random-vibration

name	random-vibration
description	Random vibration analysis — PSD, RMS response, Miles equation, SDOF response, fatigue from PSD, three-sigma rule, Grms, ASD spectra, MIL-STD-810 vibration.
metadata	{"priority":7,"promptSignals":{"phrases":["random vibration","PSD","power spectral density","Grms","Miles equation","ASD","random response"],"minScore":3}}

Random Vibration Analysis — Complete Skill

Power Spectral Density (PSD)

Definition

G(f) = PSD of acceleration [g²/Hz] Mean square value: σ² = ∫₀^∞ G(f) df [g²] RMS value: Grms = σ = √(∫₀^∞ G(f) df) [g]

Three-sigma rule: Peak acceleration ≈ ±3σ (99.73% of time within ±3σ for Gaussian process)

Typical aerospace PSD (MIL-STD-1540): 20-80 Hz: ramp up from 0.01 to 0.04 g²/Hz (+6 dB/oct) 80-350 Hz: flat 0.04 g²/Hz 350-2000 Hz: ramp down (+6 dB/oct downward): 0.04 to 0.0007 g²/Hz Grms ≈ 6.8 g_rms (typical)

Octave dB conversion: slope [dB/oct] = 10 log(G₂/G₁)/log(f₂/f₁) Integral over ramp: ∫ G(f)df = G₁ × f₁/(m+1) × [(f₂/f₁)^(m+1) - 1] where m = log(G₂/G₁)/log(f₂/f₁)

Miles Equation (SDOF Response)

Mean Square Response

For lightly damped SDOF (ζ < 0.1) excited by white noise or nearly flat PSD at fn: σ² = π/2 × fn × G(fn) / ζ = π fn G(fn) / (2ζ)

Grms response: σ = √(π × fn × G(fn) / (4ζ)) [Miles, flat PSD assumption]

Full expression (non-flat PSD): σ² = ∫₀^∞ G(f) × |H(f)|² df |H(f)|² = 1/[(1-(f/fn)²)² + (2ζf/fn)²] × (k/m)^(-2)

SDOF Frequency Response

H(jω) = 1/(k - mω² + jcω) = (1/k) / [(1-r²) + j2ζr] r = f/fn, ζ = c/(2√km) |H|² = 1/k² / [(1-r²)² + (2ζr)²]

Q factor: Q = fn/(Δf_3dB) = 1/(2ζ) [amplification at resonance: Q = 1/(2ζ)] At resonance: |H| = Q/k = 1/(2ζk)

Vibration Fatigue from PSD

Rainflow from PSD

For Gaussian response, stress cycles amplitudes follow Rayleigh distribution: p(S) = (S/σ_s²) exp(-S²/(2σ_s²)) [Rayleigh PDF for stress amplitude S] σ_s = RMS stress

Bandwidth parameter ε: ε = √(1 - m₂²/(m₀ m₄)) Narrow-band: ε → 0; wide-band: ε → 1

Fatigue damage (narrow-band, S-N: N = C/S^m): D = T × f₀ × σ_s^m × 2^(m/2) × Γ(1 + m/2) / C T = exposure time, f₀ = zero-crossing rate ≈ fn

Dirlik method (broad-band): empirical PDF for peak distribution; most widely used in software

ASD (Acceleration Spectral Density) Test Levels

MIL-STD-810H Method 514.8

Random vibration test for general equipment Functional test level (during vibration): typically equipment operating Endurance level: cumulative duration

Typical ground vehicle (MIL-STD-810): 1-10 Hz: ramp; 10-40 Hz: 0.04-0.06 g²/Hz; 40-100 Hz: flat; 100-500 Hz: ramp down Grms ≈ 3-7 g_rms depending on application

Duration: equal-damage approach using inverse power law: T₁/T₂ = (Grms,2/Grms,1)^m (m = S-N slope, typically 3-6)

Shock Response Spectrum (SRS)

Different from PSD: SRS = max response of SDOF filters to a transient shock SRS(fn) = max acceleration of SDOF at fn when base-excited by shock Units: g vs. Hz; compared to shock damage potential

Pyro shock: SRS shape 20-100 Hz: +12 dB/oct slope; 100-1000 Hz: flat plateau 1000g; > 1000 Hz: 0 dB/oct (notch)

Output

Provide: Grms input [g], σ response [g] (Miles equation), peak 3σ [g], natural frequency fn [Hz] and Q, fatigue life T [hrs] from Miner's rule, ASD profile adequacy vs. test standard.