Flywheel Energy Calculator

Formula

• Inertia: I = k_I · m · r². Annular disk: I = ½ m (ri² + ro²) → k_I = ½(1 + (ri/ro)²).
• Common k_I: solid disk 0.5 · thin ring / rim 1.0 · solid sphere 0.4.
• Stored energy: ω = 2πN/60; E = ½ I ω²; tip speed = ω · r.
• Rim hoop-stress screen: σ ≈ ρ ω² r²; SF = σ_allow / σ.
• Fluctuation: ΔE = ½ I (ω1² − ω2²); Cs = |N1 − N2| / N_mean.
• Material limit: e = k_e · σ / ρ; E_max = e · m. k_e ≠ inertia k_I (flat-disk k_e ≈ 0.606 = 2/(3+ν)).

How it works

Three modes: stored energy E = ½ I ω² from inertia shape k_I, mass, and RPM; machine-design speed fluctuation ΔE and Cs between N1 and N2; and material-limited specific energy e = k_e·σ/ρ. Inertia k_I is not the same as stress mass-efficiency k_e. Optional rim hoop stress uses σ = ρ ω² r².

Pick a Quick preset or Mode. Storage needs geometry, mass, outer radius, and RPM; annular disks also need inner radius. Fluctuation needs N1 and N2. Material mode needs mass-efficiency k_e and a material (or Custom ρ/σ). Optional material on Modes 0/1 screens rim hoop stress — Custom ρ/σ fields appear only when you pick Custom. Results stay dashed until CALCULATE. Play/Pause controls the spinning sketch. RESET restores the solid-disk 3000 rpm storage defaults.

Stored energy scales with ω²

Default CALCULATE path: I = 2.25 kg·m², E ≈ 111 kJ at 3000 rpm.

Leave Material on None if you only need KE; turn Steel/Al/Composite on when tip speed starts to look aggressive.

Speed fluctuation for machine design

Use Mode 1 when the job is sizing I for a known cycle energy and an allowed speed band.

Full shaft-plus-disk inertia from density and thickness stays out of scope — enter an equivalent k_I or measured I via custom k_I.

k_I is not k_e

Worked example

Default storage: solid disk k_I = 0.5, m = 50 kg, r = 0.3 m, N = 3000 rpm. Reproduce on CALCULATE.

1. Leave Quick preset on Storage · solid disk 50 kg · 3000 rpm (or enter the same values).
2. CALCULATE → I = 2.25 kg·m²; ω ≈ 314.16 rad/s; E ≈ 111 033 J; tip ≈ 94.25 m/s.
3. Fluctuation example: preset Fluctuation · 300 → 280 rpm → ΔE ≈ 143 J, Cs ≈ 0.069.
4. Annular check: shape Annular, ri = 0.15 m, same m/r/N → k_I = 0.625, I = 2.8125 kg·m².
5. Material example: Mode Material · steel · k_e 0.5 → e ≈ 25 478 J/kg ≈ 7.08 Wh/kg; E_max ≈ 1.27 MJ at 50 kg.

Result: Default storage ≈ 111 kJ. Legacy fluctuation ΔE ≈ 143 J. Material steel screen ≈ 7.08 Wh/kg.

When to use

• Estimating stored flywheel energy from mass, radius, shape, and RPM
• Sizing inertia to limit speed variation on presses and engines
• Comparing solid disk vs rim vs annular inertia for the same mass
• Screening rim hoop stress and tip speed before a detailed rotor study
• First-pass material-limited specific energy (Wh/kg) from σ and ρ

Limitations

• Inertia k_I and stress mass-efficiency k_e are different constants — do not swap them.
• Hoop stress uses a rim / thin-ring screen σ = ρ ω² r²; real disks have radial and tangential stress fields.
• Does not model bearing, windage, or vacuum enclosure losses.
• Does not size burst containment or composite layups.
• Does not build I from density × thickness × (ro⁴ − ri⁴) shaft-disk geometry.
• Assumes rigid-body rotation about a fixed central axis.

FAQ

What is the difference between k_I and k_e?

k_I sets inertia: I = k_I·m·r² (0.5 solid disk, 1.0 thin ring). k_e sets stress-limited specific energy: e = k_e·σ/ρ (≈ 0.606 for a flat disk with ν ≈ 0.3). This pad keeps them in separate modes.

What is the speed fluctuation coefficient?

Cs = |N1 − N2| / N_mean. It compares speed swing to mean speed (always ≥ 0). Lower Cs means smoother operation for the same mean RPM. ΔE stays signed so you can see which speed is higher.

Solid disk vs thin ring?

A uniform solid disk uses k_I = 0.5. A thin ring with mass at the rim uses k_I = 1.0, so it stores twice the energy for the same mass, outer radius, and RPM — until tip-speed / hoop-stress limits intervene.

When should I use Material mode?

When you care about the best-case Wh/kg a material can reach at its tensile (or yield) limit, not about a specific RPM and geometry inertia. Use Mode 0 when you already have m, r, and N.

Does this replace a full FESS or rotor design?

No. It is a screening and teaching pad. High-speed composite flywheels, magnetic bearings, vacuum housings, and containment need specialist design and FEA.