Library / Automotive and vehicle dynamics / VEH-50
Automotive and vehicle dynamics
Direct yaw moment from unequal longitudinal tyre forces
Yaw moment about the vertical axis from different forward forces on the right and left tyres of an axle with track t.
- Ref
- VEH-50
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Mz [N m]
Forward forces FR and FL on the right and left tyres of an axle with track t act at lever arms of t/2 on either side of the centreline. Their moment about the vertical axis is Mz = (t/2)(FR − FL), positive for left yaw when the right side pushes harder. Equal forces give no yaw moment, and the same relation describes differential braking and the steering of skid-steer vehicles.
Yaw moment about the vertical axis from different forward forces on the right and left tyres of an axle with track t. Forces are positive forwards in the body frame and Mz is positive for left yaw. Add the contributions of each axle separately.
Calculator
- Force difference, FR − FL1,000 N
- Net forward force, FR + FL2,000 N
- Force difference needed for 1 kN m of yaw moment, 2000/t1,250 N
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.
- A 1,000 N force difference across a 1.6 m track gives 800 N m of yaw moment, while the net forward force is 2,000 N.
- Yaw moment and forward thrust are independent: shifting force from one side to the other changes the first without changing the second.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| FR Right tyre longitudinal force | −20,000 to 20,000 N | −16,400 | 15,600 | 32,000 | +800 N | +80 % |
| FL Left tyre longitudinal force | −20,000 to 20,000 N | −14,800 | 17,200 | 32,000 | +800 N | −80 % |
| t Track width | 0.3 to 3 m | 150 | 1,500 | 1,350 | +10% | +10 % |
Sweep of Right tyre longitudinal force
| Right tyre longitudinal force (N) | Yaw moment (N m) |
|---|---|
| −20,000 | −16,400 |
| −15,000 | −12,400 |
| −10,000 | −8,400 |
| −5,000 | −4,400 |
| 0 | −400 |
| 5,000 | 3,600 |
| 10,000 | 7,600 |
| 15,000 | 11,600 |
| 20,000 | 15,600 |
Sweep one variable
Variables and how each one contributes
Yaw moment
What the equation gives you. Everything else feeds it.
Track width
The track width, in m. A wider track gives a bigger lever arm and raises the moment in direct proportion. Raising it by 10 % (to 1.76) moves the result by +10 %, from 800 to 880 N m.
Right tyre longitudinal force
The forward force on the right tyre, in N. Raises the left-yaw moment in proportion to t/2. Raising it by 10 % (to 1,650) moves the result by +15 %, from 800 to 920 N m.
Left tyre longitudinal force
The forward force on the left tyre, in N. Lowers the left-yaw moment in proportion to t/2. Raising it by 10 % (to 550) moves the result by −5 %, from 800 to 760 N m.
Worked example with the values above
- Force difference: FR − FL = 1,500 − 500 = 1,000 N.
- Lever arm: t/2 = 0.8 m.
- Mz = 0.8 × 1,000 = 800 N m.
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