Technical Note · Vibration Fundamentals
Power Spectral Density (PSD) and Grms
What a random vibration profile really specifies, how to turn g²/Hz into Grms, and why two profiles with the same Grms can be very different tests.
Key takeaways
- A random signal never repeats, so it is specified by how its power is distributed across frequency, not by a waveform. That distribution is the power spectral density (PSD), also called acceleration spectral density (ASD), in g²/Hz.
- The area under the PSD curve is the mean-square acceleration (g²). Its square root is Grms, the overall level of the test.
- "Per hertz" is what makes the PSD independent of analysis resolution, so a profile means the same thing on any controller at any line count.
- Grms is one standard deviation (1σ) of a Gaussian signal. Instantaneous peaks of 3σ are routine, and 4σ to 5σ peaks occur in longer tests.
- Grms alone does not describe severity. Where the energy sits relative to the test article's resonances, and how much shaker stroke the low-frequency end demands, matter just as much.
Why random vibration needs a spectrum
A sine test is easy to describe: one frequency, one amplitude. Real environments such as road transport, jet and rocket engine noise, or gunfire on a vehicle structure contain every frequency at once, with amplitudes that change continuously and never repeat. Figure 1 shows a quarter of a second of such a signal.
Because the waveform is unpredictable, a random test specifies two things that are stable: the amplitude statistics (normally Gaussian, discussed later) and the frequency content, described by the power spectral density. The controller's job is to generate a never-repeating signal whose measured PSD at the control point matches the reference profile within tolerance.
What g²/Hz actually means
The PSD answers the question: how much of the signal's mean-square value lives in each 1 Hz slice of the spectrum?
- "Power" is borrowed from electrical engineering, where power is proportional to voltage squared. In vibration it simply means the square of acceleration, so the unit is g².
- "Spectral density" means per unit of frequency. The g² is spread over the band, so we express it per hertz: g²/Hz.
A useful analogy is population density. A map shaded in people per square mile tells you nothing about any one person, but integrate it over a county and you get the county's population. Integrate a PSD over a frequency band and you get the mean-square acceleration in that band.
| Term | Units | Meaning |
|---|---|---|
| PSD / ASD | g²/Hz | Mean-square acceleration per hertz. PSD and ASD are used interchangeably for acceleration; ASD is the more precise name. |
| Mean square | g² | Area under the PSD over the test band. |
| Grms | g | Square root of the mean square. The overall level of the test, and one standard deviation (1σ) of a zero-mean signal. |
| Breakpoint | Hz, g²/Hz | A corner of the profile. Profiles are defined by breakpoints joined by straight lines on log-log axes. |
| Slope | dB/octave | How fast the level changes between breakpoints. +3 dB/octave doubles the PSD for each doubling of frequency. |
| Frequency resolution, Δf | Hz | Width of each analysis line: sample rate ÷ block size. Sets the detail visible in the measured PSD. |
From profile to Grms: area under the curve
Figure 2 shows a widely used profile, the NAVMAT P-9492 random screening spectrum. It rises at +3 dB/octave from 20 to 80 Hz, holds 0.04 g²/Hz to 350 Hz, then falls at −3 dB/octave to 2000 Hz.
On log-log axes each segment is a straight line, which means the PSD follows a power law P(f) = P1(f/f1)m. Each segment's area therefore has an exact closed form:
| Segment | Breakpoints | Slope | Calculation | Area (g²) |
|---|---|---|---|---|
| 1 | 20 Hz, 0.01 → 80 Hz, 0.04 | +3 dB/oct (m = 1) | (0.04·80 − 0.01·20) / 2 | 1.50 |
| 2 | 80 Hz → 350 Hz, 0.04 flat | 0 dB/oct | 0.04 × (350 − 80) | 10.80 |
| 3 | 350 Hz, 0.04 → 2000 Hz, 0.007 | −3 dB/oct (m = −1) | 0.04·350·ln(2000/350) | 24.40 |
| Total mean square | 36.70 | |||
| Grms = √36.70 | 6.06 g | |||
Two things are worth noticing. First, the sloping 350–2000 Hz segment contributes two-thirds of the total even though its level is lower, simply because it is so wide. Second, a common shortcut of "average level × bandwidth" is wrong on sloped segments; use the formula above or let the controller compute it.
Reading dB/octave
Slopes are given in decibels per octave. Because PSD is a power quantity, dB = 10·log10(P2/P1), and an octave is a doubling of frequency.
| Slope | Exponent m | PSD change per octave | PSD change per decade (×10 in f) |
|---|---|---|---|
| +12 dB/oct | 4 | × 16 | × 10,000 |
| +6 dB/oct | 2 | × 4 | × 100 |
| +3 dB/oct | 1 | × 2 | × 10 |
| 0 dB/oct | 0 | × 1 (flat) | × 1 |
| −3 dB/oct | −1 | × ½ | × 1/10 |
| −6 dB/oct | −2 | × ¼ | × 1/100 |
Why "per hertz" matters
A controller estimates the spectrum with an FFT, which divides the band into lines of width Δf. The raw result is power per line, in g². Analyze the same signal with lines 16 times wider, and each line collects 16 times as much power. Dividing by Δf removes that dependence.
Resolution still matters, just in a different way. Coarse resolution smears narrow features: a sharp resonance peak in the response, or a narrowband component in a field-measured spectrum, is averaged into a wider line and appears lower than it is. Fine resolution shows those features but needs longer data blocks, so each PSD estimate takes longer and has more statistical scatter for the same test time. Our note on Statistical Degrees of Freedom covers that trade-off.
Grms and peaks: the Gaussian connection
Most random tests specify a Gaussian (normal) amplitude distribution, because many natural random processes approximate one. For a zero-mean Gaussian signal, Grms is exactly one standard deviation, σ. That links the overall level directly to how often given peak levels occur.
| Level | Acceleration | Time spent within ±level | Time spent beyond ±level |
|---|---|---|---|
| 1σ (= Grms) | 6.06 g | 68.27% | 31.73% |
| 2σ | 12.1 g | 95.45% | 4.55% |
| 3σ | 18.2 g | 99.73% | 0.27% |
| 4σ | 24.2 g | 99.994% | 0.0063% |
| 5σ | 30.3 g | 99.99994% | 0.00006% |
0.27% sounds negligible, but for the NAVMAT profile it works out to more than a thousand excursions beyond ±3σ every minute. This is why shakers and amplifiers are sized for peak force, not rms force, and why the drive signal's peak-to-rms ratio is controlled. Limiting peaks to a set multiple of σ is sigma clipping; deliberately raising the proportion of high peaks to match non-Gaussian field data is kurtosis control.
Velocity and displacement: the low-frequency end
A PSD in g²/Hz can be converted to velocity or displacement spectra by dividing by (2πf)² or (2πf)⁴. Those divisors are tiny at high frequency and large at low frequency, so the three quantities are dominated by completely different parts of the profile.
| Quantity | rms | 3σ peak-to-peak | Dominated by |
|---|---|---|---|
| Acceleration | 6.06 g | 36.4 g | Mid and high frequencies |
| Velocity | 2.07 in/s (52.6 mm/s) | 12.4 in/s (315 mm/s) | Low to mid frequencies |
| Displacement | 0.0077 in (0.19 mm) | 0.046 in (1.17 mm) | The lowest frequencies |
The NAVMAT profile is gentle on stroke because it starts at 20 Hz and rises from there. Extend a flat profile down to 5 Hz and the displacement requirement grows by more than an order of magnitude, which is why transportation and some vehicle profiles are limited by shaker stroke long before they are limited by force. Checking a new profile's displacement, velocity and peak force against the shaker's ratings before running it is a basic safety step.
Same Grms, different test
It is tempting to compare random tests by Grms alone. Figure 6 shows why that is risky: two profiles with identical 6.06 Grms, one the NAVMAT shape and one flat across the band.
Using Miles' equation from our Resonance note, consider two parts, each with Q = 20:
| Test article | Profile 1: NAVMAT | Profile 2: flat 0.0185 g²/Hz | More severe for this part |
|---|---|---|---|
| Part A, fn = 100 Hz | 11.2 Grms | 7.6 Grms | Profile 1, by 47% |
| Part B, fn = 1000 Hz | 21.0 Grms | 24.1 Grms | Profile 2, by 15% |
Neither profile is "harder" in general. Severity depends on the profile and the test article, which is why test tailoring starts from measured field spectra rather than from a single overall number.
Common misconceptions
| Misconception | Reality |
|---|---|
| "Grms is the peak acceleration." | Grms is 1σ. Peaks of 3σ are routine and higher peaks occur unless the drive is clipped. |
| "Equal Grms means equal severity." | Spectral shape relative to resonances matters more (Table 6), and low-frequency content sets stroke demand. |
| "Average PSD × bandwidth gives the area." | Only for flat segments. On sloped segments use the power-law formula or the controller's calculation. |
| "Finer resolution is always better." | It reveals narrow features but slows the control loop and increases statistical scatter for a given test time. |
| "A PSD can be compared directly with a sine level." | They have different units (g²/Hz vs g peak) and describe different signals. Comparisons need a stated bandwidth and an equivalence method such as fatigue damage. |
Quick reference
| Quantity | Formula | Rule of thumb |
|---|---|---|
| Grms | √(area under PSD) | Flat 0.01 g²/Hz over 1000 Hz → 3.16 Grms |
| Flat segment area | P·(f2 − f1) | |
| Sloped segment area | (P2f2 − P1f1)/(m+1) | m = dB/oct ÷ 3.01 |
| dB between two levels | 10·log10(P2/P1) | +3 dB ≈ ×2 PSD ≈ ×1.41 Grms |
| Scaling a whole profile | Grms scales with √(PSD) | Double the Grms → ×4 PSD (+6 dB) |
| Gaussian peaks | 3σ = 3 × Grms | Exceeded 0.27% of the time |
| Velocity PSD | PSD·(g/2πf)² | Dominated by low–mid frequencies |
| Displacement PSD | PSD·(g/(2πf)²)² | Dominated by the lowest breakpoint |
| Frequency resolution | Δf = sample rate / block size | Halving Δf doubles the time per estimate |
How Panther helps
- Profile entry: Panther Random accepts profiles as breakpoints with levels or dB/octave slopes, and computes the overall Grms, so the arithmetic in Table 2 is done for you and checked against the specification.
- Pre-test safety: the demanded displacement, velocity and force can be checked against the shaker's limits before any energy reaches the test article.
- Accurate PSD estimates: 24-bit acquisition with more than 110 dB of dynamic range keeps the low-level ends of sloped profiles (like the 0.007 g²/Hz at 2000 Hz above) well above the noise floor, and selectable resolution, averaging and overlap processing balance detail against loop speed.
- Peak and distribution control: Sigma Clipping, Ultra Clip and kurtosis control set the amplitude statistics as well as the spectrum.
References and further reading
- J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures, 4th ed., Wiley.
- C. M. Harris and A. G. Piersol (eds.), Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill.
- NAVMAT P-9492, Navy Manufacturing Screening Program, Department of the Navy, 1979.
- MIL-STD-810H, Method 514.8, Vibration.
- J. W. Miles, "On Structural Fatigue Under Random Loading," Journal of the Aeronautical Sciences, 1954.
- Spectral Dynamics Technical Library: Resonance; Statistical Degrees of Freedom; Sigma Clipping, Ultra Clip and Kurtosis.
