fea-random-vibration

name	fea-random-vibration
description	FEA random vibration analysis — PSD input, modal superposition, response PSD, RMS stress, Dirlik fatigue, ASTM E1049, MIL-STD-810, electronic packaging vibration qualification.
metadata	{"priority":7,"promptSignals":{"phrases":["random vibration FEA","PSD analysis","random vibration","power spectral density","RMS stress","Dirlik fatigue","MIL-STD-810 vibration"],"minScore":3}}

FEA Random Vibration Analysis — Complete Skill

Power Spectral Density (PSD) Fundamentals

PSD definition: S_xx(f) = lim_{T→∞} |X(f)|²/T [units: g²/Hz or m²/s⁴/Hz]

RMS from PSD: x_rms = √∫₀^∞ S_xx(f) df (= √(area under PSD curve))

PSD input levels (MIL-STD-810G Method 514.7 — Road transport): 0.015 g²/Hz from 20–500 Hz (broad-band; vehicle transportation) MIL-STD-810G Jet aircraft transport: 0.02–0.04 g²/Hz from 20–2000 Hz

ASTM E1049: fatigue from PSD; spectral moments method

Random Vibration FEA — Modal Superposition Method

Step 1: Modal Analysis

Extract natural frequencies ω_n and mode shapes {φ}_n Need modes up to at least 2× highest significant PSD frequency Typically 50–200 modes for complex structures

Step 2: Modal Participation

Modal response PSD: H_r(ω) = transfer function for mode r Modal frequency response function: H_r(ω) = 1 / (ω_r² - ω² + 2iζ_r ω_r ω)

Power spectral density of modal response: S_q_r(ω) = |H_r(ω)|² × S_base(ω)

Step 3: Response PSD

Cross-correlation between modes: if modes well-separated, cross terms ≈ 0 (SRSS combination) Stress PSD at location k: S_σ,k(ω) = Σ_r |σ_k,r|² × |H_r(ω)|² × S_base(ω) [SRSS]

σ_k,r = modal stress at location k from mode r (from modal analysis)

Step 4: RMS Stress

σ_rms = √∫₀^∞ S_σ,k(ω) dω [MPa]

3σ stress: σ_3σ = 3 × σ_rms (99.73% probability of not exceeding; used for peak estimation) Design criterion: σ_3σ < S_endurance (fatigue limit check)

ABAQUS / ANSYS Random Vibration Setup

ABAQUS: *Step, name=RandomVibration, perturbation *Random Response *Correlation, type=NODAL Load definition: *Base Motion, type=ACCELERATION, DIRECTION=1 *PSD-Definition: frequency vs. PSD table *Node Print or *Output for RMS values

ANSYS Mechanical: Analysis type: Random Vibration (under Modal → Random Vibration) PSD input: G²/Hz table vs. frequency Directions: X, Y, Z accelerations (can combine) Output: 1σ (RMS) values for stress, displacement, acceleration 3σ values: multiply 1σ × 3 for practical design use

Critical settings: Rigid response correction: important when significant response at f < first natural frequency Mode significance threshold: include modes with mass participation > 0.1%

Spectral Moments for Fatigue

Moments of PSD: m_n = ∫₀^∞ f^n × S_σ(f) df

m₀ = variance = σ_rms² m₁ = first moment; m₂ = second moment; m₄ = fourth moment

Bandwidth parameter: α₂ = m₂ / √(m₀ × m₄) [0 < α₂ < 1] Narrow-band: α₂ → 1; wide-band: α₂ → 0

Irregularity factor: γ = m₂ / √(m₀ × m₄) (same as α₂)

Expected number of peaks per second: E[P] = √(m₄/m₂)

Expected number of zero crossings per second: E[0] = √(m₂/m₀)

Fatigue Damage — Frequency Domain Methods

Narrow-Band Approximation (conservative)

Assumes all cycles have amplitude equal to RMS × √2 (Rayleigh distribution): D_NB = E[P] × T × C_material × (2√2 × σ_rms)^m / K

m, K from S-N curve: σ^m × N = K (double log; m = 3 for steel)

Dirlik Method (preferred for wide-band)

Empirical probability density function from Monte Carlo simulation: f_D(S) = (D_1/Q × exp(-Z/Q) + D_2 × Z/R² × exp(-Z²/2R²) + D_3 × Z × exp(-Z²/2)) / (2 σ_rms)

D_1, D_2, D_3, Q, R from m₀, m₁, m₂, m₄ (polynomial fit to moments) Z = S / (2σ_rms) (normalized amplitude)

Damage rate: D = E[P] × ∫₀^∞ f_D(S) × S^m / K dS

More accurate than narrow-band for wide-band PSD; ASTM E1049 standard method

Zhao-Baker, Tovo-Benasciutti Methods

Alternative frequency-domain methods; each valid for different bandwidth regimes Zhao-Baker: bimodal PSD (two frequency peaks)

MIL-STD-810 Vibration Testing

Method 514.7 (random vibration): Procedure I: general minimum integrity test Procedure II: tailored test (measured data from actual field)

Common test profiles: Truck cargo (Category 4): 0.01 g²/Hz from 5–1000 Hz; duration 4 hrs/axis Aircraft (Category 14): higher levels; 0.04 g²/Hz typical

Equivalence of test and analysis: T_test × G²_test ≈ T_field × G²_field (equal damage; slope from S-N m value) Test acceleration from Miner's rule

MIL-HDBK-5 / MMPDS: aluminum/steel fatigue data for aerospace

Electronic Packaging Vibration (IPC-9704 / JEDEC)

Board-level solder joint fatigue from random vibration Steinberg method: σ_critical = S_y × 0.055 × (lead area penalty) [for BGA/SMT solder joints] Board natural frequency: f_n = (π²/L²) × √(EI/ρA) [for simply-supported PCB] Board RMS displacement = 3σ value must be < 0.006 × L [allowable for solder joint fatigue; Steinberg]

IPC-7351: PCB land pattern design; influences modal shape JEDEC JESD22-B111: board-level random vibration test; 7.7 g_rms for telecom/computing

ASTM E1049 (Cycle Counting for Variable Amplitude)

Rainflow counting for time-domain fatigue; for FEA, use PSD approach directly If time-history available: rainflow → histogram → sum Miner's rule E1049 defines rainflow algorithm for irregular time histories

Output

Provide: 1σ (RMS) and 3σ stress at critical locations [MPa], response PSD S_σ(f) at critical locations, frequency content summary (dominant response frequencies), m₀–m₄ spectral moments, bandwidth parameter α₂, Dirlik damage accumulation D [per hour of exposure], equivalent cycles to failure N_f at 3σ level, comparison to fatigue endurance [MPa], and qualification test profile (g²/Hz vs. frequency, duration, axes) per applicable standard (MIL-STD-810G, IPC-9704, ASTM E1049).