12.96 | 28.8 | +1200 N | +240 % |
| a CG to front axle | 0.2 to 5 m | −1.16 | 5.56 | 6.72 | +10% | +70 % |
| Izz Yaw inertia | 100 to 100,000 kg m² | 0.006 | 6 | 5.994 | +10% | −9.09 % |
| Mz Applied yaw moment | −5,000 to 5,000 N m | −1.76 | 2.24 | 4 | +200 N m | +33.3 % |
| b CG to rear axle | 0.2 to 5 m | −2.32 | 1.52 | 3.84 | +10% | −60 % |
Sweep of Front-axle lateral force
| Front-axle lateral force (N) | Yaw acceleration (rad/s²) |
|---|---|
| −30,000 | −15.84 |
| −22,500 | −12.24 |
| −15,000 | −8.64 |
| −7,500 | −5.04 |
| 0 | −1.44 |
| 7,500 | 2.16 |
| 15,000 | 5.76 |
| 22,500 | 9.36 |
| 30,000 | 12.96 |
Sweep one variable
Variables and how each one contributes
Yaw acceleration
What the equation gives you. Everything else feeds it.
CG to front axle
The distance from the centre of gravity to the front axle, in m. A longer front lever arm raises the moment from the front axle. Raising it by 10 % (to 1.32) moves the result by +70 %, from 0.24 to 0.408 rad/s².
CG to rear axle
The distance from the centre of gravity to the rear axle, in m. A longer rear lever arm raises the opposing moment. Raising it by 10 % (to 1.98) moves the result by −60 %, from 0.24 to 0.096 rad/s².
Front-axle lateral force
The front-axle lateral force, in N. Raises the yaw acceleration in proportion to a/Izz. Raising it by 10 % (to 3,850) moves the result by +70 %, from 0.24 to 0.408 rad/s².
Rear-axle lateral force
The rear-axle lateral force, in N. Lowers the yaw acceleration in proportion to b/Izz. Raising it by 10 % (to 2,200) moves the result by −60 %, from 0.24 to 0.096 rad/s².
Applied yaw moment
An applied yaw moment, in N m. Adds to the yaw acceleration in proportion to 1/Izz. Raising it by 10 % (to 0) moves the result by +0 %, from 0.24 to 0.24 rad/s².
Yaw inertia
The yaw moment of inertia, in kg m². A larger inertia lowers the yaw acceleration for the same moment. Raising it by 10 % (to 2,750) moves the result by −9.09 %, from 0.24 to 0.2182 rad/s².
Worked example with the values above
- Front moment: a Fyf = 1.2 × 3,500 = 4,200 N m; rear moment: b Fyr = 1.8 × 2,000 = 3,600 N m.
- Net moment: 4,200 − 3,600 + 0 = 600 N m.
- ṙ = 600 / 2,500 = 0.24 rad/s².
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
- Mixing up a and b: a is measured to the front axle and b to the rear axle.
- Using the track-averaged force for an axle without checking the total.
- Entering the path radius without its sign, which flips the sideslip sign.
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