fea-dynamics

name	fea-dynamics
description	FEA dynamics — modal analysis (Block Lanczos, DDAM), harmonic response, transient dynamics, random vibration (PSD), response spectrum, pre-stressed modal, fatigue from FEA.
metadata	{"priority":7,"promptSignals":{"phrases":["FEA modal","FEA dynamics","harmonic response","random vibration FEA","PSD analysis","response spectrum FEA","modal analysis FEA","transient FEA"],"minScore":4}}

FEA Dynamics — Complete Skill

Modal Analysis

Governing Equation (Undamped, Free Vibration)

[M]{ẍ} + [K]{x} = {0} Assume {x} = {φ}sin(ωt) → ([K] - ω²[M]){φ} = {0} Eigenvalue problem: det([K] - ω²[M]) = 0 → eigenvalues ω_i², eigenvectors {φ_i}

Mass Matrix Options

Consistent mass: [M] = ∫[N]ᵀρ[N]dV (higher accuracy, full matrix) Lumped mass: diagonal only, total mass concentrated at nodes (faster, lower accuracy for bending) ANSYS: LUMPM,OFF (consistent default); Abaqus: ELEMENT TYPE, MASS = CONSISTENT

Eigensolvers

Block Lanczos (ANSYS): most robust for large models (>10,000 DOF, many modes) PCG Lanczos: memory-efficient, very large models QR Damped (ANSYS): includes damping in eigenvalue extraction Subspace iteration: small models, few modes

Shift: extract modes near frequency f_shift (mass matrix shift) Sturm sequence check: verify no missed modes (ANSYS: MODOPT with STURMCK)

Mode Shape Normalization

Mass-normalized: {φ_i}ᵀ[M]{φ_i} = 1 Maximum displacement normalized: max|φ_i| = 1 (ANSYS default)

Mass Participation Factor

MPF_i = ({φ_i}ᵀ[M]{ι})² / (M_total) ι = unit vector in direction of excitation Sum of MPF for all modes ≥ 80% → sufficient modes extracted (ASCE 7, IBC) Critical: always verify cumulative MPF before trusting modal response results

Pre-Stressed Modal (Stress Stiffening)

Structures under tension: stiffness increases → frequencies increase Under compression: stiffness decreases → frequencies decrease (approaching buckling) Procedure: run static solution first → PSTRES,ON → modal analysis uses K + K_σ

Applications: strings under tension, centrifugal stiffening (turbine blades), pressurized membranes

Harmonic Response Analysis

Methods

Full method: solve full system (-ω²[M] + iω[C] + [K]){X(ω)} = {F₀} for each frequency Expensive but exact (includes all modes, no truncation)

Mode superposition: project onto mode shapes → NDOF problems → n modal ODFs Faster; requires modes from prior modal analysis Accuracy: governed by modes included (need MPF > 90%)

MSUP method with residual: adds static correction for missing high-freq modes

Damping in Harmonic

Viscous damping ratio ζ_i per mode (modal damping, most common) Rayleigh damping: c = α[M] + β[K] → ζ_i = α/(2ω_i) + β ω_i/2 Material damping loss factor η: equivalent ζ = η/2 ANSYS: DMPRAT (constant), MDAMP (per mode), ALPHAD/BETAD (Rayleigh)

Output

Complex displacement {X} = {X_R} + i{X_I} Amplitude: |X| = √(X_R² + X_I²) Phase: φ = arctan(X_I/X_R)

Response at resonance (ω = ω_n): X = F₀/(2ζk) ← limited by damping only Peak frequency: ω_peak = ω_n√(1-2ζ²)

Transient Dynamic Analysis

Newmark-β Integration

{d}_{n+1} = {d}_n + Δt{v}n + Δt²[(0.5-β){a}n + β{a}{n+1}] {v}{n+1} = {v}_n + Δt[(1-γ){a}n + γ{a}{n+1}]

Stability: γ=0.5, β=0.25 → unconditionally stable (constant average acceleration) Numerical damping: γ > 0.5 → overdamped high frequencies (algorithm damping) HHT-α method: better numerical damping control

Time step: Δt ≤ T_min_mode/20 where T_min = 1/f_max_of_interest Mode superposition transient: faster, good for linear problems with many load histories

Initial Conditions

ANSYS: IC command for initial displacement/velocity at nodes Abaqus: *INITIAL CONDITIONS, TYPE=VELOCITY

Random Vibration (PSD Analysis)

Power Spectral Density

S(f) [g²/Hz or (m/s²)²/Hz] = input base excitation spectrum RMS acceleration: a_rms = √(∫S(f)df) over frequency range

FEA PSD Procedure (ANSYS, Abaqus)

1. Modal analysis (extract modes up to 1.5× highest PSD frequency)
2. Define PSD input spectrum (base excitation or force)
3. Compute cross-power spectral density between modes (mode correlation)
4. Output: RMS stress, displacement, acceleration at each node σ_rms = √(Σσ_i_rms² + cross-correlation terms)

SRSS: σ_rms = √(Σσ_i²) [for well-separated modes, ignores correlation] CQC: σ_rms = √(ΣΣρ_ij σ_i σ_j) [full correlation matrix]

Mil-Std-810H PSD profiles (common inputs)

Aircraft: 0.04 g²/Hz from 15-1000 Hz Truck transport: higher at low frequencies (5-20 Hz peak) 3σ rule: peak response ≈ 3× RMS (for Gaussian, 99.73% of peaks below this)

Fatigue from PSD (Dirlik, Steinberg)

Steinberg 3-band: material below 1σ: 68.3%, 1σ-2σ: 27.1%, 2σ-3σ: 4.33% N_1σ + N_2σ/0.1 + N_3σ/0.01... based on S-N curve

Dirlik: probability density of peaks → Miner's rule integral More accurate than Steinberg for broadband PSD

Response Spectrum Analysis (Seismic)

Procedure

1. Modal analysis → extract frequencies ω_i, mode shapes {φ_i}
2. Modal participation: Γ_i = {φ_i}ᵀ[M]{ι} / {φ_i}ᵀ[M]{φ_i}
3. Peak modal response: d_i = S_d(ω_i) × Γ_i × {φ_i}
4. Combine modal peaks by CQC or SRSS

SRSS: D = √(ΣD_i²) [adequate for well-separated modes] CQC: D = √(ΣΣρ_ij D_i D_j) [required when modes close]

Input: ASCE 7 design spectrum, IBC site-specific, IEEE 693, NRC (nuclear)

Buckling (Linear, Eigenvalue)

[K]{Φ} + λ_cr[K_g]{Φ} = {0} λ_cr = buckling load factor (BLF); P_cr = λ_cr × P_applied

Limitations of linear buckling:

• Assumes perfect geometry
• Actual buckling load typically 50-80% of linear prediction (imperfection sensitivity)
• For design: P_allowable = P_cr/φ with φ = 2-5 (depends on imperfection sensitivity)

Nonlinear buckling (Riks): apply imperfections, trace load-displacement into post-buckling Imperfection amplitude: typically 1/1000 to 1/100 of critical length dimension

Output

Provide: natural frequencies ω_i [Hz], cumulative MPF by direction [%], peak RMS stress from PSD [MPa], response spectrum modal combination method, time step Δt [s] for transient, BLF for buckling.