Library / Automotive and vehicle dynamics / VEH-65

Automotive and vehicle dynamics

Road elevation RMS over a spatial band

RMS elevation of a zero-mean road profile over the band n₁ to n₂ cycles per metre, for a one-sided power-law spectrum G(n) = G₀(n/n₀)⁻² with G₀ in m³.

Ref: VEH-65
Rev: v1
Field: Automotive and vehicle dynamics
Result: z_RMS [m]

z_{RMS} = \sqrt{G_{0} {n_{0}}^{2} (1 / n_{1} - 1 / n_{2})}

For a one-sided spatial PSD with a power-law shape G(n) = G₀(n/n₀)⁻² (waviness w = 2), the variance over the band n₁ to n₂ is ∫G dn = G₀n₀²(1/n₁ − 1/n₂), and the RMS elevation is its square root. The integral is dominated by the lowest spatial frequencies, so the long wavelengths carry most of the roughness.

RMS elevation of a zero-mean road profile over the band n₁ to n₂ cycles per metre, for a one-sided power-law spectrum G(n) = G₀(n/n₀)⁻² with G₀ in m³. Other spectral slopes need a different integral.

Calculator

Solve for

z_RMSBand-limited RMS elevationComputed

G₀PSD at the reference frequency

m³

1e-90.01

n₀Reference spatial frequency

cycles/m

0.011

n₁Lower band limit

cycles/m

0.0050.05

n₂Upper band limit

cycles/m

0.55

Variance of the elevation, z²4.608×10⁻⁵ m²
Longest wavelength in the band, 1/n₁20 m
Shortest wavelength in the band, 1/n₂2 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.

Band-limited RMS elevation (m) current value 0.006788 m

n₀Reference spatial frequencyKey driver

6.79e-40.0679

G₀PSD at the reference frequency

1.34e-50.0424

n₁Lower band limit

0.0225

n₂Upper band limit

0.00712

The road RMS elevation is 6.788 mm over wavelengths from 2 m to 20 m.
The long-wavelength limit 1/n₁ contributes 111 % of the band integral, so extending the lower limit moves the answer more than the upper limit does.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m)	Local step	Change in result
n₀ Reference spatial frequency	0.01 to 1 cycles/m	6.788e-4	0.06788	0.0672	+10%	+10 %
G₀ PSD at the reference frequency	1e-9 to 0.01 m³	1.342e-5	0.04243	0.04241	+10%	+4.88 %
n₁ Lower band limit	0.005 to 0.05 cycles/m	0.006788	0.02251	0.01573	+10%	−5.18 %
n₂ Upper band limit	0.5 to 5 cycles/m	0.006788	0.00712	3.313e-4	+10%	+0.504 %

Sweep of Lower band limit

Lower band limit (cycles/m)	Band-limited RMS elevation (m)
0.005	0.02251
0.01063	0.01536
0.01625	0.01235
0.02188	0.01058
0.0275	0.009379
0.03313	0.008495
0.03875	0.007807
0.04438	0.007251
0.05	0.006788

Sweep one variable

Vary

Band-limited RMS elevation (m)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

z_RMS

Band-limited RMS elevation

Result · m

What the equation gives you. Everything else feeds it.

G₀

PSD at the reference frequency

m³ · typical 1e-9 to 0.01

The PSD at the reference spatial frequency, in m³. Raises the RMS as its square root. Raising it by 10 % (to 2.816×10⁻⁴) moves the result by +4.88 %, from 0.006788 to 0.00712 m.

n₀

Reference spatial frequency

cycles/m · typical 0.01 to 1

The reference spatial frequency, in cycles/m. Raises the RMS in direct proportion, since G₀ is fixed at n₀. Raising it by 10 % (to 0.11) moves the result by +10 %, from 0.006788 to 0.007467 m.

n₁

Lower band limit

cycles/m · typical 0.005 to 0.05

The lower band limit, in cycles/m. Lowering it adds long wavelengths and raises the RMS. Raising it by 10 % (to 0.055) moves the result by −5.18 %, from 0.006788 to 0.006436 m.

n₂

Upper band limit

cycles/m · typical 0.5 to 5

The upper band limit, in cycles/m. Raising it adds short wavelengths, which adds little. Raising it by 10 % (to 0.55) moves the result by +0.504 %, from 0.006788 to 0.006822 m.

Worked example with the values above

Band term: 1/n₁ − 1/n₂ = 20 − 2 = 18 m.
Variance: G₀ n₀² × that = 2.56×10⁻⁴ × 0.01 × 18 = 4.608×10⁻⁵ m².
RMS = √variance = 0.0067882 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

Related formulas

Practise this formula