Sine vs Random Vibration Comparison: Maximum Response Spectrum and Fatigue Damage Spectrum for a Sine Sweep and a Random PSD
There is no direct conversion between a sine level and a random level. MIL-STD-810H says to compare them through the response of the part. This page does that for a part with one resonance, at every natural frequency: the largest response each test causes, and the fatigue damage each one does.
The two tests
The list has MIL-STD-810H, DO-160G and ISO 16750-3 sine profiles. Units: g, m/s², in/s, mm/s, in DA, mm pk-pk. Build the profile on the sine sweep page and use its "Compare" button to bring it here.
The list has MIL-STD-810H, DO-160G and ISO 16750-3 random profiles, with each standard's test time. Paste two columns from a spec, or build the profile on the random PSD page and use its "Compare" button.
Q = 10 (5% damping) is the value MIL-STD-810H uses in its examples. b is the slope of the part's S-N curve: 810H says values from 5 to 8 are common. The result depends strongly on b, so the table at the bottom shows all four.
Which test is more severe?
| Peak response | Random is higher at 46% of natural frequencies |
|---|---|
| Fatigue damage (b = 6) | Random does more damage at 29% of natural frequencies |
| To cover the sine | Random ASD ×6.93 (+8.4 dB), or random time ×333, worst at 51.7 HzFatigue damage, from 20 to 500 Hz, where both tests have input. For peak response alone: random ASD ×3.17 (+5.0 dB). |
| To cover the random | Sine levels ×1.13 (+1.1 dB)Fatigue damage, same sweep rate and number of sweeps. |
The sine starts at 5 Hz but the random only at 20 Hz: no random level covers a resonance below that. The random goes on to 2000 Hz, past the sine's 500 Hz.
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Response and damage spectra
Sine: the largest response as the sweep passes. Random: the level crossed once on average in the test time, and 3σ for reference.
Relative scale: the larger curve peaks at 1. Where one curve is above the other, that test does more fatigue damage to a part resonating there.
At selected natural frequencies
| fn, Hz | Sine peak response, g | Random largest response, g | Damage ratio, sine ÷ random | More damaging |
|---|---|---|---|---|
| 5 | 2.56 | 0.1483σ 0.101 | 9.2×10⁷ | Sine |
| 10.3 | 10.8 | 0.7473σ 0.488 | 6.8×10⁷ | Sine |
| 19.8 | 20.0 | 10.43σ 6.61 | 450 | Sine |
| 48.7 | 43.7 | 26.83σ 16.4 | 220 | Sine |
| 100 | 50.1 | 39.83σ 23.6 | 53 | Sine |
| 205 | 50.1 | 58.83σ 34.0 | 6 | Sine |
| 504 | 49.7 | 95.43σ 53.3 | 0.17 | Random |
| 975 | 6.77 | 1333σ 72.6 | 1.1×10⁻⁶ | Random |
| 2000 | 5.33 | 90.03σ 48.0 | 3.3×10⁻⁶ | Random |
How much to scale the random test to cover the sine
Damage grows with ASD to the power b/2 and in proportion to time. These are the factors that make the random damage equal or exceed the sine damage at every natural frequency where both tests have input, for four values of b.
| Fatigue exponent b | Random ASD factor | Or random time factor |
|---|---|---|
| 5 | ×6.55 (+8.2 dB) | ×110 |
| 6 | ×6.93 (+8.4 dB) | ×333 |
| 7 | ×7.13 (+8.5 dB) | ×967 |
| 8 | ×7.19 (+8.6 dB) | ×2,675 |
Why There Is No Direct Conversion
A sine test puts one frequency on the part at a time. A random test puts the whole band on at once, with a level that changes from moment to moment. MIL-STD-810H (Method 514.8, Annex A, paragraph 2.3.5) says a general equivalence is not feasible, and that comparing sine peak g with random Grms compares nothing but the units.
What both tests do to the part. A part that matters has resonances. At each possible natural frequency you can compare the largest response each test causes, and the fatigue damage each one does.
Largest Response (Maximum Response Spectrum)
The part is a mass on a spring with natural frequency fₙ and amplification Q = 1/(2ζ). Driven at f with base acceleration A, its response (relative displacement × ω₀², in g) is
y = A / √[(1 − r²)² + (2ζr)²], r = f/fₙ.
It peaks at A/(2ζ√(1−ζ²)) ≈ Q·A when f = fₙ. These are the formulas of MIL-STD-810H Annex F, Appendix E.
Sine sweep: the largest y as the sweep passes every frequency.
Random: the rms response σ is the square root of ∫ y² · ASD df. For a flat ASD this is Miles' equation, σ = √(π/2 · fₙ · Q · ASD). The largest response in test time T is the level crossed once on average: σ · √(2 ln(fₙ·T)), as in Annex F, Appendix E. For one hour at 100 Hz that is 5.1σ, not 3σ.
Fatigue Damage Spectrum
With an S-N curve N·Sb = constant and Miner's rule, each cycle of response amplitude y adds damage in proportion to yb. Stress in the part is taken as proportional to its response.
Sine sweep: D = Σ cycles × yb. For a log sweep at R octaves per second, D = ∫ yb df / (R·ln 2), per sweep.
The response is narrow-band near fₙ, with Rayleigh-distributed peaks:
D = fₙ · T · (√2·σ)b · Γ(1 + b/2)
Assumptions and Limits
One linear resonance at a time, with the same Q at every natural frequency. Real parts have several modes, and damping changes with level.
The sweep is slow enough for the response to build to steady state at each frequency. Fast sweeps through a high-Q resonance give less than this.
Response peaks are counted at fₙ cycles per second for random. Fatigue uses a single S-N slope b and Miner's rule. The answer depends strongly on b and Q, so use the sensitivity table and try other Q values.
Sources
MIL-STD-810H, Environmental Engineering Considerations and Laboratory Tests, Method 514.8 Vibration (31 January 2019; public release). Annex A, paragraph 2.2 (fatigue exponents) and 2.3.5 (random versus sinusoidal equivalence). Annex E, Figures 514.8E-1 and 514.8E-2 (the default profiles). Annex F, Appendix C (fatigue damage spectrum, after Henderson and Piersol, 1995) and Appendix E (maximum response spectrum).
C. Lalanne, Mechanical Vibration and Shock Analysis, Volume 5, Specification Development (cited by MIL-STD-810H): the response and fatigue damage spectrum method.
J. W. Miles, "On Structural Fatigue Under Random Loading", Journal of the Aeronautical Sciences, 21(11), 1954, pp. 753–762.
The integrals are computed numerically on this page. The tests in this site's code check them against Miles' equation and the closed-form damage of a slow sweep through one resonance.