flywheel

name	flywheel
description	Flywheel design — energy storage, coefficient of fluctuation, rim flywheel, solid disk, stress at speed, material limits, Watt governor, energy density.
metadata	{"priority":5,"promptSignals":{"phrases":["flywheel","energy storage","coefficient of fluctuation","moment of inertia flywheel","flywheel design","rotary energy"],"minScore":4}}

Flywheel Design — Complete Skill

Purpose

Flywheel stores rotational kinetic energy to:

• Smooth cyclic speed variations (punch presses, engines)
• Provide peak power (hybrid vehicles, UPS)
• Store energy (grid scale, renewable integration)

Energy Storage

Kinetic energy: E = ½ I ω²

For flywheel with mass moment of inertia I: E = ½ I ω²_avg [J]

Energy released over speed variation Δω: ΔE = I ω_avg × Δω = I × (ω_max² - ω_min²)/2

Coefficient of fluctuation of speed: C_s = (ω_max - ω_min)/ω_avg = Δω/ω_avg

Required moment of inertia: I = ΔE / (C_s × ω_avg²)

Typical C_s values: Punch press: 0.03-0.20 (large variation acceptable) Machine tool: 0.005-0.020 Electric generator: 0.001-0.010 (strict frequency control) Diesel engine: 0.002-0.030

Geometry and I

Solid Disk (radius R, thickness t, density ρ)

m = ρ × π × R² × t I = ½ m R² = ρπtR⁴/2

Rim Flywheel (rim mass M at mean radius R, thin)

I ≈ M × R² [all mass at radius R] m_rim = 2π × R × A_cross × ρ (A = cross-section area of rim) Spoke inertia ≈ 10-15% of rim; I_total = I_rim × 1.1 (correction)

Annular Disk (inner radius r_i, outer radius r_o)

I = ½ m (r_i² + r_o²) where m = ρ π (r_o² - r_i²) t

Stress Analysis (Rotating Disk)

Rotating Disk (flat, uniform, σ_z = 0)

Radial stress: σ_r = ((3+ν)/8) × ρω²(r_o² + r_i² - r_i²r_o²/r² - r²) Hoop stress: σ_θ = ((3+ν)/8) × ρω²(r_o² + r_i² + r_i²r_o²/r² - (1+3ν)/(3+ν)×r²)

Solid disk (r_i = 0): σ_r = σ_θ = (3+ν)/8 × ρω² × (r_o² - r²) [at center: max and equal] σ_max = (3+ν)/8 × ρω² × r_o² [at center, r=0]

Rim/annular disk (r_i → r_o): σ_hoop,max ≈ ρω²R² (rim flywheel under centrifugal tension) Same as a spinning ring: σ = ρv² = ρ(ωR)²

Critical/burst speed: σ = S_y → v_max = √(S_y/ρ) [peripheral velocity for stress = yield]

Material Performance Index

Energy density per mass: E/m = ½ × I/m × ω² = ¼ r_o² × ω² (solid disk) Maximize energy density: need high S_y/ρ (specific strength)

Flywheel merit index: S_y/ρ [J/kg]

Material	S_y [MPa]	ρ [kg/m³]	S_y/ρ [kJ/kg]	Max v_tip [m/s]
AISI 4340	1100	7850	140	375
Ti-6Al-4V	880	4430	199	446
CFRP (unidirectional)	1500	1600	938	969
E-Glass/epoxy	600	2000	300	548
Maraging 300 steel	2000	8100	247	497

CFRP flywheels: E/m ≈ 100-150 Wh/kg (best commercially available) Steel flywheels: E/m ≈ 5-30 Wh/kg

Design Process

1. Determine ΔE [J] from torque-angle diagram or load cycle
2. Select C_s from application table
3. Compute I_required = ΔE / (C_s × ω²)
4. Select geometry (rim or disk): I = f(geometry)
5. Check σ_hoop or σ_r < S_y/n (n ≥ 2)
6. Check n_burst = (v_max/v_operating)^2 ≥ 1.5 for containment

Energy Diagram from Torque-Angle Data

For cyclic machine: plot T_driving - T_resisting vs. crank angle θ ΔE = ∫T_net dθ between speed max and min If torque data discrete: ΔE = Σ (area under positive excess, subtract negative)

Watt Governor (Speed Control)

Conical pendulum governor: fly balls on pivoted arms Equilibrium: mg = m×ω²×r × tan(θ) ... actually: Height h = g/ω² Speed: ω = √(g/h) → n_rpm = 60/(2π) × √(g/h) [h in meters]

Sensitivity: Δn/n for a given speed change Hunting (oscillation): too sensitive + load change → instability Porter governor (weighted sleeve): reduces hunting tendency

Output

Provide: I [kg·m²], ΔE [J], flywheel mass [kg], OD [mm], σ_hoop [MPa], safety factor n, peripheral speed v_tip [m/s].