Library / Automotive and vehicle dynamics / VEH-58
Automotive and vehicle dynamics
Coupled body and wheel natural frequencies
Lower (body) natural frequency of the undisturbed two-mass quarter-car model with a fixed road, from the eigenvalue equation Aλ² − Bλ + C = 0 with A = ms mu, B = ks mu + ms(ks + kt) and C = ks kt.
- Ref
- VEH-58
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- f1 [Hz]
For the two-mass model with a fixed road, the natural frequencies come from the eigenvalue equation Aλ² − Bλ + C = 0, with A = ms·mu, B = ks·mu + ms(ks + kt) and C = ks·kt. The lower root is the body bounce and the higher is the wheel hop, and f = √λ/(2π). If the masses are well separated, these reduce to the familiar decoupled values.
Lower (body) natural frequency of the undisturbed two-mass quarter-car model with a fixed road, from the eigenvalue equation Aλ² − Bλ + C = 0 with A = ms mu, B = ks mu + ms(ks + kt) and C = ks kt. The higher (wheel) frequency is in the derived rows. The lower root uses the stable product form.
Calculator
- Wheel-hop (higher) frequency, f₂10.56 Hz
- Body frequency with a rigid tyre, √(ks kt/(ks + kt)/ms)/(2π)1.214 Hz
- Wheel frequency with a rigid body, √((ks + kt)/mu)/(2π)10.56 Hz
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 body bounces at 1.213 Hz and the wheel hops at 10.56 Hz, a ratio of 8.71.
- Comfort is mostly set by the low frequency, which should stay in the roughly 1 Hz to 1.5 Hz band, while the high one should stay away from the frequencies that road wavelengths excite.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ks Suspension spring rate | 1,000 to 200,000 N/m | 0.3173 | 2.927 | 2.61 | +10% | +4.39 % |
| ms Sprung mass | 50 to 5,000 kg | 0.2714 | 2.705 | 2.434 | +10% | −4.65 % |
| kt Tyre vertical stiffness | 20,000 to 1,000,000 N/m | 0.9338 | 1.263 | 0.3294 | +10% | +0.427 % |
| mu Unsprung mass | 5 to 500 kg | 1.202 | 1.214 | 0.0117 | +10% | −0.00679 % |
Sweep of Suspension spring rate
| Suspension spring rate (N/m) | Body natural frequency (Hz) |
|---|---|
| 1,000 | 0.3173 |
| 25,880 | 1.5 |
| 50,750 | 1.966 |
| 75,630 | 2.262 |
| 100,500 | 2.47 |
| 125,400 | 2.627 |
| 150,300 | 2.749 |
| 175,100 | 2.847 |
| 200,000 | 2.927 |
Sweep one variable
Variables and how each one contributes
Body natural frequency
What the equation gives you. Everything else feeds it.
Sprung mass
The sprung mass, in kg. A heavier body lowers the body frequency, roughly as 1/√ms. Raising it by 10 % (to 275) moves the result by −4.65 %, from 1.213 to 1.157 Hz.
Unsprung mass
The unsprung mass, in kg. A heavier wheel lowers the wheel-hop frequency, with little effect on the body frequency. Raising it by 10 % (to 44) moves the result by −0.00679 %, from 1.213 to 1.213 Hz.
Suspension spring rate
The suspension spring rate, in N/m. Raises the body frequency, roughly as √ks. Raising it by 10 % (to 17,600) moves the result by +4.39 %, from 1.213 to 1.266 Hz.
Tyre vertical stiffness
The vertical tyre stiffness, in N/m. Raises mainly the wheel-hop frequency. Raising it by 10 % (to 176,000) moves the result by +0.427 %, from 1.213 to 1.218 Hz.
Worked example with the values above
- Coefficients: A = 10,000, B = 4.464×107, C = 2.56×109.
- Roots: λ₁ = 58.104 rad²/s², λ₂ = 4,405.9 rad²/s².
- f₁ = √λ₁/(2π) = 1.2132 Hz, and f₂ = 10.564 Hz.
Assumptions and limits
- A linear two-mass model with a point-contact tyre that never leaves the road and with no tyre damping.
- Displacements measured from static equilibrium, so gravity cancels, and a sinusoidal or sampled input as stated.
Common mistakes
- Using a total displacement instead of one measured from static equilibrium.
- Mixing displacement and velocity inputs in the force terms.
- Assuming the body and wheel frequencies decouple when the masses and stiffnesses are not well separated.
Statistics and uncertainty
Tyre stiffness varies by 10 % or more with pressure, load and speed, and damper behaviour is usually nonlinear, so computed responses give trends and frequency estimates rather than exact amplitudes.
Where it is used
- Ride, comfort and road-holding studies of passenger and off-road vehicles.
- Choosing spring and damper values for a quarter-car model.
Maths you need
- Newton’s second law: m z̈ = Σ forces
- Eigenvalue problem: natural frequencies from the system matrices
- Complex frequency response: amplitude and phase at a driven frequency