| 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
- Determine ΔE [J] from torque-angle diagram or load cycle
- Select C_s from application table
- Compute I_required = ΔE / (C_s × ω²)
- Select geometry (rim or disk): I = f(geometry)
- Check σ_hoop or σ_r < S_y/n (n ≥ 2)
- 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].
