Random Vibration Analysis

Random Vibration Response Analysis (structures/loads/random-vibration-analysis)

Use when a structure or equipment item must be checked against a random vibration environment described by a base-input acceleration power spectral density (PSD, in g^2/Hz). The item is modeled as a single-degree-of-freedom (SDOF) oscillator at its natural frequency f_n with damping ratio zeta; the analysis computes the response PSD through the acceleration transmissibility, the RMS response in g from the Miles equation (flat input at resonance) or by numerical integration of the response PSD (arbitrary spectrum), the 3-sigma peak level, and the equivalent static load factor used for equipment qualification screening. This is the other dynamic loads input alongside discrete gust and maneuver loads, feeding the same structural sizing and fatigue flows. The response-level model here is deliberately confined to SDOF random vibration response; cycle counting and cumulative fatigue damage of the vibration response are owned by the fatigue pack.

Domain quick reference

• SDOF base-excitation model: frequency ratio r = f/f_n and the acceleration transmissibility (response to base acceleration)

  |H(f)|^2 = 1 / ((1 - r^2)^2 + (2*zeta*r)^2)
  

  |H(f)|^2 equals 1 at f = 0 (rigid-body limit) and Q^2 = 1/(2*zeta)^2 at f = f_n, and rolls off as (f_n/f)^4 well above resonance.

• Response PSD: G_out(f) = |H(f)|^2 * G_in(f), with G_in in g^2/Hz and G_out in g^2/Hz. The response PSD peaks near the damped resonance f_nsqrt(1 - 2zeta^2), which is the dominant response frequency.

• Amplification factor at resonance: Q = 1/(2*zeta) (about 10 at zeta = 0.05, about 5 at zeta = 0.10).

• Miles equation (flat, band-limited input at the resonance):

  sigma = sqrt((pi/2) * f_n * Q * G_in(f_n))
  

  in g_rms, with f_n in Hz. It follows from integrating |H(f)|^2 over all frequencies: the integral equals (pi/2)f_nQ, so sigma^2 = G_in * (pi/2) * f_n * Q.

• Numerical path (input not flat at f_n): sigma^2 = integral of G_out(f) df, trapezoidal rule on the supplied spectrum points. The grid must resolve the response peak, whose half-power width is about 2zetaf_n (4 Hz at f_n = 40 Hz, zeta = 0.05); a spectrum sampled coarser than that underestimates the integral.

• 3-sigma peak and equivalent static load factor: for a narrowband Gaussian random response the peak accelerations reach about 3sigma, so peak = n_eq = 3sigma in g. The load factor n_eq is a screening value only: it assumes the response at resonance dominates and the item behaves as the SDOF oscillator modeled here.

• Units: f, f_n in Hz; zeta dimensionless in (0, 1); G in g^2/Hz; sigma in g_rms; peaks and load factors in g.

• Context: transport certification FAR 25.301/25.303 frames the limit loads context (reference only); equipment random vibration test levels follow MIL-STD-810H (random vibration test method) and DO-160G Section 8 (vibration tests of airborne equipment), which this analysis converts into item response levels. Standard text is named and paraphrased, never reproduced.

Workflow

1. Gather the item's first natural frequency f_n (Hz) and damping ratio zeta, from the modal analysis or a resonance survey, and the input base-acceleration PSD spectrum as (f, G_in) points in g^2/Hz over the test band.
2. Confirm the amplification factor with quality_factor(zeta): Q = 1/(2*zeta). Typical equipment damping runs 2-10% of critical.
3. Get the transmissibility at any frequency with transmissibility(f, f_n, zeta) or transmissibility_squared for the PSD amplification |H(f)|^2.
4. For a flat band-limited input at resonance, use the Miles closed form miles_sigma(f_n, zeta, g_in_at_resonance): sigma = sqrt((pi/2)f_nQ*G_in(f_n)).
5. For an arbitrary spectrum, build the response PSD with response_psd(spectrum, f_n, zeta) and integrate with numerical_sigma(spectrum, f_n, zeta) (trapezoidal on the provided points; at least two points required). Check the grid resolves the response peak (step 1 quick reference).
6. Run the full analysis with random_vibration_analysis(f_n, zeta, spectrum): returns sigma_rms_g (numerical), sigma_miles_g (Miles value with the input level interpolated at f_n, None when f_n lies outside the spectrum coverage), psd_response_points, f_n, q, dominant_response_frequency, peak_3sigma_g and n_eq_g.
7. Report the 3-sigma level with peak_three_sigma(sigma) and the equivalent static load factor with equivalent_static_load_factor(sigma) for the qualification screening against the item's test-withstand level.
8. Feed the response levels into the fatigue flow (Miner damage) when the vibration environment contributes to the load spectrum; cycle counting and damage accumulation live in the fatigue pack, not here.

Worked example

Equipment item at f_n = 40 Hz with zeta = 0.05 (Q = 10) under a flat base-input PSD of G = 0.01 g^2/Hz over 20-500 Hz.

• Miles: sigma = sqrt((pi/2)40100.01) = sqrt(6.2831853) = 2.5066 g_rms; the 3-sigma peak is 32.5066 = 7.52 g and the equivalent static load factor is n_eq = 7.52 g for screening.
• Response PSD at resonance: G_out(40) = 0.01 * Q^2 = 0.01 * 100 = 1.0 g^2/Hz; the dominant response frequency is 40sqrt(1 - 20.05^2) = 39.90 Hz.
• Numerical cross-check: trapezoidal integration of G_out over the 20-500 Hz band on a 1 Hz grid gives sigma = 2.458 g_rms, about 2% below the Miles value (the deficit is the input band truncation below 20 Hz, far from the resonance); a 2 Hz grid gives 2.462 g_rms, still within a few percent. Both confirm the Miles result when the input is flat around resonance.
• An avionics box with f_n = 90 Hz and zeta = 0.03 (Q = 16.7) under the same flat 0.01 g^2/Hz input responds at sigma = sqrt((pi/2)9016.67*0.01) = 4.854 g_rms, a 3-sigma level of 14.6 g, which would fail a 10 g equipment screening; the case demonstrates how strongly the response level scales with f_n and Q.

Pitfalls

• Using Miles on a non-flat spectrum: the Miles equation assumes a flat band-limited input level at the resonance; for an arbitrary spectrum the response PSD must be integrated numerically, and the module returns None for the Miles value when f_n lies outside the input coverage.
• Sampling the spectrum coarser than the response peak: the half- power width of the peak is about 2zetaf_n (4 Hz at f_n = 40 Hz, zeta = 0.05), so a grid coarser than that underestimates the integrated response.
• Quoting a 3-sigma level as a hard peak: for a narrowband Gaussian random response the peak reaches about 3sigma and n_eq = 3sigma is a screening load factor that assumes the response at resonance dominates and the item is SDOF.
• Reading the transmissibility at resonance as the amplification: |H(f)|^2 peaks at Q^2 = 1/(2*zeta)^2 (100 at zeta = 0.05) while Q itself is 10; the response PSD scales with the squared factor.
• Feeding an out-of-range damping: zeta must lie in (0, 1); zeta = 0 makes the amplification infinite and zeta >= 1 is not an oscillator, and both raise ValueError.
• Doing the fatigue damage here: this leaf stops at response levels (sigma, 3-sigma, n_eq); cycle counting and cumulative Miner damage of the vibration response belong to the fatigue pack.

Verification

• miles_sigma(40, 0.05, 0.01) returns 2.5066 g_rms and peak_three_sigma gives 7.52 g, both within 1% of the hand calc.
• quality_factor(0.05) returns 10 and transmissibility_squared at f = f_n returns Q^2 = 100.
• numerical_sigma over the flat 20-500 Hz band (2 Hz grid) returns a value within 3% of the Miles sigma (worked example cross-check).
• random_vibration_analysis returns the dict with sigma_rms_g, psd_response_points, f_n, q and dominant_response_frequency, and reports sigma_miles_g = None when f_n falls outside the input band.
• Every non-physical input raises ValueError: non-positive f_n, zeta <= 0 or zeta >= 1, negative PSD ordinates, empty or single-point spectra for integration, non-increasing frequencies, and negative frequencies.
• Deterministic contract: scripts/test_random_vibration_analysis.py.

Related leaves

• structures/loads/gust-maneuver-loads: discrete gust and maneuver load factors, the other dynamic loads input that joins random vibration in the loads pack.
• structures/fem/modal-analysis: natural frequencies and damping used as the f_n and zeta inputs of this response analysis.
• structures/fatigue/miner-damage: cumulative damage of the cycles implied by the vibration response; this leaf stops at response levels, the fatigue pack owns the damage estimate.
• structures/thermal-structures/thermal-stress-analysis: thermal environment loads assessed alongside the vibration qualification.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_random_vibration_analysis.py

The test covers the worked example (Miles sigma 2.5066 g_rms and 3-sigma level 7.52 g within 1% of the hand calc), the numerical cross-check on the flat band within a few percent of Miles, the Q^2 = 100 response amplification at resonance, the (f_n/f)^4 roll-off, the response PSD construction, sigma scaling with sqrt(G), sqrt(f_n) and 1/sqrt(zeta), the Miles round-trip that recovers the input PSD, interpolation of the input level at f_n, the analysis dict contents, and ValueError rejection of non-positive f_n, out-of-range zeta, negative ordinates, and empty, single-point or non-monotonic spectra.

Compliance

• FAR 25 is US government work (public domain); summary and physics values only, per standards-map.yaml. MIL-STD-810H and DO-160G Section 8 are named as equipment test-level context and paraphrased, never reproduced.
• compliance: STANDARDS-REF, gated: false.