Random Vibration PSD Calculator: Grms, Peak Acceleration, Velocity and Displacement from a Breakpoint Profile (g²/Hz and dB/octave)
Type a random vibration profile as breakpoints. Each row fills in once it has two of frequency, ASD (g²/Hz) and slope (dB/octave). You get the overall Grms, the 3σ peak acceleration, and the velocity and displacement the shaker must make, in US and metric units.
Breakpoint table
Printed overall level: 7.7 Grms. Test time: 60 min per axis.
| # | Frequency Hz |
ASD g²/Hz |
Slope dB/oct |
Area g² |
Grms to here |
Delete row |
|---|---|---|---|---|---|---|
| 1 | 20 | 0.04 | — | |||
| 2 | 1000 | 0.04 | 0.00 | 39.20 | 6.26 | |
| 3 | 2000 | 0.01005 | −6.00 | 20.04 | 7.70 |
Keyboard and paste
- Tab moves along the row, then to the next row. Tab from the last cell adds a row.
- Enter, ↓ and ↑ move down and up the same column, like a spreadsheet.
- Calculated cells (grey italic) stay editable. Type over one and it becomes an input; the cell you typed least recently is calculated instead.
- Ctrl+Delete removes the current row. Esc undoes the cell you are in.
- Paste two columns (frequency and ASD) from Excel or a spec into any cell.
4x10^-2and4E-2both work.
| RMS (1σ) | Peak (3σ) | |
|---|---|---|
| Acceleration | 7.70 g75.5 m/s² | 23.1 g226 m/s² |
| Velocity | 2.73 in/s69.3 mm/s | 8.19 in/s208 mm/s |
| Displacement | 0.0126 in0.321 mm | 0.0758 in DA1.92 mm pk-pk |
Range 20–2000 Hz. Displacement peak is double amplitude (peak to peak) at 3σ.
+3 dB doubles every ASD value and raises Grms by √2 (×1.41). +6 dB raises Grms ×2.
Opens the sine vs random page with this profile. It is passed in your browser's storage, not sent anywhere.
Profile chart
Log-log axes. Straight lines between breakpoints, as a vibration controller draws them. The dashed red lines are the ±3 dB control tolerance of IEC 60068-2-64.
The same profile as velocity and displacement PSD
Some specs give velocity or displacement spectra instead of acceleration. At each frequency the acceleration ASD is divided by (2πf)² for velocity and by (2πf)⁴ for displacement. Each segment stays a straight line on log-log axes, 6 dB/oct lower for velocity and 12 dB/oct lower for displacement, so converting the breakpoints is exact.
| Frequency, Hz | Acceleration, g²/Hz | Velocity, (in/s)²/Hz | Velocity, (mm/s)²/Hz | Displacement, in²/Hz | Displacement, mm²/Hz |
|---|---|---|---|---|---|
| 20 | 0.04 | 0.3776 | 243.6 | 2.391×10⁻⁵ | 0.01543 |
| 1000 | 0.04 | 1.510×10⁻⁴ | 0.09743 | 3.826×10⁻¹² | 2.468×10⁻⁹ |
| 2000 | 0.01005 | 9.484×10⁻⁶ | 0.006119 | 6.006×10⁻¹⁴ | 3.875×10⁻¹¹ |
Standard profiles, checked
Every profile in the "Start from" list comes from a standard that prints its overall level. This page's math reproduces every printed value, which is how it is tested. MIL-STD-810H also prints peak velocity and displacement for its common-carrier profile.
| Profile (MIL-STD-810H, Method 514.8) | Breakpoints (Hz, g²/Hz) | Grms printed / here | Peak velocity, in/s, printed / here | Displacement DA, in, printed / here |
|---|---|---|---|---|
| Fig. 514.8E-1, general minimum integrity (1 h per axis) | 20, 0.04 · 1000, 0.04 · 2000 at −6 dB/oct | 7.7 / 7.70 | not printed / 8.19 | not printed / 0.076 |
| Fig. 514.8C-2, common carrier (US highway truck), vertical | 5, 0.015 · 40, 0.015 · 500, 0.00015 | 1.08 / 1.08 | 9.68 / 9.68 | 0.37 / 0.371 |
| Fig. 514.8C-2, common carrier, transverse | 5, 0.00013 · 10, 0.00013 · 20, 0.00065 · 30, 0.00065 · 78, 0.00002 · 79, 0.00019 · 120, 0.00019 · 500, 0.00001 | 0.21 / 0.205 | 1.23 / 1.23 | 0.04 / 0.037 |
| Fig. 514.8C-2, common carrier, longitudinal | 5, 0.0065 · 20, 0.0065 · 120, 0.0002 · 121, 0.003 · 200, 0.003 · 240, 0.0015 · 340, 0.00003 · 500, 0.00015 | 0.76 / 0.762 | 6.11 / 6.11 | 0.24 / 0.244 |
Peaks are at 3σ, as the standard's notes to Table 514.8C-I state. The standard says its printed peaks are approximate and depend on the control system and spectral resolution.
RTCA DO-160G, Section 8 (aircraft equipment)
| Curve | Test | Grms printed | Grms here |
|---|---|---|---|
| B2 | Standard (Fig. 8-1), 1 h per axis min. | 0.74 | 0.743 |
| B | Standard (Fig. 8-1), 1 h per axis min. | 1.48 | 1.488 |
| B3 | Standard (Fig. 8-1), 1 h per axis min. | 1.55 | 1.554 |
| C | Standard (Fig. 8-1), 1 h per axis min. | 4.12 | 4.127 |
| D | Standard (Fig. 8-1), 1 h per axis min. | 8.92 | 8.920 |
| E | Standard (Fig. 8-1), 1 h per axis min. | 7.94 | 7.952 |
| B12 | Robust (Fig. 8-4), 3 h per axis | 1.48 | 1.488 |
| B1 | Robust (Fig. 8-4), 3 h per axis | 2.10 | 2.105 |
| B4 | Robust (Fig. 8-4), 3 h per axis | 2.20 | 2.198 |
| C1 | Robust (Fig. 8-4), 3 h per axis | 5.85 | 5.853 |
| D1 | Robust (Fig. 8-4), 3 h per axis | 12.61 | 12.615 |
| E1 | Robust (Fig. 8-4), 3 h per axis | 11.33 | 11.336 |
DO-160G prints Grms cut off rather than rounded (1.488 is printed 1.48). Table 8-1 of DO-160G says which curve applies to each aircraft type and equipment zone: for a jet, for example, C for the fuselage, B, B2 or B3 for the instrument panel, D for the nacelle and E for the wing.
ISO 16750-3:2012 (road vehicles)
| Test | Time per plane | m/s² rms printed | m/s² rms here |
|---|---|---|---|
| Test I, passenger car engine (random part) | 22 h | 181 | 180.6 |
| Test IV, passenger car body | 8 h | 27.1 | 27.1 |
| Test V, passenger car wheel and suspension | 8 h | 107.3 | 107.3 |
| Test VI, commercial vehicle engine and gearbox | 94 h | 177 | 177.0 |
| Test VI, additional (fn below 30 Hz) | 32 h | 28.6 | 28.6 |
| Test VII, commercial vehicle body | 32 h | 57.9 | 57.9 |
| Test VII, additional (fn below 30 Hz) | — | 33.7 | 33.7 |
ISO 16750-3 gives ASD in (m/s²)²/Hz; the calculator switches to those units when you pick an ISO profile. Tests I and VI are mixed sine and random tests: the sine part is on the sine sweep page.
Reading a Random Vibration Profile
Random vibration has every frequency in the band at once. The profile says how much of the total power sits at each frequency. Its unit is g² per hertz. It is called the acceleration spectral density (ASD) or, loosely, the power spectral density (PSD).
A single ASD value is not an acceleration. You cannot read "0.04 g²/Hz at 100 Hz" as some number of g at 100 Hz. Only the area under the whole curve is an acceleration: its square root is the overall Grms.
Specs give the profile as a short table of breakpoints. Between them the line is straight on log-log axes. Each segment is set by its two ends, or by one end and a slope in dB per octave (an octave is a doubling of frequency).
dB per Octave Slopes
Slope from two breakpoints: S = 10·log₁₀(P₂/P₁) / log₂(f₂/f₁) dB/oct.
ASD after a slope: P₂ = P₁·(f₂/f₁)S/3.0103.
Frequency where a slope reaches an ASD: f₂ = f₁·(P₂/P₁)3.0103/S.
3.0103 is 10·log₁₀2: the number of dB in a doubling of power.
| Slope, dB/oct | ASD per octave | ASD per decade |
|---|---|---|
| +12 | ×15.8 | ×9,690 |
| +6 | ×3.98 | ×98 |
| +3 | ×1.995 | ×9.9 |
| 0 | ×1 | ×1 |
| −3 | ×0.501 | ×0.101 |
| −6 | ×0.251 | ×0.0102 |
| −12 | ×0.0631 | ×0.000103 |
Grms: the Area Under the Curve
On a log-log straight line the ASD is P = P₁(f/f₁)b, with b = S/3.0103. The area of the segment is
A = P₁·f₁/(b+1) · [(f₂/f₁)b+1 − 1]
When b = −1 (a slope of −3.01 dB/oct) that formula divides by zero. Use A = P₁·f₁·ln(f₂/f₁) instead.
Add the segment areas and take the square root: Grms = √(ΣA).
Width × height is only right for a flat segment. On a sloped segment it can be far off: from 1000 to 2000 Hz at −6 dB/oct from 0.04 g²/Hz, the true area is 20.0 g². The average of the two ends times 1000 Hz gives 25.0 g².
Velocity, Displacement and Peak Values
At a single frequency, a = (2πf)·v = (2πf)²·d. So the velocity spectrum is the ASD divided by (2πf)², and the displacement spectrum by (2πf)⁴, with g converted to 386.09 in/s² (9.80665 m/s²). Integrate those the same way to get the rms velocity and displacement.
Dividing by f⁴ makes displacement almost all low frequency. That is why the lowest breakpoints decide whether a shaker has enough stroke.
Random vibration has no fixed peak. With a Gaussian signal the rms value is one standard deviation (1σ), and values beyond 3σ happen about 0.27% of the time. Test labs and MIL-STD-810H quote peaks at 3σ.
Can a PSD Be Converted to a Sine Level?
A sine test puts all its energy at one frequency at a time. A random test spreads it over the band at once. There is no single number that makes one equal the other.
A formula often quoted for this, apsd = apk²/(3·Δf), is about how a sine tone shows up on an FFT analyzer. Δf there is the analyzer's frequency resolution (for example 4 Hz), not the test frequency. It does not turn a sine requirement into a random one.
The usual method is to compare what each test does to a part with a resonance. At each natural frequency you compare the peak response (sine: Q × input g; random: 3σ from Miles' equation), or the fatigue damage. The sine vs random comparison does this for any two profiles. For single sine points, use the sine vibration converter.
Sources
MIL-STD-810H, Environmental Engineering Considerations and Laboratory Tests, Method 514.8 Vibration (31 January 2019; public release). Annex C: Figure 514.8C-2 and Table 514.8C-I (common carrier break points, rms, peak velocity and displacement). Annex E: Figure 514.8E-1 (general minimum integrity exposure).
RTCA DO-160G, Environmental Conditions and Test Procedures for Airborne Equipment (2010), Section 8 Vibration: Figures 8-1 and 8-4 (break point tables and Grms), Table 8-1 (curves by aircraft type and zone), paragraphs 8.5.2 and 8.7.2 (test times).
ISO 16750-3:2012, Road vehicles — Environmental conditions and testing for electrical and electronic equipment — Part 3: Mechanical loads, 4.1.2 (Tests I, IV, V, VI and VII, Tables 3, 5, 7, 8, 10 to 13).
IEC 60068-2-64:2008, Environmental testing — Part 2-64: Tests — Test Fh: Vibration, broadband random and guidance, 4.6 (±3 dB ASD and ±10% rms tolerances, drive clipping and crest factor).
J. W. Miles, "On Structural Fatigue Under Random Loading", Journal of the Aeronautical Sciences, 21(11), 1954, pp. 753–762 (Miles' equation).
Standard gravity g = 9.80665 m/s² (exact, CGPM 1901); 1 in = 25.4 mm (exact).
The segment-area, velocity and displacement formulas are derived on this page from the straight log-log line and checked against every value MIL-STD-810H prints for the examples above.