Library / Automotive and vehicle dynamics / VEH-53
Automotive and vehicle dynamics
Vertical ride rate including tyre compliance
Stiffness of the suspension in series with the tyre, from the wheel rate kw and the vertical tyre stiffness kt.
- Ref
- VEH-53
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- kr [N/m]
The suspension and the tyre are in series: the same force passes through both, and their deflections add. So 1/kr = 1/kw + 1/kt, which gives kr = kw·kt/(kw + kt). The ride rate is always below the smaller of the two stiffnesses, and the softer one takes the larger share of the deflection.
Stiffness of the suspension in series with the tyre, from the wheel rate kw and the vertical tyre stiffness kt. This quasi-static ride rate gives the body deflection relative to the road for a load change.
Calculator
- Share of the deflection in the tyre, kw/(kw + kt)16.67 %
- Share of the deflection in the suspension, kt/(kw + kt)83.33 %
- Ride rate as a share of the wheel rate83.33 %
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 ride rate is 25,000 N/m, 83.3 % of the wheel rate, because the tyre adds about 20 % extra compliance.
- The suspension takes 83.3 % of the total deflection and the tyre 16.7 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N/m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| kw Wheel rate | 1,000 to 500,000 N/m | 993.4 | 115,400 | 114,400 | +10% | +8.2 % |
| kt Tyre vertical stiffness | 20,000 to 1,000,000 N/m | 12,000 | 29,130 | 17,130 | +10% | +1.54 % |
Sweep of Wheel rate
| Wheel rate (N/m) | Ride rate (N/m) |
|---|---|
| 1,000 | 993.4 |
| 63,380 | 44,550 |
| 125,800 | 68,400 |
| 188,100 | 83,460 |
| 250,500 | 93,820 |
| 312,900 | 101,400 |
| 375,300 | 107,200 |
| 437,600 | 111,700 |
| 500,000 | 115,400 |
Sweep one variable
Variables and how each one contributes
Ride rate
What the equation gives you. Everything else feeds it.
Wheel rate
The wheel rate of the suspension, in N/m. Raises the ride rate, with diminishing returns once it far exceeds the tyre stiffness. Raising it by 10 % (to 33,000) moves the result by +8.2 %, from 25,000 to 27,050 N/m.
Tyre vertical stiffness
The vertical tyre stiffness, in N/m. Raises the ride rate, with diminishing returns once it far exceeds the wheel rate. Raising it by 10 % (to 165,000) moves the result by +1.54 %, from 25,000 to 25,380 N/m.
Worked example with the values above
- Product: kw kt = 30,000 × 150,000 = 4.5×109.
- Sum: kw + kt = 180,000.
- kr = 4.5×109 / 180,000 = 25,000 N/m.
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