Library / Automotive and vehicle dynamics / VEH-43
Automotive and vehicle dynamics
Pneumatic aligning moment
Self-aligning moment about the vertical axis of a tyre in pure slip, from the pneumatic trail tp and the signed lateral force Fy.
- Ref
- VEH-43
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Mz [N m]
The lateral force on a slipping tyre acts behind the centre of the contact patch by the pneumatic trail tp, so it gives a self-aligning moment Mz ≈ −tp·Fy that turns the tyre back towards its heading. The trail is largest at small slip angles and shrinks towards zero as the tyre approaches saturation, which is why steering feels lighter near the limit of grip.
Self-aligning moment about the vertical axis of a tyre in pure slip, from the pneumatic trail tp and the signed lateral force Fy. The force acts behind the centre of the contact patch, so the moment opposes the slip. It leaves out mechanical trail, camber and residual moments.
Calculator
- Moment per 1 kN of side force−35 N m
- Pneumatic trail in millimetres35 mm
- Moment if the trail halves, as it does near the grip limit−42 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.
- A 2,400 N side force with a 35 mm trail gives an aligning moment of −84 N m, the sign showing that it opposes the slip.
- Because the trail falls as the tyre saturates, the moment peaks and then drops even while the lateral force is still rising, which the driver feels as lightening steering.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Fy Lateral tyre force | −20,000 to 20,000 N | −700 | 700 | 1,400 | +800 N | +33.3 % |
| tp Pneumatic trail | 0 to 0.2 m | −480 | 0 | 480 | +10% | +10 % |
Sweep of Lateral tyre force
| Lateral tyre force (N) | Aligning moment (N m) |
|---|---|
| −20,000 | 700 |
| −15,000 | 525 |
| −10,000 | 350 |
| −5,000 | 175 |
| 0 | 0 |
| 5,000 | −175 |
| 10,000 | −350 |
| 15,000 | −525 |
| 20,000 | −700 |
Sweep one variable
Variables and how each one contributes
Aligning moment
What the equation gives you. Everything else feeds it.
Pneumatic trail
The pneumatic trail, in m. A longer trail raises the moment in direct proportion. Raising it by 10 % (to 0.0385) moves the result by +10 %, from −84 to −92.4 N m.
Lateral tyre force
The lateral tyre force, in N, positive to the left. A larger force raises the moment in direct proportion and opposes the force direction. Raising it by 10 % (to 2,640) moves the result by +10 %, from −84 to −92.4 N m.
Worked example with the values above
- Trail: tp = 0.035 m.
- Force: Fy = 2,400 N.
- Mz = −tp Fy = −0.035 × 2,400 = −84 N m.
Assumptions and limits
- A rolling tyre on a flat road with a point contact, small angles where a linearised form is used, and constant properties during the event.
- Velocities resolved in the body or tyre axes named on the page, with the sign convention stated there.
Common mistakes
- Mixing the SAE and ISO sign conventions for slip angle and lateral force.
- Entering steer angle in radians when degrees are expected, or the reverse.
- Using the relaxation-length formula for a step that is not small enough to stay in the linear tyre range.
Statistics and uncertainty
Tyre forces are known only to about 5 % to 15 % in practice, and slip angle and slip ratio are small differences of large velocities, so small velocity errors give large relative errors in slip.
Where it is used
- Computing tyre slip in handling simulations from measured vehicle motion.
- Estimating steering feel and wheel-speed response in control design.
Maths you need
- Rigid-body velocity: v_wheel = v_cg + ω × r
- Slip angle: the angle between a tyre’s heading and its velocity
- Relaxation length: the rolling distance over which tyre force builds up