Library / Automotive and vehicle dynamics / VEH-62
Automotive and vehicle dynamics
Front-to-rear axle road-input delay
Time delay between the front and rear axle meeting the same road feature, for a straight-running vehicle at speed U with both axles on the same wheel path.
- Ref
- VEH-62
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Δt [s]
The rear axle meets a road feature a time Δt = L/U after the front axle if both follow the same wheel path at constant speed. This lag turns one road input into two, and it is what excites pitch: at frequencies where the delay is half a period, the front and rear are driven in antiphase. The relation fails in turns or when the axles follow different tracks.
Time delay between the front and rear axle meeting the same road feature, for a straight-running vehicle at speed U with both axles on the same wheel path. Turning and different wheel tracks invalidate the relation.
Calculator
- Frequency at which the inputs are in antiphase, 1/(2Δt)2.5 Hz
- Frequency at which the inputs are in phase, 1/Δt5 Hz
- Phase lag at 1 Hz, 360 Δt72 °
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 rear axle meets each feature 0.2 s after the front axle.
- The two inputs are in antiphase at 2.5 Hz, which drives pitch most, and in phase at 5 Hz, where only bounce is driven.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Vehicle speed | 0.5 to 60 m/s | 0.05 | 6 | 5.95 | +10% | −9.09 % |
| L Wheelbase | 0.5 to 10 m | 0.03333 | 0.6667 | 0.6333 | +10% | +10 % |
Sweep of Vehicle speed
| Vehicle speed (m/s) | Road-input delay (s) |
|---|---|
| 0.5 | 6 |
| 7.938 | 0.378 |
| 15.38 | 0.1951 |
| 22.81 | 0.1315 |
| 30.25 | 0.09917 |
| 37.69 | 0.0796 |
| 45.13 | 0.06648 |
| 52.56 | 0.05707 |
| 60 | 0.05 |
Sweep one variable
Variables and how each one contributes
Road-input delay
What the equation gives you. Everything else feeds it.
Wheelbase
The wheelbase, in m. A longer wheelbase raises the delay in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 0.2 to 0.22 s.
Vehicle speed
The vehicle speed, in m/s. A higher speed lowers the delay in inverse proportion. Raising it by 10 % (to 16.5) moves the result by −9.09 %, from 0.2 to 0.1818 s.
Worked example with the values above
- Wheelbase: L = 3 m; speed: U = 15 m/s.
- Δt = L/U = 3 / 15.
- Δt = 0.2 s.
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