MATERIALS FOR FLYWHEELS

Flywheels store energy. Small ones—the sort found in children’s toys—are made of lead. Old steam engines have flywheels; they are made of cast iron. Cars have them too (though you cannot see them) to smooth power-transmission. More recently flywheels have been proposed for power storage and regenerative braking systems for vehicles; a few have been built, some of high strength steel, some of composites. Lead, cast iron, steel, composites—there is a strange diversity here.

An efficient flywheel stores as much energy per unit weight as possible. As the flywheel is spun up, increasing its angular velocity, ω, it stores more energy. The limit is set by failure caused by centrifugal loading: if the centrifugal stress exceeds the tensile strength (or fatigue strength), the flywheel flies apart. One constraint, clearly, is that this should not occur. The flywheel of a child’s toy is not efficient in this sense. Its velocity is limited by the pulling-power of the child, and never remotely approaches the burst velocity. In this case, and for the flywheel of an automobile engine—we wish to maximize the energy stored per unit volume at a constant (specified) angular velocity. There is also a constraint on the outer radius, R, of the flywheel so that it will fit into a confined space. The answer therefore depends on the application. The strategy for optimizing
flywheels for efficient energy-storing systems differs from that for children’s toys. The two alternative sets of design requirements are listed in Table 1(a) and (b).

The model

An efficient flywheel of the first type stores as much energy per unit weight as possible, without failing. Think of it as a solid disk of radius R and thickness t, rotating with angular velocity ω (Figure 1). The energy U stored in the flywheel is

Here J= (π/2)ρR⁴t is the polar moment of inertia of the disk and ρ the density of the material of which it is made, giving

  
Table 1 Design requirements for maximum-energy flywheel and fixed velocity
(a) For maximum-energy flywheel

Function	Flywheel for energy storage
Constraints	-  Outer radius, R, fixed -  Must not burst -  Adequate toughness to give crack-tolerance
Objective	Maximize kinetic energy per unit mas
Free variables	Choice of material

(b) For fixed velocity

Function	Flywheel for child’s toy
Constraints	Outer radius, R, fixed
Objective	Maximize kinetic energy per unit volume at fixed angular velocity
Free variables	Choice of material

Figure 1 A flywheel. The maximum kinetic energy it can store is limited by its strength.

The mass of the disk is

m = πR⁴tρ

The quantity to be maximized is the kinetic energy per unit mass, which is the ratio of the last two equations:

U/m = 0.25 R²ω²

As the flywheel is spun up, the energy stored in it increases, but so does the centrifugal stress. The maximum principal stress in a spinning disk of uniform thickness is

where v is Poisson’s ratio (v ≈ 1/3). This stress must not exceed the failure stress σf (with an appropriate factor of safety, here omitted). This sets an upper limit to the angular velocity, ω, and disk radius, R (the free variables). Eliminating Rω between the last two equations gives

U/m = ½ (σ_f / ρ )

The best materials for high-performance flywheels are those with high values of the material index

M = σ_f / ρ

It has units of kJ/kg.
And now the other sort of flywheel—that of the child’s toy. Here we seek the material that stores the most energy per unit volume V at constant velocity, ω. The energy per unit volume at a given ω is

U/V = ¼ ρR²ω²

Both R and ω are fixed by the design, so the best material is now that with the greatest value of

M₂ =  ρ

The selection

Figure 2 shows the strength—density chart. Values of M1 correspond to a grid of lines of slope 1. One such is plotted as a diagonal line at the value M1 = 200 kJ/kg. Candidate materials with high values of M1 lie in the search region towards the top left. The best choices are unexpected ones: composites, particularly CFRP, high strength titanium alloys and some ceramics, but these are ruled out by their low toughness. But what of the lead flywheels of children’s toys? There could hardly be two more different materials than CFRP and lead: the one, strong and light, 

Figure 2. Materials for flywheels. Composites are the best choices. Lead and cast iron, traditional for flywheels, are good when performance is limited by rotational velocity, not strength.

the other, soft and heavy. Why lead? It is because, in the child’s toy, the constraint is different. Even a super-child cannot spin the flywheel of his toy up to its burst velocity. The angular velocity ω is limited instead by the drive mechanism (pull-string, friction drive). Then as we have seen, the best material is that with the largest density. The second selection line on Figure 2 shows the index M2 at the value 10 Mg/m3. We seek materials in Search Area 2 to the right of this line. Lead is good. Cast iron is less good, but cheaper. Gold, platinum, and uranium (not shown on the chart) are better, but may be thought unsuitable for other reasons.

Table 2 Energy density of power sources

Source	Energy density (kJ/kg)	Comment
Gasoline	20,000	Oxidation of hydrocarbon—mass of oxygen not included
Rocket fuel	5000	Less than hydrocarbons because oxidizing agent forms part of fuel
Flywheels	Up to 400	Attractive, but not yet proven
Lithium-ion battery	Up to 350	Attractive but expensive, and with limited life
Nickel-cadmium battery	170–200
Lead-acid battery	50–80	Large weight for acceptable range
Springs rubber bands	Up to 5	Much less efficient method of energy storage than flywheel