Random vibration analysis

The formula

The environment is a power spectral density (PSD), acceleration squared per hertz, usually plotted g²/Hz versus frequency. The single most-used number falls out of the area under it:

Grms = sqrt( ∫ W(f) df )      integrated over the test band

Reading: the overall RMS acceleration is the square root of the area under the PSD curve. That is the whole thing — RMS is an area, and area lives under the curve, not at any one peak.

For a flat PSD W (constant g²/Hz) from f1 to f2:

Grms = sqrt( W · (f2 − f1) )

For a segment sloped in the usual log-log sense, the area integral is a little uglier but it is still just area; you sum the areas of the segments and take one square root at the end.

Miles' equation answers the harder question: given a base-input PSD, how hard does a single lightly-damped resonance respond? For a single-degree-of-freedom (SDOF) system with natural frequency fn, quality factor Q, driven by a flat input PSD level W (in g²/Hz) at fn:

Grms_response = sqrt( (π/2) · fn · Q · W )

Reading: the response of a resonance is set by three things — where it rings (fn), how sharply it rings (Q), and how much energy the environment has right there (W). Everywhere else in the spectrum barely matters. Q ≈ 1/(2ζ), so a Q of 10 is 5% critical damping.

The peak load an analyst carries forward is the 3-sigma value:

peak (3σ) = 3 · Grms

Reading: for a Gaussian response, three sigma bounds 99.73% of the instantaneous excursions. That is the number that goes on the equivalent static load and into the stress margin.

Where you meet it

Writing a shaker test spec on a bench at Redstone. The customer hands you a qual environment as a PSD table — breakpoints in Hz and g²/Hz. Before the article ever touches the slip table, you compute the overall Grms to sanity-check the level and to size the amplifier and fixture.
A vibration test stand running a random profile. The controller drives the shaker to match a demanded PSD inside a tolerance band. You read the achieved Grms off the console and compare it to the demand. A number that drifts high means a fixture resonance is dumping energy into the wrong band.
A structural review board for a bracket, box, or avionics tray. Somebody asks "what static load did you design the mount to?" The answer traces back through Miles' equation: find the mode, grab Q, read the PSD at fn, get response Grms, multiply by 3, hang that as an equivalent static g on the mass.
Fatigue life on a component under a random duty cycle. Miles wrote his paper about fatigue, not static strength. You take the response Grms, the stress it produces, and the number of stress cycles at fn over the mission, and run it against an S-N curve.

How it works

Start with the PSD itself. It is not a signal — it is a statistical description of one. A random vibration environment has no repeatable time history; two runs at the same PSD produce different waveforms with the same statistics. The PSD tells you how the mean-square acceleration is distributed across frequency. Integrate it and you recover the total mean square; square-root that and you have Grms. Everything downstream is bookkeeping on that one integral.

Miles' equation is the SDOF response to a flat input. The trick that makes it a one-liner: a lightly-damped resonance acts like a razor-thin filter centered on fn. It only "hears" the environment in a narrow slice around its own frequency. So instead of integrating the full response spectrum, Miles replaced the input with a constant equal to the PSD level right at fn, integrated the SDOF transfer function analytically, and got (π/2)·fn·Q·W. The π/2 and the fn·Q are literally the area of that resonant peak.

Run the numbers to see how tight it is. Take fn = 200 Hz, Q = 10, W = 0.04 g²/Hz. Miles gives sqrt((π/2)(200)(10)(0.04)) = 11.21 g RMS. Integrate the full SDOF base-input transmissibility numerically over a wide band and you get 11.27 g — under half a percent apart. Miles is not a rough cut; when its assumptions hold, it is nearly exact.

Now the assumptions, because that is where people get hurt.

The input must be flat near the resonance. Miles assumes white noise — a constant PSD stretching to infinity. Real environments roll off. The working rule: the equation is trustworthy if the input PSD is roughly flat within about one octave on either side of fn. Put a resonance right on a steep slope or in a notch and the equation reads the wrong W. Read the PSD at the resonant frequency, not at some convenient breakpoint.

You have to know Q, and you usually don't. Q swings the answer as sqrt(Q) — double the damping estimate and the response drops 30%. Analysts default to Q = 10 (5% damping) or sometimes Q = sqrt(fn), but a bolted joint might be Q = 5 and a clean machined bracket might be Q = 30. This single guess dominates the margin. Measure it in a sine sweep if the schedule allows.

Three sigma is a floor, not a ceiling. The 3σ multiplier comes straight from the Gaussian: 99.73% of excursions fall inside ±3σ. That still leaves 0.27% outside, and over millions of cycles at fn the peaks will exceed 3σ many times. For fatigue-critical parts, analysts often carry 4σ or run a proper rainflow count. Do not treat 3σ as a hard maximum load — it is a percentile.

Single mode only. Miles is SDOF. A real box has dozens of modes. If they are well separated in frequency you can apply Miles mode-by-mode and root-sum-square the responses. If they are packed together and coupled, Miles undercounts and you need a full modal or finite-element random response run.

The most common mistake, seen on more than one review board: pulling the PSD level off the wrong breakpoint. An analyst reads W at 100 Hz because that is where the table lists a number, but the mode is at 175 Hz sitting on a rising slope. The equation then reports a level that is real, precise, and wrong. Miles' equation is only as good as the W you feed it, and the W it wants is the one directly under the resonance.

History

John Wilder Miles was an applied mathematician, born December 1920, who spent the first stretch of his career in electrical and aeronautical engineering before turning to fluid dynamics and, eventually, the physics of wind-driven ocean waves ^[1]^[2]. In 1954 he was an associate professor at UCLA ^[1]. Jet aircraft were new, and their exhaust and boundary-layer turbulence were shaking structures apart in a way steady aerodynamic loads never had — the loading was broadband and random, and the classical tools assumed loads you could write down as a function of time.

Miles published "On Structural Fatigue Under Random Loading" in the Journal of the Aeronautical Sciences in 1954 ^[3]. The paper treated the stress in an elastic structure driven by random forcing as a statistical quantity, worked out the response of a resonant structure to a broadband input, and connected that response to fatigue life. Out of that analysis came the compact result the test community now calls Miles' equation — the resonant response of an SDOF system to a flat input PSD, sqrt((π/2)·fn·Q·W).

The name stuck to the equation more firmly than it stuck to Miles himself, who went on to a long career in geophysical fluid dynamics at Scripps and is at least as well known there for his wind-wave growth theory ^[2]. But every aerospace test engineer who sizes a bracket off a shaker spec is using a shortcut he wrote down for jet fatigue seventy years ago. It has held up because the assumption behind it — a sharp resonance only feels the environment in its own narrow band — is true of almost everything that rings.

Related tools

/tools/vibration-natural-freq
/tools/rms-peak
/tools/lc-resonance
/tools/shaft-critical-speed
/tools/stress-strain

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →