modal-analysis

name	modal-analysis
description	Modal analysis — SDOF/MDOF modal parameters, mode shape visualization, FRF curve fitting, model correlation (MAC/COMAC), structural dynamic modification, experimental vs FEA correlation.
metadata	{"priority":7,"promptSignals":{"phrases":["modal analysis","mode shape","natural frequency","damping ratio","modal parameters","MAC","model correlation","COMAC"],"minScore":4}}

Modal Analysis — Complete Skill

Modal Parameters (Per Mode r)

ω_r = natural frequency [rad/s] = 2π f_r ζ_r = viscous damping ratio [dimensionless] {φ_r} = mode shape vector [N×1, N = number of DOF measured] m_r = modal mass [kg], k_r = modal stiffness [N/m], c_r = modal damping [N·s/m]

Relationships: k_r = ω_r² × m_r c_r = 2 × ζ_r × ω_r × m_r = 2 × ζ_r × √(k_r × m_r) ω_dr = ω_r × √(1 - ζ_r²) [damped natural frequency]

Mass normalization: {φ_r}ᵀ [M] {φ_r} = m_r → {φ_r_hat} = {φ_r}/√m_r → {φ_r_hat}ᵀ [M] {φ_r_hat} = 1

FRF in Terms of Modal Parameters

Receptance FRF (MDOF): H_jk(ω) = Σ_r [φ_jr × φ_kr] / [k_r - ω²m_r + iωc_r] = Σ_r A_jkr / (ω_r² - ω² + 2iζ_rω_rω)

Where A_jkr = {φ_jr × φ_kr}/m_r = modal residue (complex) j = response DOF, k = excitation DOF

At resonance (ω = ω_r, lightly damped): H_jk(ω_r) ≈ A_jkr / (2iζ_rω_r²) |H_jk(ω_r)| ≈ |A_jkr| / (2ζ_rω_r²) — amplitude at resonance

Half-Power Bandwidth Method (Damping)

Identify: ω₁ and ω₂ at amplitude = A_max/√2 (half power = -3 dB) ζ_r = (ω₂ - ω₁)/(2ω_r) = Δω/(2ω_r)

Accuracy: only valid for well-separated modes (no modal overlap) Modal overlap factor: η = Σ 2ζ_r ω_r / (modal spacing) < 0.3 for good separation Alternative: Nyquist circle fit (more robust for overlapping modes)

Mode Shape Visualization

Animation

Mode shape deflection: u_j(t) = Re({φ_j} × e^(iω_r t)) [harmonic oscillation] Scale amplitude to visible; superimpose on undeflected geometry Direction: forward/backward whirl for rotating systems

Classification

Rigid body modes: ω ≈ 0 (rigid body translation/rotation — 6 for free-free) Elastic modes: ω > 0, structural deformation (bending, torsion, breathing) Local modes: high-frequency, small component (fastener, bracket) Global modes: full structure deflects (usually lowest frequencies of interest)

Model Correlation (EMA vs. FEA)

MAC (Modal Assurance Criterion)

MAC_ij = |{φ_i}ᵀ{φ_j}|² / ({φ_i}ᵀ{φ_i} × {φ_j}ᵀ{φ_j})

Interpretation: MAC = 1.0: identical shapes (well-correlated mode pair) MAC > 0.9: good correlation MAC = 0.7-0.9: acceptable for industrial purposes MAC < 0.7: poor — investigate sensor placement or FEA error Off-diagonal MAC > 0.3: modes contaminated (linear dependence)

MAC matrix (N_modes × N_modes): diagonal = auto-MAC (1.0 if modes distinct), off-diagonal = cross-MAC Target: strong diagonal, weak off-diagonal (MAC > 0.9 on diagonal, < 0.1 off)

COMAC (Coordinate Modal Assurance Criterion)

COMAC_j = [Σ_r MAC_r × |φ_jr^EXP × φ_jr^FEA|]² / [Σ_r |φ_jr^EXP|² × Σ_r |φ_jr^FEA|²] Identifies which DOF j has poor correlation (poor sensor, FEA model error at that location)

Frequency Correlation

Δf% = (f_EXP - f_FEA)/f_FEA × 100% Target: Δf < 5% (good), < 10% (acceptable) Systematic error: all FEA frequencies too high → mass too low in model; too stiff → wrong boundary conditions

Paired Mode Identification

Sort EMA and FEA modes by frequency Use MAC to pair: FEA mode i paired with EXP mode j where MAC_ij is maximum Frequency pairing alone can be wrong (crossing modes, missed modes)

Model Updating

Manual updating: identify physical cause of discrepancy → adjust model

• Too stiff: check connection modeling (rigid bonds vs. flexible joints)
• Too heavy: check added masses, attachments
• Wrong mode shape: check BCs, connection stiffness at joints

Automated model updating: minimize J = ||{f_EXP - f_FEA}||² + ||{MAC - I}||² Parameters: stiffness multipliers, mass correction factors Risk: overfitting → validate with other modes not used in updating

Structural Dynamic Modification (SDM)

Adding Mass/Stiffness

Effect on frequency: Δω_r²/ω_r² ≈ -Δm × φ_jr²/m_r + Δk × φ_jr²/(m_r ω_r²) Mass at antinode (max |φ_jr|): maximum frequency reduction Stiffness at node (min |φ_jr|): minimal benefit

Mass-on-spring modification: New eigenvalues from augmented system [K + ΔK - ω²(M + ΔM)]{φ} = {0} Perturbation formula valid for Δm/m_r < 10% and Δk/k_r < 10%

Tuned Mass Damper (TMD)

Secondary mass m_a + spring k_a + damper c_a attached to primary Optimal tuning: ω_TMD = ω_primary/(1 + μ) where μ = m_a/m Optimal damping: ζ_opt = √(3μ/(8(1+μ)³)) Typical mass ratio: μ = 1-5% (bridges, floors, tall buildings)

Common Modal Anomalies

Close Modes

Modes within 10% of each other in frequency Appear as single peak in FRF → need more sensors/averages or MDOF curve fitter Phase quadrature plot helps separate close modes

Nonlinearity

Amplitude-dependent: resonant frequency shifts with excitation level Distorted FRF: asymmetric peaks, jump phenomenon Test: FRF at multiple excitation levels → overlay → check consistency Characterize with Volterra series or Hilbert transform method

Harmonic Contamination (OMA)

Rotating machinery: tonal excitation (multiples of RPM) appears as spurious modes Identification: does this frequency follow RPM linearly? → harmonic, not mode Removal: filter or track harmonics in OMA algorithms

Damping Types and Measurement

Viscous (velocity-proportional): c×ẋ → loss factor η = 2ζ at resonance Structural (hysteretic): force proportional to displacement but in phase with velocity → complex stiffness k(1+iη) Coulomb (dry friction): amplitude-dependent; FRF shapes distorted Structural damping most common for metals: η = 0.001-0.01 (steel), 0.001-0.003

Software Tools

Tool	Capability
ME'scope VES	FRF display, curve fitting, ODS animation
LMS Test.Lab (Simcenter)	Full EMA, correlation, SDM
Brüel & Kjær Connect	Measurement + analysis
pyEMA (Python, open source)	EMA with PolyMAX
MATLAB	Custom signal processing, FRF computation
MATCONT, NNM toolbox	Nonlinear modal analysis

Output

Provide: natural frequencies [Hz], damping ratios [%], mode shape description (bending/torsion/local), MAC table vs. FEA, frequency error Δf% for each mode, SDM recommendation for frequency shift target.