shell-vibration

name	shell-vibration
description	Thin shell vibration analysis — cylindrical shell frequencies (Flügge/Donnell equations, circumferential wave number n, axial wave number m), ring frequency, breathing mode, beam mode, Rayleigh-Ritz method, fluid-filled shell (added mass), shell-fluid interaction (acoustic coupling), Love and Donnell shell theories, free vibration of spherical shells, and applications (pressure vessels, pipelines, aircraft fuselages, submarine hulls).
metadata	{"priority":7,"promptSignals":{"phrases":["shell vibration","cylindrical shell frequency","ring frequency","shell natural frequency","shell modes","vibration of cylinders"],"minScore":3}}

Thin Shell Vibration Analysis — Complete Skill

Shell Theory Fundamentals

Coordinate System and Assumptions

Cylindrical shell coordinates: x = axial; θ = circumferential; r = radial; R = mean radius; t = thickness; L = length Displacements: u(x,θ,t) axial; v(x,θ,t) circumferential; w(x,θ,t) radial (positive outward)

Love-Kirchhoff assumptions:

• Shell thickness small vs. R and wavelength: t/R << 1, t/λ << 1
• Normals remain normal after deformation (Kirchhoff plate analog)
• Stresses across thickness small vs. in-plane stresses

Shell theories (accuracy order): Love (1888): strain tensor based; accurate for thin shells (t/R < 1/20) Donnell (1933): simplified; drops some curvature terms; good for high n (n > 4); simplest math Flügge (1934): more complete than Love; recommended for n < 4 (beam modes, ring modes) Timoshenko: includes transverse shear and rotary inertia; for thicker shells (t/R > 1/20)

Free Vibration Frequencies — Cylindrical Shell

Donnell-Mushtari Equations (Simplified)

Modal form — mode (m,n): u = A_mn × cos(mπx/L) × cos(nθ) × e^(iωt) v = B_mn × sin(mπx/L) × sin(nθ) × e^(iωt) w = C_mn × sin(mπx/L) × cos(nθ) × e^(iωt) [m = number of half-waves axially; n = number of full waves circumferentially]

Substituting into equations of motion yields 3×3 eigenvalue problem: [K_mn - ω² M_mn] × {A, B, C} = 0 Three natural frequencies per (m,n) pair: two in-plane (high freq) + one bending (low freq)

Simplified bending frequency (dominant, lowest for each (m,n)): ω²_(m,n) = (E/(ρ R²)) × [(1-ν²)/1] × [λ²(m,n) + n² - 1]² / [λ²(m,n) + n²]² + [κ/12] × [λ²(m,n) + n²]² [λ = mπR/L; κ = t²/R²; ρ = density]

Useful approximation (Leissa 1973, for thin shells): Ω² = (1-ν²) × λ_m⁴/(λ²_m + n²)⁴ + κ(λ²_m + n²)² + n²(n²-1)²/(λ²_m+n²)² [approximate; Ω = ωR√(ρ(1-ν²)/E)]

Special Modes

Ring Frequency (m=0, n=0 Breathing Mode)

Ring frequency: axisymmetric radial breathing of entire cylinder f_ring = (1/(2πR)) × √(E/(ρ(1-ν²))) [Hz; all cross-sections expand/contract uniformly] For steel R = 0.5 m: f_ring = 1/(2π×0.5) × √(200×10⁹/(7,850×0.91)) = 0.318 × 5,276 = 1,678 Hz Below ring frequency: structural (mass-controlled); above ring frequency: acoustic (radiation controlled)

Significance: Pipe vibration analysis: resonances above ring frequency attenuated more rapidly Acoustic radiation: efficient radiation for modes above ring frequency (bending wavelength > ring circumference)

Beam Mode (n=1)

n=1 mode: entire cylinder displaces as rigid beam (translational + bending) Lowest frequency for most shells; controlled by axial stiffness and support conditions f_(1,1) from Euler-Bernoulli beam formula: f = (β_i L)²/(2πL²) × √(EI/(ρA)) [but shell bending stiffness I = π R t³/4 + π R³ t for thin shell; A = 2πRt] For simply supported: β₁L = π; f_bend = (π/2L²) × √(EI_shell/ρA_shell) / (2π)

Ovaling Mode (n=2)

Lowest shell (not beam) mode: n=2: shell cross-section deforms into oval shape; called lobar or ovaling mode Critical for piping elbows (Bourdon effect); also for seismic slug flow

n=2 frequency (simplified for long shell L >> R): f_(m,2) ≈ (1/2π) × (t/R²) × √(E/(3ρ(1-ν²))) [approximate for n=2 ring oscillation] For steel pipe R=0.5m, t=10mm: f ≈ (1/2π)×(0.01/0.25)×√(200e9/(3×7850×0.91)) = 0.0127Hz → very low for thin shells

Numerical Values — Natural Frequencies

Thin steel cylindrical shell example: R = 0.3 m; t = 5 mm; L = 2.0 m; E = 200 GPa; ν = 0.3; ρ = 7,850 kg/m³ κ = (t/R)² = (0.005/0.3)² = 2.78×10⁻⁴

Ring frequency: f_ring = (1/(2π×0.3)) × √(200e9/(7850×0.91)) = 0.531 × 5,310 = 2,820 Hz

Beam mode (n=1, SS): governed by Euler beam formula with I_shell = πR³t f_beam = (π/2)²/(2π×L²) × √(E×πR³t/(ρ×2πRt)) = (π/(4L²)) × √(ER²/(2ρ)) = (π/16)×√(200e9×0.09/(2×7850)) = 138 Hz

Ovaling (n=2, m=1): from Rayleigh quotient f_2 ≈ 3t/(2π√3 × R²) × √(E/(ρ(1-ν²))) = 3×0.005/(2π×1.732×0.09) × 5,310 = 0.0833 × 5,310 / (0.976) ≈ 450 Hz (this formula is for ring only)

Min shell frequency (typically n=3 to 5 for moderate R/t): Minimize over n: dω/dn = 0 → n_min = (R/t)^(1/2) / (12(1-ν²))^(1/4) ≈ (R/t)^(1/2) / 1.86 [approximate] For R/t = 60: n_min ≈ √60/1.86 ≈ 4.2 → n=4 lowest shell mode

Fluid-Filled Shell — Added Mass

Added mass effect (incompressible fluid inside): Effective density: ρ_eff = ρ_shell + ρ_fluid × (J_n(k_n R))² × R / (t × J'_n(k_n R) × k_n R) Simplified (non-radiating mode): ρ_added/ρ_fluid ≈ 1/n [per unit circumferential wave for n>>1]

Frequency reduction factor: f_filled/f_empty = 1/√(1 + ρ_fluid × R / (ρ_shell × t × n)) [approximate; n = circumferential wave number] For n=2 steel pipe water-filled (ρ_fluid/ρ_shell = 1/8, R/t = 40): f_filled/f_empty = 1/√(1 + 1000×0.3/(7850×0.0075×2)) = 1/√(1+2.55) = 1/1.88 = 0.53 (53% of in-vacuo)

Shell-Fluid Acoustic Coupling

Breathing mode radiates sound efficiently: Radiation impedance Z_rad = ρ_fluid × c_fluid × (ka)² / (1 + (ka)²) [for n=0; a = R] k = ω/c_fluid; ka < 1: radiation efficiency poor; ka > 1: efficient radiator

Structural-acoustic coupling: Coupled system eigenvalues differ from uncoupled; use FEM with acoustic elements (ABAQUS, NASTRAN) For light coupling (ρ_fluid × R << ρ_shell × t): perturbation correction adequate

Experimental Modal Analysis of Shells

Roving impact or shaker test: Identify (m,n) mode shapes; fit with SDOF or MDOF modal model Ring of accelerometers at one axial station to identify n; axial array to identify m

Typical damping for steel shells: Air: ζ = 0.001–0.005 (0.1–0.5%); water: ζ = 0.005–0.02 (added fluid damping) Welded: ζ = 0.01–0.03; bolted: ζ = 0.02–0.05

Applications

Pressure vessel vibration: Nozzle excitation at resonance → fatigue crack at welds (primary failure mode for vibrating vessels) Design: force vessel frequencies away from pump/compressor operating speeds ± 20% Stiffening rings: raise f_(m,n) above excitation range

Pipeline slug flow: Slugs excite n=2 ovaling and beam modes; check n=2 frequency above slug frequency f_slug = v_slug / L_slug; typically 0.5–5 Hz (very low vs. shell modes for most pipes)

Submarine hulls: n=2 breathing mode critical for underwater sound radiation Design hull ring-stiffener spacing to keep f_2 below sonar operating frequencies

Standards and References

Standard	Scope
Leissa "Vibration of Shells" (NASA SP-288)	Comprehensive frequency equations for shells
Soedel "Vibrations of Shells and Plates"	Textbook with worked examples
ASME STS-1	Steel stacks (includes shell vibration criteria)
API 650	Tanks — seismic sloshing and shell vibration
Junger & Feit "Sound, Structures, and Their Interaction"	Fluid-structure coupling

Output

Provide: shell geometry (R [mm]; t [mm]; L [mm]; material: E [GPa]; ν; ρ [kg/m³]; end conditions: SS/C/F), frequency parameters (κ = (t/R)²; R/t; L/R; ring frequency f_ring [Hz]), natural frequencies (beam mode n=1 [Hz]; lowest shell mode n_min: f_(1,n_min) [Hz]; table of f_(m,n) for m=1,2 and n=1,2,3,4,5), mode shapes (critical (m,n) combinations; Ω = ωR√ρ(1-ν²)/E; dimensionless frequency), fluid-filled correction (if applicable: ρ_fluid; added mass factor; f_filled [Hz]), forced response (excitation frequency [Hz]; distance to nearest resonance; resonance risk assessment), FEM recommendation (ABAQUS or NASTRAN; meshing: 6–8 elements through thickness wave pattern; minimum 20 elements per half-wave), and applicable standard (NASA SP-288 Leissa; Soedel textbook; ASME STS-1 if stack).