Library / Automotive and vehicle dynamics / VEH-64
Automotive and vehicle dynamics
Temporal elevation PSD from a spatial road PSD
One-sided temporal road elevation PSD at frequency f for a vehicle at constant speed U, from the one-sided spatial PSD Gzz (m³) evaluated at f/U cycles per metre.
- Ref
- VEH-64
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Szz [m²/Hz]
Along a profile with a one-sided spatial PSD Gzz(n) in m³ (n in cycles per metre), a vehicle at speed U sees frequency f = U·n, so the variance in a band is preserved if Szz(f)·df = Gzz(n)·dn. Since df = U dn, the temporal PSD is Szz(f) = Gzz(f/U)/U, in m²/Hz. A faster vehicle sees the same road roughness spread over a wider band, so the PSD at each frequency falls.
One-sided temporal road elevation PSD at frequency f for a vehicle at constant speed U, from the one-sided spatial PSD Gzz (m³) evaluated at f/U cycles per metre. The division by U keeps the integrated variance the same.
Calculator
- Temporal PSD at twice the speed1×10−5 m²/Hz
- Temporal PSD at half the speed4×10−5 m²/Hz
- Speed that gives a PSD of 1×10⁻⁵ m²/Hz at this spatial level40 m/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 temporal level is 2×10−5 m²/Hz at 20 m/s.
- The spatial PSD value also changes with speed, because the frequency f maps to a different n = f/U; here the sweep varies U at a fixed Gzz, so it shows only the 1/U scaling.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²/Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Vehicle speed | 0.5 to 60 m/s | 6.667e-6 | 8e-4 | 7.933e-4 | +10% | −9.09 % |
| Gzz Spatial PSD at f/U | 1e-9 to 0.01 m³ | 5e-11 | 5e-4 | 5e-4 | +10% | +10 % |
Sweep of Vehicle speed
| Vehicle speed (m/s) | Temporal elevation PSD (m²/Hz) |
|---|---|
| 0.5 | 8e-4 |
| 7.938 | 5.039e-5 |
| 15.38 | 2.602e-5 |
| 22.81 | 1.753e-5 |
| 30.25 | 1.322e-5 |
| 37.69 | 1.061e-5 |
| 45.13 | 8.864e-6 |
| 52.56 | 7.61e-6 |
| 60 | 6.667e-6 |
Sweep one variable
Variables and how each one contributes
Temporal elevation PSD
What the equation gives you. Everything else feeds it.
Spatial PSD at f/U
The one-sided spatial PSD at n = f/U, in m³. Raises the temporal PSD in direct proportion. Raising it by 10 % (to 4.4×10−4) moves the result by +10 %, from 2×10−5 to 2.2×10−5 m²/Hz.
Vehicle speed
The vehicle speed, in m/s. Lowers the PSD in inverse proportion at a fixed spatial level, because the spectrum is spread over a wider band. Raising it by 10 % (to 22) moves the result by −9.09 %, from 2×10−5 to 1.818×10−5 m²/Hz.
Worked example with the values above
- Spatial PSD: Gzz = 4×10−4 m³.
- Speed: U = 20 m/s.
- Szz = Gzz/U = 4×10−4 / 20 = 2×10−5 m²/Hz.
Assumptions and limits
- Constant vehicle speed along a straight wheel path, and a stationary, zero-mean road profile.
- One-sided PSDs, with spatial frequency in cycles per metre and temporal frequency in hertz.
Common mistakes
- Mixing cycles per metre with radians per metre, which differ by 2π.
- Mixing one-sided and two-sided spectra.
- Applying a spectral relation outside the wavelength band over which the road was measured.
Statistics and uncertainty
Measured road PSDs scatter by a factor of two or more between road sections, and a power-law fit describes only a limited band. Treat spectral levels as indicative of a road class, not of a specific road.
Where it is used
- Converting measured road roughness into inputs for ride simulations.
- Predicting which road wavelengths excite body bounce, pitch and wheel hop.
Maths you need
- Wavelength and frequency: f = U/λ
- Power spectral density: variance per unit frequency
- Parseval’s theorem: the area under the PSD equals the variance