plate-vibration

name	plate-vibration
description	Plate vibration — thin plate equation (biharmonic), natural frequency of rectangular and circular plates (Leissa), Kirchhoff plate theory, boundary conditions (SSSS/CCCC/CFFF), mode shapes, vibration of stiffened plates, parametric excitation, experimental modal analysis of plates, Chladni patterns, acoustic radiation from vibrating plates (radiation efficiency), and application in aerospace panels/automotive body panels/MEMs.
metadata	{"priority":7,"promptSignals":{"phrases":["plate vibration","vibration of plates","plate natural frequency","Kirchhoff plate","Chladni pattern","panel flutter"],"minScore":3}}

Plate Vibration — Complete Skill

Governing Equation

Kirchhoff (Thin) Plate Theory

Equation of motion (transverse vibration): D × ∇⁴w + ρ_s × h × ẅ = q(x,y,t) [D = plate flexural rigidity; w = transverse displacement; ρ_s = density; h = thickness; q = distributed load]

Biharmonic operator: ∇⁴w = ∂⁴w/∂x⁴ + 2 × ∂⁴w/∂x²∂y² + ∂⁴w/∂y⁴

Flexural rigidity: D = E × h³ / (12(1-ν²)) [N·m; E = Young's modulus; h = thickness; ν = Poisson's ratio]

Free vibration (q = 0): D × ∇⁴w = ρ_s × h × ω² × w [ω = natural frequency] w(x,y,t) = W(x,y) × e^(iωt) [modal separation] D × ∇⁴W = ρ_s × h × ω² × W [eigenvalue problem]

Thin plate validity: h/a ≤ 0.1 (h = thickness; a = in-plane dimension); for h/a > 0.1 use Mindlin (thick plate; shear deformation + rotary inertia)

Frequency Parameter λ

Non-dimensional frequency parameter: λ_mn = ω_mn × √(ρ_s × h / D) × a² [a = reference dimension; tabulated by boundary condition and mode]

Natural frequency: ω_mn = λ_mn × √(D / (ρ_s × h)) / a² [rad/s] f_mn = ω_mn / (2π) [Hz]

General plate: combine x and y contributions: For simply supported (SS) on all edges: λ_mn² = π² × (m² + n² × a²/b²) [exact; m,n = mode numbers] ω_mn = π² × √(D/(ρ_s × h)) × (m²/a² + n²/b²) [exact for SSSS]

Natural Frequencies for Standard Cases

Rectangular Plate (SSSS — Simply Supported All Edges)

Exact solution (Navier): ω_mn = π² × √(D/(ρ_s × h × a²)) × (m² + n² × (a/b)²) [a, b = plate dimensions; m, n = mode indices] f_mn = ω_mn / (2π)

First mode (m=n=1, square plate a = b): f₁₁ = π²/(2π × a²) × √(D/ρ_s h) = π/(2a²) × √(D/ρ_s h)

Steel plate example (a = 0.5 m, b = 0.7 m, h = 3 mm): E = 200 GPa; ν = 0.30; ρ_s = 7,850 kg/m³ D = 200×10⁹ × (0.003)³ / (12 × (1-0.09)) = 4.945 N·m f₁₁ = π²/(2π) × √(4.945 / (7850 × 0.003)) × (1/0.5² + 1/0.7²) = 57 Hz (approximate)

Mode shape: W_mn(x,y) = sin(mπx/a) × sin(nπy/b) [SSSS; exactly satisfies zero deflection and zero moment at all boundaries]

Rectangular Plate — Other Boundary Conditions

Leissa (1969) NASA SP-160 tables: definitive reference for all BCs Boundary condition notation: S = simply supported; C = clamped; F = free

CCCC (clamped all edges): λ²₁₁ = 35.99 (square plate); f₁₁ ≈ 2.28 × f₁₁_SSSS [clamped has higher frequency; more constrained] λ²_mn from Leissa Table 4.21 for square plate; interpolate for a/b ≠ 1

CFFF (cantilever plate — clamped one edge, three free): λ²₁₁ = 3.47 (square plate); f₁₁ ≈ 0.59 × f₁₁_SSSS [lower; free edges reduce stiffness] First bending mode like cantilever beam + transverse curvature

CSCS (simply supported on short edges; clamped on long edges): Intermediate between SSSS and CCCC

SFSF (simply supported on two parallel edges; free on others): One-dimensional bending (like a beam if a >> b); mode frequencies approach beam formula

Circular Plate

Circular plate (radius R; clamped edge, CCCC): f_mn = λ_mn² × √(D / (ρ_s × h × R⁴)) / (2π) λ₀₁ (clamped) = 10.22; λ₀₂ = 39.77; λ₁₁ = 21.26; λ₁₂ = 60.82

Clamped center, free edge (opposite; annular plate): f depends on inner/outer radius ratio; Leissa Section 2 tables

Mode shapes (circular): W_mn(r,θ) = [A × J_m(k_mn r) + B × Y_m(k_mn r)] × cos(mθ) [m = circumferential mode; n = radial node] k_mn from boundary condition; Bessel functions J_m, Y_m

Stiffened Plate Vibration

Equivalent Orthotropic Plate

For plate with unidirectional ribs (ribs in x-direction only): Treat as orthotropic plate with D_x >> D_y (ribs stiffen x-bending much more than y) D_x ≈ (E × I_rib) / s [I_rib = moment of inertia of rib + plate strip; s = rib spacing] D_y ≈ D [plate only; no rib stiffness in y-direction]

Frequency from orthotropic plate formula: ω²_mn = (π/a)⁴ × [D_x × m⁴ + 2D_t × m²n²(a/b)² + D_y × n⁴(a/b)⁴] / (ρ_s × h) (SSSS orthotropic; D_t = D_1 + 2D_xy = effective twisting stiffness)

Smeared Stiffness Method

Applicability: many closely-spaced ribs (s/a < 0.1); otherwise use discrete stiffener FEA Smearing: distribute rib stiffness uniformly across plate → equivalent uniform orthotropic plate Accuracy: good for average frequency; poor for localized mode shapes between stiffeners

Experimental Modal Analysis

Chladni Patterns

Historical visualization: Spread fine sand on plate; vibrate at resonance frequency → sand migrates to nodal lines (zero motion) Modern: scanning laser Doppler vibrometry (LDV) → full-field mode shapes without contact Chladni patterns: visually striking; used for education and quick mode identification

Experimental Methods

Impact hammer test (plate FRF measurement): Excite plate with instrumented hammer at grid of points Measure response with fixed accelerometer (or roving hammer with fixed accelerometer) Compute FRF H(ω) = X(ω)/F(ω) at each location → assemble FRF matrix Modal curve fitting: identify ω_n, damping ζ_n, mode shape φ_n from FRF peaks

Scanning LDV: Non-contact; measure velocity at grid of points (automated optical scan) Full-field velocity maps at each frequency → FRF and mode shapes without mass loading Spatial resolution: 0.1–1 mm; frequency range: DC to > 100 kHz

SIMO vs. MIMO: SIMO: multiple response locations, single excitation point; standard MIMO: multiple inputs, multiple outputs; better for complex mode separation (closely-spaced modes)

Damping in Plates

Sources: Material damping (η = 0.001–0.01 for metals; 0.05–0.1 for composites with viscoelastic layers) Air radiation damping (acoustic coupling; significant for thin plates at low frequency) Joint damping (bolted connections; most practical source of damping)

Typical measured ζ_n: Bare aluminum plate in air: ζ_n = 0.1–0.5% Steel with welded stiffeners: ζ_n = 0.2–1% Sandwich panel (CLD): ζ_n = 2–10% (effective damping treatment)

Acoustic Radiation from Vibrating Plates

Radiation Efficiency

Radiation efficiency σ: W_radiated = σ × ρ_air × c_air × A_plate × <v²>_space_time σ = 1 for rigid piston (no bending); σ < 1 for bending plate below coincidence; σ >> 1 at coincidence

Critical (coincidence) frequency: f_c = c_air² / (1.8 × h × c_L) [c_L = √(E/(ρ_s(1-ν²))) = longitudinal wave speed in plate] Below f_c: σ << 1 (plate radiates inefficiently — subcritical) Above f_c: σ → 1 (plate radiates efficiently — supercritical)

Example (3 mm steel plate): c_L = √(200×10⁹/(7850×0.91)) = 5,300 m/s; f_c = 343² / (1.8 × 0.003 × 5300) = 7,500 Hz

Sound Power Level

Sound power from vibrating plate: W [W] = σ × ρ_air × c_air × A_plate × <v²> [<v²> = mean-square velocity of plate surface] SWL = 10 × log₁₀(W / 10⁻¹²) [dB re 10⁻¹² W] For structural noise control: reduce <v²> (damping, stiffening) or reduce σ (decoupling plate from radiating surface)

Panel Flutter (Aeronautical)

Supersonic Panel Flutter

Flutter criterion (von Kármán plates in supersonic flow): Dimensionless dynamic pressure: λ = q_∞ × a³ / D [q_∞ = ½ρ_∞U²; a = panel length; D = flexural rigidity] Flutter onset: λ_cr ≈ 500 (simply supported panel; aerodynamic pressure alone); depends on M, panel mass Damping effect: reduces λ_cr; in-plane compression further reduces it

Thermal post-buckling (hypersonic): High-speed panels heat up → thermal compression → buckling → coupled flutter-buckling (complex) NASA-SP-8002: panel flutter design guide

Standards and References

Source	Scope
Leissa (1969) NASA SP-160	Definitive plate vibration frequency tables
Leissa (1993) Vibration of Shells	Curved plates and shells
Ewins "Modal Testing"	Experimental modal analysis of plates
NASA-SP-8002	Panel flutter design
ISO 7626	Mechanical vibration — frequency response measurement
ANSI S2.31	Mechanical impedance and mobility

Output

Provide: plate geometry (a × b × h [m and mm]; boundary conditions e.g. SSSS/CCCC/CFFF), material (E [GPa]; ν; ρ [kg/m³]), flexural rigidity D [N·m], natural frequencies f_mn for m,n = 1,2,3 [Hz] (table), mode shape description (number of nodal lines in x and y), method used (Navier exact / Ritz approximation / FEA), orthotropic correction (if stiffened: D_x, D_y, D_t; revised frequencies), damping ratio ζ_n [%] (measured or estimated), coincidence frequency f_c [Hz] and acoustic radiation efficiency σ at operating frequency, experimental modal test recommendation (hammer/LDV; grid spacing for required mode resolution), panel flutter check if aeronautical (λ vs. λ_cr), and applicable reference (Leissa 1969 NASA SP-160, Ewins Modal Testing).