Library / Automotive and vehicle dynamics / VEH-51
Automotive and vehicle dynamics
Vertical wheel rate from spring installation
Vertical stiffness at the wheel from the spring rate ks and the motion ratio MR, the spring travel per unit vertical wheel travel.
- Ref
- VEH-51
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- kw [N/m]
A spring working through a linkage sees only part of the wheel’s travel: a wheel movement z moves the spring by MR·z, and the wheel feels the spring force divided by the same ratio, so the wheel rate is kw = ks·MR². The square comes from the ratio acting twice, once on displacement and once on force. The result holds for small motions about the design position and for a ratio that stays nearly constant.
Vertical stiffness at the wheel from the spring rate ks and the motion ratio MR, the spring travel per unit vertical wheel travel. It assumes a locally constant motion ratio and no geometric stiffness from preload. Use the ratio at the specified ride height.
Calculator
- Wheel rate in N/mm25.6 N/mm
- Fraction of the spring rate felt at the wheel, MR²0.64
- Spring travel for 1 mm of wheel travel0.8 mm
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 wheel feels 25.6 N/mm from a 40 N/mm spring, only 64 % of its rate, because the spring is installed at a motion ratio of 0.8.
- A motion ratio below 1 means the spring must be stiffer than the wheel rate you want, and its force and travel are higher than they appear at the wheel.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N/m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ks Spring rate | 1,000 to 500,000 N/m | 640 | 320,000 | 319,400 | +10% | +10 % |
| MR Motion ratio (spring travel per wheel travel) | 0.2 to 1.5 | 1,600 | 90,000 | 88,400 | +10% | +21 % |
Sweep of Motion ratio (spring travel per wheel travel)
| Motion ratio (spring travel per wheel travel) | Wheel rate (N/m) |
|---|---|
| 0.2 | 1,600 |
| 0.3625 | 5,256 |
| 0.525 | 11,030 |
| 0.6875 | 18,910 |
| 0.85 | 28,900 |
| 1.012 | 41,010 |
| 1.175 | 55,230 |
| 1.337 | 71,560 |
| 1.5 | 90,000 |
Sweep one variable
Variables and how each one contributes
Wheel rate
What the equation gives you. Everything else feeds it.
Spring rate
The spring rate, in N/m. Raises the wheel rate in direct proportion. Raising it by 10 % (to 44,000) moves the result by +10 %, from 25,600 to 28,160 N/m.
Motion ratio (spring travel per wheel travel)
The motion ratio, the spring travel per unit wheel travel. Raises the wheel rate with its square, so a 10 % change in MR changes the wheel rate by 21 %. Raising it by 10 % (to 0.88) moves the result by +21 %, from 25,600 to 30,980 N/m.
Worked example with the values above
- Square the ratio: MR² = 0.8² = 0.64.
- kw = ks × MR² = 40,000 × 0.64.
- kw = 25,600 N/m, or 25.6 N/mm.
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