Library / Automotive and vehicle dynamics / VEH-54
Automotive and vehicle dynamics
Axle roll stiffness from two wheel springs
Roll stiffness of an axle from two equal wheel rates kw acting at half the track t/2 each side, for small roll about the centreline.
- Ref
- VEH-54
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- K_φ [N m/rad]
For small roll angle φ, the two wheels move in opposite directions by ±(t/2)φ, so each wheel spring gives a force kw(t/2)φ at a lever arm t/2. The resisting moment is kw·t²φ/2, which gives the spring roll stiffness Kφ = kw·t²/2 of the axle. It depends on the square of the track, so wide tracks are stiff in roll for the same wheel rate.
Roll stiffness of an axle from two equal wheel rates kw acting at half the track t/2 each side, for small roll about the centreline. Spring contribution only; add anti-roll bar stiffness separately.
Calculator
- Roll stiffness per degree670.2 N m/deg
- Roll moment at 2° of roll1,340 N m
- Spring force change at each wheel per degree of roll418.9 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.
- The axle springs alone give 38,400 N m/rad, which resists 670.2 N m per degree of roll.
- Doubling the track would quadruple this, so roll stiffness is often added with an anti-roll bar rather than by stiffening the wheel springs and spoiling ride.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N m/rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| kw Wheel rate | 1,000 to 500,000 N/m | 1,280 | 640,000 | 638,700 | +10% | +10 % |
| t Track width | 0.3 to 3 m | 1,350 | 135,000 | 133,700 | +10% | +21 % |
Sweep of Wheel rate
| Wheel rate (N/m) | Axle roll stiffness (N m/rad) |
|---|---|
| 1,000 | 1,280 |
| 63,380 | 81,120 |
| 125,800 | 161,000 |
| 188,100 | 240,800 |
| 250,500 | 320,600 |
| 312,900 | 400,500 |
| 375,300 | 480,300 |
| 437,600 | 560,200 |
| 500,000 | 640,000 |
Sweep one variable
Variables and how each one contributes
Axle roll stiffness
What the equation gives you. Everything else feeds it.
Wheel rate
The wheel rate of each wheel, in N/m. Raises the roll stiffness in direct proportion. Raising it by 10 % (to 33,000) moves the result by +10 %, from 38,400 to 42,240 N m/rad.
Track width
The track width, in m. Raises the roll stiffness with its square. Raising it by 10 % (to 1.76) moves the result by +21 %, from 38,400 to 46,460 N m/rad.
Worked example with the values above
- Half track: t/2 = 0.8 m; t² = 2.56 m².
- Kφ = kw t²/2 = 30,000 × 2.56 / 2.
- Kφ = 38,400 N m/rad, or 670.21 N m/deg.
Assumptions and limits
- A linear suspension with small motion about the design ride height and a constant motion ratio.
- Rigid links and negligible bush compliance, and no preload or geometric stiffness unless stated.
Common mistakes
- Forgetting to square the motion ratio.
- Using the inverse motion ratio, wheel travel per spring travel, in place of spring travel per wheel travel.
- Adding spring and anti-roll-bar stiffness at different points without converting them to the same reference.
Statistics and uncertainty
Spring rates and installation ratios are usually known to about 3 % to 5 %, but the motion ratio enters squared, so a 3 % error in it becomes about 6 % in wheel rate.
Where it is used
- Converting spring and damper settings to wheel rate and wheel damping.
- Setting roll stiffness and front-to-rear roll balance.
Maths you need
- Motion ratio: spring travel per unit wheel travel
- Series springs: 1/k = 1/k₁ + 1/k₂
- Roll stiffness: roll moment per radian of roll