Moment of inertia
-Energy swing at Cs
-Peak hoop stress
-Required inertia for dE
-All results
| Mass | - |
| Mean angular speed | - |
| Kinetic energy at mean speed | - |
| Maximum and minimum speed | - |
| Rim speed at maximum speed | - |
| Allowable stress (strength / safety factor) | - |
| Stress / allowable | - |
| Maximum safe speed for this stress limit | - |
| Inertia provided / required | - |
Method and formulas
Standard rotating-disc theory for a uniform-thickness disc or ring, so every figure can be checked by hand.
| Quantity | Formula |
|---|---|
| Mass | m = ρ · π/4 (Do2 − Di2) t |
| Moment of inertia | I = m (Do2 + Di2) / 8 |
| Angular speed | ω = 2π n / 60 |
| Kinetic energy | E = ½ I ω2 |
| Speed limits | ωmax = ω(1 + Cs/2), ωmin = ω(1 − Cs/2) |
| Energy swing | ΔE = ½ I (ωmax2 − ωmin2) = Cs I ω2 |
| Required inertia | Ireq = ΔE / (Cs ω2) |
| Peak hoop stress, bored disc (at bore) | σ = (3 + ν)/4 · ρω2Ro2 + (1 − ν)/4 · ρω2Ri2 |
| Peak stress, solid disc (at centre) | σ = (3 + ν)/8 · ρω2Ro2 |
| Allowable stress | strength / safety factor (yield for steel and aluminum, ultimate for cast iron) |
Material values are commonly published figures: carbon steel ρ 7850 kg/m³, ν 0.29, yield 250 MPa; alloy steel 7850, 0.29, 655 MPa; gray cast iron 7200, 0.26, 214 MPa ultimate; aluminum 6061-T6 2700, 0.33, 276 MPa. Use your supplier's certified values for a real design. Stress is evaluated at the maximum speed and ignores keyways, stress raisers and temperature.
Worked example: carbon steel disc, 500 mm outside, 100 mm bore, 60 mm thick, 1200 rpm mean, Cs 0.05
m = 7850 × π/4 (0.52 − 0.12) × 0.06 = 88.8 kg. I = 88.8 × (0.52 + 0.12) / 8 = 2.885 kg·m². ω = 125.7 rad/s, so E = 22782 J and the usable swing ΔE = 0.05 × I ω2 = 2278 J. To absorb 1500 J per cycle at the same Cs the inertia must be at least 1.900 kg·m². At ωmax the rim speed is 32.2 m/s and the peak bore stress is 6.8 MPa against an allowable of 83.3 MPa (250 MPa / 3).
Questions
How do you calculate flywheel moment of inertia?
For a solid or bored disc of uniform thickness, I = m (Do^2 + Di^2) / 8, where m is the mass, Do the outside diameter and Di the bore diameter. The mass is m = density x pi/4 (Do^2 - Di^2) x thickness. A solid disc (Di = 0) gives I = m Do^2 / 8.
How much energy does a flywheel store?
Kinetic energy is E = 1/2 I w^2 with w in rad/s (w = 2 pi n / 60 for n in rpm). Only the part released while the speed drops from its maximum to its minimum is usable: dE = I w^2 Cs for a coefficient of fluctuation Cs.
What is the coefficient of speed fluctuation?
Cs = (w_max - w_min) / w_mean. Typical targets are about 0.01 to 0.03 for generators and machine tools, 0.05 for general machinery and up to 0.10-0.20 for punch presses and crushers, where the flywheel mainly carries a peak load.
How do I size a flywheel for a given energy swing?
Work out the energy to absorb or release per cycle, dE, from the torque-versus-angle curve, choose Cs and the mean speed, then I = dE / (Cs w^2). Enter dE in the calculator to get the required inertia and compare it with the geometry you chose.
What limits flywheel speed?
Centrifugal hoop stress. For a thin rim it is rho v^2 with v the rim speed. For a disc the peak stress occurs at the bore: (3 + nu)/4 x rho w^2 Ro^2 + (1 - nu)/4 x rho w^2 Ri^2. It rises with the square of speed, so a 10 % overspeed adds 21 % stress.
What safety factor should a flywheel use?
Flywheel failures are violent, so use a generous factor on yield (ductile steel) or on ultimate strength (cast iron). Many designs use 3 or more, plus an overspeed margin; always follow the applicable code, guarding rules and a qualified engineer's review.
What does this calculator not cover?
It models a uniform-thickness disc or ring at constant temperature. It does not cover spoked wheels, stepped hubs, keyways, shaft fits, bearing loads, fatigue from start-stop cycles, fracture of cast iron, vibration or containment. Verify critical designs.
Engineering aid for guidance. It models a uniform disc or ring only and does not cover spokes, stepped hubs, keyways, fatigue, brittle fracture, vibration or containment. Rotating machinery can fail violently: verify critical designs.