A rim-weighted flywheel is a heavy ring on light spokes, and its limit is the hoop stress in the ring, density times rim speed squared, plus bending between the spokes. A 500 kg steel wheel with a 450 kg rim at 0.6 m radius and 1,500 rpm stores about 2.1 MJ but loads its rim to 69.7 MPa, only 4.9 times below the weakest mild steel, which is under the safety factor of 8 this wiki recommends, so the speed should come down to about 1,178 rpm and the energy to about 1.3 MJ.
Part of Flywheel Energy Storage
A built-up flywheel with its mass in a rim and a light hub and spokes: proportions, stresses, and the joints that decide whether it holds.
Mass At The Rim Stores The Most Per Kilogram
Rim Mass vs Solid Disc shows why the rim shape stores the most for a bored wheel: the energy per kilogram is a function of ½ × v², and a kilogram at the rim moves at the full tip speed while a kilogram at the hub barely moves. The cost is that the rim is a tension ring with spokes pulling inward on it.
The Proportions The Corpus Gives
The corpus page Flywheel gives these proportions for iron engine wheels: rim width about 1/8 to 1/6 of the rim diameter; radial thickness about equal to the width; 6 spokes up to 3 ft diameter and 8 above; spokes oval or I-section, wide at the hub and narrowing toward the rim. They are from cast iron practice at 100 rpm, not from a stress analysis at storage speeds.
Worked Example: A 500 kg Steel Wheel
Inputs: rim 450 kg (90% of the 500 kg) at a mean radius of 0.6 m; hub and 50 kg of spokes treated as slender rods from 0.10 to 0.55 m; mild steel 7,850 kg/m³, tensile 345 to 483 MPa (corpus); 1,500 rpm, ω = 157.1 rad/s.
- Rim: I = 450 × 0.6² = 162 kg·m².
- Spokes: I = 50 × (0.10² + 0.10×0.55 + 0.55²) ÷ 3 = 50 × 0.3675 ÷ 3 = 6.1 kg·m².
- Total I = 168 kg·m²; E = ½ × 168 × 157.1² = 2,072 kJ, 2.1 MJ.
- Rim speed = 157.1 × 0.6 = 94.2 m/s; thin-ring stress = 7,850 × 94.2² = 69.7 MPa.
- Margin = 345 ÷ 69.7 = 4.9 against the weakest steel, and 483 ÷ 69.7 = 6.9 against the strongest.
Neither margin reaches 8. Slow the wheel to 74 m/s (the 345 ÷ 8 limit from Speed vs Mass Trade-offs): ω = 74 ÷ 0.6 = 123.3 rad/s (1,178 rpm), E = ½ × 168 × 123.3² = 1,277 kJ (1.3 MJ) = 355 Wh.
Real Rims Are Thick And Carry Extra Stress
The ring formula assumes a thin rim. With the corpus proportions a 1.2 m wheel’s rim is 0.15 m thick on the radius, 25% of the mean radius. For a thick rotating annulus the standard result (quoted from memory of the elasticity texts, not derived here) puts the peak hoop stress on the inner surface at ρ × ω² ÷ 4 × [(3 + ν) × b² + (1 − ν) × a²], with outer radius b = 0.675 m, inner radius a = 0.525 m and ν = 0.3:
- (3.3 × 0.4556 + 0.7 × 0.2756) ÷ 4 = 0.424 against r² = 0.36 for the thin ring.
- Stress = 7,850 × 157.1² × 0.424 = 82 MPa, 18% above the 69.7 MPa thin-ring figure.
That cuts the weakest-steel margin at 1,500 rpm from 4.9 to 4.2. On top of this come the bending of the rim between spokes, the stress concentrated at each spoke root, and the stress in a joint if the rim is made in segments. None of these has a sourced factor in the corpus, so none is added here; they all push the real margin lower, not higher.
Wound, Banded And Timber Rims
- Iron bands on a timber rim. Flywheel Design suggests iron straps around a wooden rim to add mass. The straps carry the hoop load only if they are tight and joined; a lapped or riveted strap joint is a weak point and the corpus gives no strength for it. Keep such a wheel at the speed of a water wheel and do not use it for storage.
- Shrunk steel hoops. A hoop heated and shrunk over the rim is held in pre-tension and rides without gaps; this is a standard way to resist the spin stress, but it needs measured interference and this page cannot source it. Treat it as a way to hold the rim together, not as extra speed margin.
- Segmented rims. Each joint carries less than the base metal. A segmented rim is limited by its weakest joint; with no sourced joint factor, run it as slowly as a cast iron wheel at the 25 to 30 m/s the corpus gives for engine flywheels.
Spokes And Hub
Spokes hold the rim against its own outward pull. Each spoke carries its own centrifugal load as well as the rim’s pull toward the hub, and the hub bore has to hold the shaft while the whole wheel grows slightly with speed. Keep spokes flat in section, round all re-entrant corners and make the spoke roots wide. Test fit the hub warm, as in Cast Iron & Steel Disc.
Common Mistakes
- Sizing for the thin-ring stress alone. A thick rim and the spoke roots both exceed it.
- Using a safety factor of 4 on a built-up rim. The corpus gives 4 to 8 for machine parts; on a wheel built from joined pieces, take 8.
- Adding rim mass without checking the spokes. Heavier rim, same spokes, higher spoke stress.
Safety
A rim failure throws large pieces in the plane of the wheel
At 94.2 m/s a rim fragment of one-eighth of the wheel’s mass (about 56 kg of rim) carries ½ × 56 × 94.2² ≈ 250 kJ. Run every rim-weighted wheel under the pit and cover described in Housing, and keep people out of the wheel’s plane at every speed.
What’s Next
- Learn next: Balancing — a rim-and-spoke wheel is easy to build off-centre
- Combine with: Burst Energy Calculation — what the pieces carry when it lets go