At the same mass, radius and speed, a thin rim stores exactly twice the energy of a solid disc, because I = m × r² against ½ × m × r². The comparison changes once strength limits the speed: a bored disc can store about 0.30 times the ratio of its strength to its density in joules per kilogram, a thin ring 0.50, so the rim wins by about 1.65 times rather than 2, and an unbored disc would edge ahead of the ring with 0.61.
Part of Flywheel Energy Storage
Two ways to arrange the same mass on a shaft, and what each stores both at equal speed and at the equal stress that actually limits real wheels.
At Equal Speed The Rim Doubles The Energy
From 2 × I × omega², a thin ring has I = m × r² and a solid disc I = ½ × m × r². With the same m, r and ω the ring stores twice as much. Example: 100 kg, 0.25 m radius, 1,500 rpm (ω = 157.1 rad/s):
- Ring: ½ × (100 × 0.0625) × 157.1² = 77.1 kJ
- Disc: ½ × (½ × 100 × 0.0625) × 157.1² = 38.6 kJ
This is the reason pottery wheels, bicycle wheels and engine flywheels put their mass at the edge. It is the right comparison when speed is set by the machine, as with a steam engine at 120 rpm.
The Comparison That Matters For Storage Is At Equal Stress
For a store, the speed is not given; it is pushed up until the rim approaches what the material can bear. The spin stress in each shape follows from the tip speed v = ω × r and the density ρ:
| Shape | Peak tensile stress | Energy per kg at that stress |
|---|---|---|
| Thin ring | ρ × v² | ½ × v² = 0.50 × σ ÷ ρ |
| Solid disc, no hole | (3 + ν) ÷ 8 × ρ × v² = 0.41 × ρ × v² | ¼ × v² = 0.61 × σ ÷ ρ |
| Disc with a small central bore | (3 + ν) ÷ 4 × ρ × v² = 0.83 × ρ × v² | ¼ × v² = 0.30 × σ ÷ ρ |
Poisson’s ratio ν is taken as 0.3. The ring result is derived in Burst Energy Calculation. The two disc factors are the standard elasticity results for a uniform rotating disc, quoted from textbook memory and not derived here.
Reading the table:
- A solid, hole-free disc would store 0.61 ÷ 0.50 = 1.21 times what a ring stores before reaching the same stress. Its centre carries the peak stress, and it spreads the load through all its material.
- Boring the shaft hole doubles the peak stress at the bore. The same disc then stores 0.30 ÷ 0.50 = 0.61 of what the ring does, so the ring is ahead by 0.50 ÷ 0.30 = 1.65 times.
- Every practical flywheel is on a shaft, and a keyway or a bore raises the stress further still.
So the folk rule “twice as much” holds at equal speed and is about 1.65 at equal stress for a bored disc. What does not change is the conclusion: put the mass at the rim, and keep the hub light.
Worked Example: Stress In The Two Shapes
Using the cast iron case above, density 7,200 kg/m³ (the corpus value) and tip speed 0.25 × 157.1 = 39.3 m/s:
- Ring: 7,200 × 39.3² = 11.1 MPa
- Solid disc, centre: 0.4125 × 11.1 = 4.6 MPa
- Bored disc, at the bore: 0.825 × 11.1 = 9.2 MPa
Against the corpus cast iron tensile range of 15,000 to 30,000 psi (103 to 207 MPa; see Tensile Strength) the smallest margin here is 103 ÷ 11.1 = 9 times, which is why a small cast iron disc at 1,500 rpm is comfortable and a half-metre-radius wheel at the same rpm is not: its tip speed is 78.5 m/s, its ring stress 7,200 × 78.5² = 44 MPa, and the margin against 103 MPa falls to 2.3.
Spokes, Rims And What They Add
A real rim-and-spoke wheel is neither shape. The rim carries the hoop stress ρ × v² and the spokes pull on it, bend it between spoke ends and carry their own centrifugal load. Arms and joints raise local stress above the ring figure. The ring value is the floor for a rim wheel, not the whole story; treat it as the minimum stress and apply the safety factors in Safety Factors.
Common Mistakes
- Quoting “twice as much energy” for a storage wheel. It holds at equal speed. At equal stress, on a bored disc, it is about 1.65.
- Thickening the hub. Mass near the axis adds weight and little energy: a kilogram at half the radius counts a quarter as much.
- Treating the disc factors as measured. They assume uniform thickness, uniform material and no flaws. A casting has flaws.
Safety
Rim speed, not diameter, sets the stress
A bigger wheel at the same tip speed has exactly the same rim stress. A 1 m radius ring at 39.3 m/s turns at 375 rpm and sees 11.1 MPa, the same as the 0.25 m wheel at 1,500 rpm. Anyone who “scales up” a design and keeps the rpm instead of the tip speed multiplies the stress by the square of the scale factor.
What’s Next
- Learn next: Speed vs Mass Trade-offs — how much steel or iron a kilowatt-hour of storage costs
- Combine with: Rim-Weighted Design — building the rim-and-spoke version