Library / Automotive and vehicle dynamics / VEH-31
Automotive and vehicle dynamics
Quarter-car ride frequency
Natural frequency of the body on its suspension: the spring rate at the wheel (spring rate times motion ratio squared) in series with the tyre stiffness, acting on the sprung mass at one corner.
- Ref
- VEH-31
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- fn [Hz]
ks is the stiffness of the spring itself, MR the motion ratio (spring travel divided by wheel travel), and kt the vertical stiffness of the tyre. The spring rate seen at the wheel is the wheel rate kw = ks MR². Spring and tyre act in series, giving the ride rate kride = kwkt/(kw + kt), and the body resting on it with sprung mass ms at one corner has natural frequency fn. It neglects the unsprung mass and the damping, and so is a low-frequency approximation.
Natural frequency of the body on its suspension: the spring rate at the wheel (spring rate times motion ratio squared) in series with the tyre stiffness, acting on the sprung mass at one corner.
Calculator
- Wheel rate, ks MR²25.6 kN/m
- Ride rate (spring and tyre in series)22.7 kN/m
- Period of the ride motion0.7224 s
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 default corner has a wheel rate of 25.6 kN/m, a ride rate of 22.7 kN/m and a natural frequency of 1.384 Hz. Road cars are usually set at 1 to 1.5 Hz, which feels comfortable, as it is near the frequency of walking; sports cars are stiffer at 1.5 to 2.5 Hz.
- Raising the wheel rate by 50 % (the spring rate to 60 kN/m) moves the frequency only to 1.649 Hz, because the frequency is proportional to the square root of the stiffness, and loading the car so that the corner mass is 400 kg lowers it to 1.199 Hz.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ks Spring stiffness at the spring | 15 to 120 kN/m | 0.8795 | 2.165 | 1.285 | +10% | +4.29 % |
| MR Motion ratio (spring travel per wheel travel) | 0.5 to 1.2 | 0.8967 | 1.943 | 1.046 | +10% | +8.71 % |
| ms Sprung mass at one corner | 200 to 600 kg | 0.9788 | 1.695 | 0.7166 | +10% | −4.65 % |
| kt Vertical tyre stiffness | 150 to 350 kN/m | 1.359 | 1.419 | 0.0604 | +10% | +0.52 % |
Sweep of Spring stiffness at the spring
| Spring stiffness at the spring (kN/m) | Ride natural frequency (Hz) |
|---|---|
| 15 | 0.8795 |
| 28.13 | 1.181 |
| 41.25 | 1.403 |
| 54.38 | 1.582 |
| 67.5 | 1.732 |
| 80.63 | 1.861 |
| 93.75 | 1.974 |
| 106.9 | 2.074 |
| 120 | 2.165 |
Sweep one variable
Variables and how each one contributes
Ride natural frequency
What the equation gives you. Everything else feeds it.
Spring stiffness at the spring
The stiffness of the spring at the spring, in kN/m. The frequency rises with its square root, so a stiffer spring raises the frequency by a smaller amount. Raising it by 10 % (to 44) moves the result by +4.29 %, from 1.384 to 1.444 Hz.
Motion ratio (spring travel per wheel travel)
The motion ratio: how much the spring compresses per unit of wheel travel. It is typically 0.5 to 1. Enters the wheel rate squared, so a small change in the linkage geometry has a bigger effect than the same change in the spring. Raising it by 10 % (to 0.88) moves the result by +8.71 %, from 1.384 to 1.505 Hz.
Vertical tyre stiffness
The vertical stiffness of the tyre, in kN/m: typically 150 to 300 for a car tyre. A stiffer tyre barely changes the ride frequency, because the softer spring dominates the series combination. Raising it by 10 % (to 220) moves the result by +0.52 %, from 1.384 to 1.391 Hz.
Sprung mass at one corner
The sprung mass supported at this corner, in kg: about a quarter of the sprung mass of a car. The frequency falls with the square root of the mass, so a heavier load makes the ride softer. Raising it by 10 % (to 330) moves the result by −4.65 %, from 1.384 to 1.32 Hz.
Worked example with the values above
- Wheel rate: kw = ks MR² = 40 × 0.8² = 25.6 kN/m.
- Ride rate: kride = kwkt / (kw + kt) = 25.6 × 200 / (25.6 + 200) = 22.7 kN/m.
- fn = (1/2π) √(22,700 / 300) = 1.384 Hz.
Assumptions and limits
- Linear springs and a constant motion ratio near the static position, with the tyre treated as a linear spring in series.
- Unsprung mass and damping neglected, so the result is a low-frequency estimate of the body bounce frequency.
Common mistakes
- Using the spring stiffness itself rather than the wheel rate. The motion ratio must be squared.
- Using the mass of the whole vehicle in place of the sprung mass at one corner.
Statistics and uncertainty
The frequency is a square root of a ratio, so relative errors are halved: a 10 % error in the spring rate gives about 5 %. The motion ratio is the uncertain input in a real linkage, because it varies with wheel travel and the correct value is the one at the static ride position. Tyre stiffness changes with pressure and speed but has little effect on the answer, since it is much stiffer than the spring.
Where it is used
- Choosing spring rates for a target ride comfort.
- Checking that the front and rear frequencies are well matched, with the rear slightly higher to limit pitching.
Maths you need
- Springs in series: 1/k = 1/k₁ + 1/k₂
- Natural frequency: f = (1/2π) √(k / m)