rotating-discs

name	rotating-discs
description	Rotating disc and ring stresses — uniform disc, disc with central hole, variable thickness disc, burst speed, thermal stresses in discs, turbine disc design.
metadata	{"priority":7,"promptSignals":{"phrases":["rotating disc","rotating disk","burst speed","disc stress","centrifugal stress","turbine disc"],"minScore":3}}

Rotating Discs — Complete Skill

Solid Disc (No Central Hole)

Stress Distribution

σ_r = (3+ν)/8 × ρ ω² (R² - r²) σ_θ = (3+ν)/8 × ρ ω² R² - (1+3ν)/8 × ρ ω² r²

ρ = density [kg/m³], ω = angular velocity [rad/s], R = outer radius [m] Maximum stress at center (r=0): σ_r = σ_θ = (3+ν)/8 × ρ ω² R² (biaxial, equal in both directions)

Radial displacement: u_r = ρ ω² r (3+ν)(1-ν) R²/(8E) × [1 - (1+ν)/(3+ν) × (r/R)²]

Disc with Central Hole (Radii a to b)

Lamé-Type Solution

σ_r = A - B/r² σ_θ = A + B/r²

For rotating disc: add centrifugal terms: σ_r = A - B/r² - (3+ν)/8 × ρ ω² r² σ_θ = A + B/r² - (1+3ν)/8 × ρ ω² r²

Boundary conditions: σ_r = 0 at r = a (inner free) and r = b (outer free, or applied p) → Solve for A and B

Maximum hoop stress at inner bore: σ_θ,max = (3+ν)/4 × ρ ω² [b² + (1-ν)/(3+ν) × a²]

As a → 0 (hole approaches zero size): σ_θ,max = 2 × σ_θ (solid disc center) — stress doubles by introducing small hole!

Rotating Ring (Thin Ring)

For thin ring (t << r_mean): σ_θ = ρ v² = ρ (ω r_mean)² v = peripheral velocity [m/s] At burst: σ_θ = S_u → v_burst = √(S_u/ρ)

Specific strength index: S_u/ρ [m²/s²]; higher = better for high-speed rotation Carbon fiber composite: ~3×10⁶ m²/s² vs. steel: ~0.07×10⁶ m²/s²

Burst Speed

Von Mises Yield at Bore (First Yield)

ω_y: σ_θ,max = S_y at bore → first yielding ω_burst ≈ 1.2-1.5 × ω_y for typical discs (plastic redistribution before burst)

Empirical Burst Margin (Turbine Discs): σ_avg = T/A [average tangential stress = T_tangential per unit area across disc] σ_avg,burst ≈ S_u (average stress approach) Safety factor against burst: SF_burst = ω_burst/ω_operating ≥ 1.5-2.0 (per AS9102B, EASA)

Variable Thickness Disc

Conical or hyperbolic disc: t(r) = t_0 × (r/r_0)^n n < 0 (thick hub, thin rim): reduces stress concentration at bore Hyperbolic: n chosen to give constant stress in disc (Donath's equal-stress disc) For equal σ_r = σ_θ = σ_const: numerical solution required (ODE in t(r))

Thermal Stresses in Discs

T = T(r) radial temperature distribution σ_r,thermal = αE/(1-ν) × [1/b²∫_a^b T r dr - 1/r²∫_a^r T r dr] σ_θ,thermal = αE/(1-ν) × [-T + 1/b²∫_a^b T r dr + 1/r²∫_a^r T r dr]

For uniform ΔT: σ_r = σ_θ = -αE ΔT/(1-ν) (hydrostatic, uniform change causes no stress in flat disc — stress arises only from temperature gradient)

Output

Provide: σ_r and σ_θ vs. radius [MPa] at key points, maximum stress location, yield speed ω_y [rpm], burst speed ω_burst [rpm], safety factors, radial displacement u_r at bore and rim [mm], temperature-stress interaction.