Random vibration is the environment almost everything real sees — a launch, a truck bed, a jet engine bay. It behaves nothing like the sine sweep people picture, and the maths trips up anyone meeting it for the first time.
It is described by a PSD, not an amplitude
In random vibration every frequency is excited at once, continuously, and the instantaneous acceleration never repeats. So you cannot quote "10 G at 200 Hz". Instead you quote a power spectral density in G²/Hz — acceleration energy per unit bandwidth. Integrating the PSD across the band and taking the square root gives the overall Grms, the single number that describes the whole environment.
A bracket fails a random vibration test. You stiffen it, doubling its natural frequency, and retest under the same flat PSD. What happens to its Grms response?
Miles' equation gives Grms = √(π/2 · fn · Q · ASD), so under a flat PSD the response goes as √fn — stiffer means a higher acceleration, not a lower one. What stiffening actually buys is smaller displacement, and usually lower stress, because the part is deflecting less even though it is accelerating more. This is the single most counter-intuitive result in random vibration, and it is why "make it stiffer" has to be checked rather than assumed — especially when the PSD is not flat and you may be moving fn into a peak.
Try it — Miles' equation and the 3σ load
Grms is 1σ — design to 3σ
The instantaneous level is Gaussian, so Grms is the 1σ value. That means about 68% of the time you are under it, 95% under 2σ, and 99.7% under 3σ — but 3σ peaks do occur, thousands of times in a qualification test. The convention is therefore to design structure to the 3σ load, and it is why a 12 Grms environment is a 36 G design case.
Miles' equation: the first-cut response
For a component whose response is dominated by one mode, Miles' equation gives the 1σ response from the PSD at its natural frequency: Grms = √((π/2)·fn·Q·ASD), with Q ≈ 1/(2ζ) — Q = 10 (5% damping) is the usual default for bolted metal structure. Run it in the Miles' Equation tool, multiply by 3, multiply by mass, and you have a design load for the mounts.
What this misses
Miles assumes a single degree of freedom and a flat PSD around fn. Multiple close modes, a sloped PSD, or a stiff item that just rides along all break the assumption — those need a real random-response FE run. Note too that fatigue matters as much as peak load: use the rainflow or three-band (Steinberg) method for life, since random vibration accumulates damage continuously. Test levels come from MIL-STD-810 method 514 or NASA-STD-7001.