Random Vibration Rms⚠ unverified
Mechanical / Vibration Analysis · Compute the RMS response for a flat (white-noise) power spectral density
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| psd | psd | units**2/Hz | 1.0 | Power spectral density of the response |
| bandwidth | bandwidth | Hz | 1.0 | Frequency bandwidth over which the PSD is integrated |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | rms | — | Root-mean-square response, in the base units of the response quantity |
The science & history
Understanding the Parameters
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Power spectral density $PSD$ — the mean-square response per unit frequency (e.g. $g^2/\text{Hz}$). It describes how the vibration energy is distributed across frequency; a "flat" or "white" PSD is constant over the band.
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Bandwidth $BW$ — the frequency span of interest. Because energy accumulates across frequency, a wider band gives a larger RMS ($x_{rms} \propto \sqrt{BW}$).
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RMS as standard deviation — for a zero-mean Gaussian random process, RMS $=$ the $1\sigma$ level. Instantaneous values exceed $1\sigma$ about 32 % of the time, $2\sigma$ about 5 %, $3\sigma$ about 0.3 % — the basis of the "$3\sigma$ peak" design rule.
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Flat-PSD assumption — $\sqrt{PSD\cdot BW}$ is exact only for a constant PSD. Real qualification spectra are shaped (sloped or peaked); their RMS is $\sqrt{\int PSD(f)\,df}$, the square root of the area under the curve.
Derivation (Approaching a Proof)
The power spectral density $S(f)$ is defined so that the total mean-square (variance, for zero mean) of a random signal is the area under it:
$$\overline{x^2} = \int_0^\infty S(f)\,df.$$
For a flat spectrum, $S(f) = PSD$ = constant over a band of width $BW$ and zero elsewhere, the integral is just height times width:
$$\overline{x^2} = PSD \cdot BW.$$
The RMS is the square root of the mean-square:
$$x_{rms} = \sqrt{\overline{x^2}} = \sqrt{PSD\cdot BW}.$$
This is the Parseval/Wiener–Khinchin result that variance equals the integrated spectrum, specialised to a rectangular (white) band. For narrow-band response near a resonance, the effective bandwidth is $\tfrac{\pi}{2}f_n/Q$, tying RMS to the system's damping.
Dimensional check. $x_{rms} = \sqrt{PSD\cdot BW} = \sqrt{\dfrac{(\text{unit})^2}{\text{Hz}}\cdot\text{Hz}} = \sqrt{(\text{unit})^2} = \text{unit}$ — the base unit of the response quantity, confirming the registry's "dimensionless" label is a mislabel.
History and Development
Statistical description of vibration by power spectral density grew out of random-process theory (Wiener, Rice, 1940s) and became the standard for environments too irregular for a single amplitude — rocket and aircraft vibration, seismic and wind loading, road-induced automotive vibration. RMS-from-PSD underlies the shaker-test specifications of MIL-STD-810, NASA GEVS, and automotive durability standards, where equipment is qualified against a specified acceleration PSD and its $g_{rms}$.
Related Concepts: Fatigue From Vibration, Quality Factor Q, Half Power Bandwidth, Magnification Factor, Response Spectrum Value, Damping Ratio
Notes: Output carries the response quantity's units ($\sqrt{PSD\cdot BW}$; e.g. $g_{rms}$) — registry mislabels dimensionless. Flat-PSD approximation — shaped spectra need $\sqrt{\int PSD\,df}$ (area under curve). RMS $= 1\sigma$ for Gaussian; design to $3\sigma$ peaks. Feeds Fatigue From Vibration.