Mechanical Energy Storage, Composite Rotordynamics & UPS Grid Frequency Regulation
Flywheel Kinetic Energy Storage & Rim Hoop-Stress Formula
Calculate High-Speed Vacuum Flywheel Usable Extractable Kinetic Energy (kWh) Across [RPM_min, RPM_max], Outer-Rim Linear Tip Speed (m/s), Thin-Rim Centrifugal Hoop Stress σ_θ = ρ*v² (MPa), and Composite Burst Safety Factor.
See why carbon-fiber rotors beat heavy steel in flywheel energy storage—σ_θ = ρ * v_tip²!
At first glance, engineers assume a flywheel rotor with high mass density ρ stores more kinetic energy E_kin = 0.5 * I * ω²! In reality, maximum rotational speed is governed by Centrifugal Thin-Rim Hoop Stress σ_θ = ρ * v_tip² (in Pa, or (ρ * v_tip²) / 10^6 in MPa, where v_tip = ω * R = (2 * π * RPM / 60) * R)! Because hoop stress scales linearly with density ρ, light high-tensile carbon-epoxy composite (ρ ≈ 1,600 kg/m³, σ_ult >= 2,400 MPa) spins to supersonic rim speeds (~785 m/s inside a vacuum containment), storing ~7.5x more specific energy per kilogram than heavy steel! Discharging from RPM_max down to RPM_min = 0.5 * RPM_max extracts 75% of total kinetic energy!
Verify on a Data-Center UPS Composite Flywheel—Rotor_Inertia_I = 12.0 kg·m² (B2), Max_Speed = 30,000 RPM (B3), Min_Cutoff_Speed = 15,000 RPM (B4 -> Usable Kinetic Energy = 12.34 kWh out of 16.45 kWh total in E2), Outer_Rim_Radius = 0.25 m (B5 -> Rim Tip Speed = 785.40 m/s in E3), Carbon_Epoxy_Density = 1,600 kg/m³ (B6 -> Centrifugal Hoop Stress = (1600 * 785.398^2) / 10^6 = 986.96 MPa in E4), and Composite_Tensile_Strength = 2,400 MPa (B7): Burst Safety Factor is 2400 / 986.96 = 2.43x (E5)!
| Composite Flywheel Metric (I=12 kg·m², 30k->15k RPM, R=0.25 m, ρ=1600, σ_u=2400 MPa) | Formula | Verified Output |
|---|---|---|
| Usable Kinetic Energy (kWh) | 0.5 * I * (ω_max^2 - ω_min^2) / 3.6e6 | 12.34 kWh |
| Outer Rim Tip Speed (m/s) | ((2 * PI() * RPM_max) / 60) * R_m | 785.40 m/s (Mach 2.3 in Vacuum) |
| Centrifugal Hoop Stress (MPa) | (ρ_kg_m3 * v_tip^2) / 1,000,000 | 986.96 MPa (4,842.27 MPa Steel!) |
| Tensile Burst Safety Factor | Tensile_Strength_MPa / Hoop_Stress_MPa | 2.43x (0.21x Burst on Steel!) |
Flywheel Usable Kinetic Energy & Hoop-Stress Safety Factor Formula
Validate Inertia_I > 0, 0 <= RPM_min < RPM_max, Radius > 0, Density > 0, and Strength > 0!
=LET(in_r, B2, rp_x, B3, rp_n, B4, rd_m, B5, rh_o, B6, st_u, B7, IF(OR(in_r <= 0, rp_x <= 0, rp_n < 0, rp_n >= rp_x, rd_m <= 0, rh_o <= 0, st_u <= 0), "Invalid Flywheel Rotor Input", LET(om_x, (2 * PI() * rp_x) / 60, vt_p, om_x * rd_m, sg_h, (rh_o * (vt_p ^ 2)) / 1000000, ROUND(st_u / sg_h, 2))))Alloy-Steel Rotor Burst Check (Set Density = 7,850 kg/m³, Yield = 1,000 MPa)
Substitute alloy steel (B6 = 7850 kg/m³, B7 = 1000 MPa) at the same 30,000 RPM: Centrifugal Hoop Stress E4 explodes nearly 5x to 4,842.27 MPa, collapsing Safety Factor E5 to 0.21 (< 1.00 catastrophic burst)!
=IF(ISTEXT(E5), E5, ROUND((0.5 * B2 * ((((2 * PI() * B3) / 60) ^ 2) - (((2 * PI() * B4) / 60) ^ 2))) / 3600000, 2))Try the example
Use the original values or change them to test your understanding.