Library / Automotive and vehicle dynamics / VEH-63
Automotive and vehicle dynamics
Amplitude of mean front and rear road input
Amplitude of the average of the road heights under the front and rear axles when both follow the same sinusoidal track of peak amplitude A and wavelength λ, a wheelbase L apart.
- Ref
- VEH-63
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Amean [m]
For a sinusoidal road of amplitude A and wavelength λ, the two axles see the same wave with a phase difference of 2πL/λ. The average of the front and rear heights is a sinusoid of amplitude A|cos(πL/λ)|, the part that drives bounce. The half-difference has amplitude A|sin(πL/λ)| and drives pitch.
Amplitude of the average of the road heights under the front and rear axles when both follow the same sinusoidal track of peak amplitude A and wavelength λ, a wheelbase L apart. A kinematic heave input, excluding the body and suspension response.
Calculator
- Fraction of the amplitude that reaches the bounce input0.5
- Half-difference amplitude (pitch input), A|sin(πL/λ)|0.01732 m
- Longest wavelength that cancels the bounce input, 2L4 m
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 bounce input is 50 % of the road amplitude, and the pitch input 86.6 %.
- A wavelength of 2L = 4 m cancels the bounce input entirely, so the vehicle only pitches, which is the wheelbase filtering effect.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Road amplitude | 0.001 to 0.2 m | 5e-4 | 0.1 | 0.0995 | +10% | +10 % |
| L Wheelbase | 0.5 to 10 m | 3.49e-4 | 0.01996 | 0.01961 | +10% | −18.7 % |
| λ Road wavelength | 0.5 to 100 m | 0.001508 | 0.02 | 0.01849 | +10% | +16 % |
Sweep of Road wavelength
| Road wavelength (m) | Mean axle input amplitude (m) |
|---|---|
| 0.5 | 0.02 |
| 12.94 | 0.01769 |
| 25.38 | 0.01939 |
| 37.81 | 0.01972 |
| 50.25 | 0.01984 |
| 62.69 | 0.0199 |
| 75.13 | 0.01993 |
| 87.56 | 0.01995 |
| 100 | 0.01996 |
Sweep one variable
Variables and how each one contributes
Mean axle input amplitude
What the equation gives you. Everything else feeds it.
Road amplitude
The amplitude of the road wave, in m. Raises the mean input in direct proportion. Raising it by 10 % (to 0.022) moves the result by +10 %, from 0.01 to 0.011 m.
Wheelbase
The wheelbase, in m. Changes the phase difference and moves the cancellation wavelengths. Raising it by 10 % (to 2.2) moves the result by −18.7 %, from 0.01 to 0.008135 m.
Road wavelength
The road wavelength, in m. Long waves give a mean input near the full amplitude; wavelengths of 2L cancel it completely. Raising it by 10 % (to 6.6) moves the result by +16 %, from 0.01 to 0.0116 m.
Worked example with the values above
- Phase term: πL/λ = π × 2 / 6 = 1.0472 rad.
- |cos| = 0.5.
- A_mean = 0.02 × 0.5 = 0.01 m.
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