Flywheel - Wikiparency

# Flywheel A **flywheel** is a rotating mass used as a store of kinetic energy. Because the energy held in a spinning body, `E = ½Iω²`, depends on its [[Moment_of_inertia|moment of inertia]] and on the square of its [[Angular_velocity|angular velocity]], a wheel turning at working speed holds a reservoir that the machine around it can draw on and refill many times a second without the speed changing much. The flywheel is therefore the standard answer to any machine whose torque arrives in pulses — a single-cylinder engine that fires once every two revolutions, a press that does all its work in the last centimetre of a stroke — and the reason such machines run smoothly at all. Two numbers govern the design, and they pull in opposite directions. The speed fluctuation a flywheel leaves behind falls as `1/(Iω²)`, so inertia and speed both buy smoothness, and speed buys it faster. But the [[Cylinder_stress|hoop stress]] in the rim also grows with the square of the rim's speed, and it is stress, not energy, that sets the limit a flywheel can never be taken past. A flywheel design is the settlement between those two facts, made in the choice of a shape, a radius and a material. The framework microsim *The flywheel: inertia smooths a pulsing engine* runs that settlement as an experiment. A one-cylinder engine delivers its torque as one pulse per turn against a steady load; changing the wheel's mass, its radius or its shape narrows or widens the speed band the trace sweeps out each revolution, while the readout gives the stored energy, the coefficient of speed fluctuation and the burst speed the rim's own tension implies. ## History The flywheel is older than any theory of it. A [[Potter's_wheel|potter's wheel]] with a heavy lower disc keeps turning between the potter's pushes, and a [[Spinning_wheel|spinning wheel]], a spindle whorl and a hand quern all work because a rotating mass carries motion through the gaps in an intermittent human effort. Histories of medieval technology credit the earliest clear written description of a flywheel to the treatise *De diversis artibus* of Theophilus Presbyter, written about 1122, where a wheel is added to a hand-turned machine to keep it going evenly between strokes.[^theophilus] The flywheel became a defining component during the age of steam. A reciprocating [[Steam_engine|steam engine]] produces no torque at all at the two dead centres of its stroke, where the connecting rod and [[Crankshaft|crank]] are in line; without a flywheel it would stall at every revolution, and with one it turns through the dead point on stored energy. Rotative engines of the later eighteenth century, associated above all with [[James_Watt|James Watt]], pair the flywheel with a centrifugal [[Governor_(device)|governor]]:[^watt] the flywheel smooths the fluctuation within each revolution, and the governor corrects the mean speed from one revolution to the next, a division of labour between inertia and control that every engine has kept since. The Victorian mill flywheel was a large [[Cast_iron|cast iron]] wheel with a heavy rim and spokes, and its size is a reminder of what the material allowed: cast iron is weak in tension, so its rim speed had to stay low, and low rim speed at useful energy meant a very large, very heavy wheel. Bursting was a real and feared failure, the whole rim releasing its energy in an instant. The same arithmetic runs the other way today. Modern high-speed rotors are small and light, and they store more, because the material permits a rim speed the mill engineer could not have contemplated. ## Physics The moment of inertia of a body about its axis can always be written `I = k m r²`, where `m` is the mass, `r` the outer radius and `k` a pure number set by how that mass is distributed. For a uniform disc `k = ½`; for a thin hoop or rim, where every particle sits at the full radius, `k = 1`.[^b080] The microsim's default is a 10 kg steel disc of 0.2 m radius, so `I = ½ × 10 × 0.04 = 0.200 kg·m²`, and at 1,000 rpm — an [[Angular_frequency|angular velocity]] of 104.72 rad/s — it holds `E = ½ × 0.2 × 104.72² = 1.10 kJ`.[^engsim-fly][^b077rot] Both factors are open to the designer, but not equally. Energy grows with the square of speed and only linearly with inertia, so doubling the running speed stores four times as much where doubling the mass stores twice. Within the inertia term, radius is the cheap variable, because `I` goes as `r²`: the same 10 kg formed as a rim rather than a disc doubles `I` to 0.4 kg·m². This is why flywheels are rims on spokes and not solid slabs. Rotation about an axis that misses the centre of mass adds a term. The [[Parallel_axis_theorem|parallel-axis theorem]] gives `I = I_cm + m d²`, and the energy divides into a spin about the centre of mass and a translation of that centre around the axis. The See-also variant *Rigid body: motion about an offset axis is a spin plus a circling centre* puts the default rim on a shaft 0.15 m off its centre and reads out both parts: 2.19 kJ rotational and 1.23 kJ translational, totalling 3.43 kJ, which is exactly `½(I_cm + m d²)ω²` at the same 104.72 rad/s.[^engsim-rigid] The [[Rigid_body|rigid body]] case is the general one; the centred flywheel is its tidy special case. What the wheel does with that energy follows from `I dω/dt = T(θ) − T_load`: speed changes only as fast as the net [[Torque|torque]] can change it, so a large `I` turns a torque pulse's surplus into a small change of speed.[^b080] The same inertia resists every wanted change of speed, and the stored [[Angular_momentum|angular momentum]] resists tilting of the axis — welcome in a spacecraft wheel, a nuisance in a machine that must start and stop quickly. *Try: step through the ten shapes with mass and radius held fixed and read `k` off the table — the hoop at `k = 1` and the solid sphere at `k = 2/5` differ by two and a half times in the same envelope; then wind the offset `d` away from zero and watch the parallel-axis term `m d²` add to `I` as the body starts to circle the pivot instead of spinning on it.* ## Design A flywheel is sized by how much speed variation the machine can tolerate, not by how much energy it holds. The measure is the coefficient of speed fluctuation, `Cs = (ω₂ − ω₁)/ω̄`, the peak-to-peak swing within one cycle as a fraction of the mean, and the energy that must be exchanged to produce it is `ΔE = Cs I ω̄²`.[^shigley16] Rearranged, `Cs = ΔE/(I ω̄²)`: the swing `ΔE` is a property of the engine or process, while the inertia and the running speed belong to the designer — and because speed enters squared, putting the flywheel on the fastest shaft available is worth more than making it heavier. Practice sets the tolerable coefficient by what the machine feeds: a few thousandths where output frequency or surface finish depends on it, several percent in [[Machine_press|presses]] and crushers, which use the dip as the working energy.[^shigley16] The microsim drives the wheel with a one-cylinder [[Internal_combustion_engine|engine]]'s torque shape, `T = T_mean(1 + A sin θ)` with a 20 N·m mean, against a steady load. That pulse is ILLUSTRATIVE, and labelled so on the sim's notes sheet and in its sources note: a smooth stand-in for a real indicator diagram, chosen so the surplus area is exactly `ΔE = 2 T_mean A`.[^engsim-fly] At full amplitude that surplus is 40 J, and with the default disc's `I ω̄² = 2,193 J` the coefficient is 0.018 — the 991 to 1,009 rpm band shaded on the trace.[^engsim-fly] Swap the disc for a rim of the same mass and radius and `I` doubles, so the band halves; take the wheel to 2,000 rpm and the coefficient falls by four. Two further numbers fall out of the same balance. The time a flywheel carries a machine after its drive is cut is `t = Iω/T` for a constant resisting torque, 1.05 s for the default wheel against its 20 N·m load — short, because a smoothing flywheel is sized to bridge milliseconds, not minutes.[^engsim-fly] And the fluctuation is periodic, not transient: the sim starts each revolution at `ω₁ = ω̄(1 − Cs/2)` so the trace is stationary and the readout is the steady-state coefficient, integrating the torque balance with a fourth-order Runge–Kutta step. The animation is slowed to about one revolution every three seconds so the motion can be watched; the differential equation and every readout are unchanged by that choice.[^engsim-fly] *Try: hold the mass at 10 kg and push the radius from 0.1 to 0.3 m — the inertia rises ninefold, the speed band closes from a wide sweep to a flat line and the `Cs` readout falls with it; then switch the shape from disc to rim and watch the band halve again at the same mass and radius.* ## Materials The limit on a flywheel is tensile. A spinning rim must supply its own centripetal force, and in a thin ring of density `ρ` at rim speed `v` the tension needed produces a hoop stress of exactly `σ = ρv²`. Setting that equal to the working strength gives the tip speed a rim can never pass: `v = √(σ/ρ)`. Radius and mass have vanished: a large rim and a small one of the same material burst at the same *speed*, not the same rotational rate, which is why rim speed rather than rpm is quoted on a flywheel drawing. For the microsim's steel at a 400 MPa working strength and 7,850 kg/m³ the limit is 226 m/s — on a 0.2 m radius, 1,128 rad/s or about 10,800 rpm.[^engsim-fly] A solid disc behaves differently, because its highest stress is at the centre, where `σ_max = (3 + ν)ρv²/8` for [[Poisson's_ratio|Poisson's ratio]] `ν`. With `ν = 0.3` the factor `8/(3 + ν)` is 2.42, so the same steel disc reaches its peak stress only at `v = √(2.42σ/ρ) = 351 m/s`, 1,757 rad/s or 16,800 rpm at the same radius — the figure the sim prints for its default.[^engsim-fly] The disc tolerates the higher tip speed, the rim holds more inertia per kilogram, and the choice is made against whichever failure matters. One honesty note belongs with those numbers. The framework's `energy.storage.flywheelBurstSpeed(σ, ρ, K)` returns `√(Kσ/ρ)`, the square root of the specific energy `E/m = Kσ/ρ` rather than the tip speed its comment claims — low by `√2` for a rim and by a factor of 2 for a disc. The sim's hooks correct it by `√(2/k)` with `k = I/(mr²)`, and the speeds above are the hoop-stress forms, not the library's raw return.[^engsim-fly] The strength-and-density table the sim shows is likewise headed ILLUSTRATIVE: typical values for material classes, not certified data for an alloy or a layup.[^engsim-fly] Written as energy, the same result explains material choice. A flywheel's energy per unit mass is `E/m = ½kv²`, and the limiting `v²` is itself a multiple of `σ/ρ`, so at the limit `E/m = Kσ/ρ`: it depends on the material's [[Specific_strength|specific strength]] and on nothing else about it.[^storage] Strength alone is not the point: a dense strong metal and a light weak one can tie. That is the argument for filament-wound [[Carbon_fiber_reinforced_polymer|carbon fibre]] and glass-fibre [[Composite_material|composite]] rotors, strong along the fibre at a fraction of steel's density; a rim with, say, four times the strength and a fifth of the density would reach `√20 ≈ 4.5` times the steel tip speed above. Composite rotors also fail by shedding fibre rather than throwing fragments, which changes what a containment housing must take. Against all of it stands [[Fatigue_(material)|fatigue]]: a rotor that is charged and discharged is cyclically loaded, so the working stress in every formula here is a fatigue allowable well below the [[Ultimate_tensile_strength|ultimate tensile strength]], and it is the cycle count, not the burst speed, that retires it. ## Applications The oldest application is still the commonest: smoothing a reciprocating engine. A single-cylinder [[Four-stroke_engine|four-stroke engine]] produces one power stroke every two revolutions and needs a large flywheel; multi-cylinder engines overlap their pulses, so `ΔE` falls and the wheel shrinks. In a car the flywheel doubles as the [[Clutch|clutch]] friction face and the starter ring gear, and the [[Dual-mass_flywheel|dual-mass flywheel]] divides it into two masses coupled by springs so that the torsional resonance of the driveline falls below idle speed instead of sitting in the driving range. Machines that work in short, violent strokes use the flywheel the other way round, as a battery rather than a smoother. A [[Punch_press|punch press]] draws its punching energy from the wheel and lets the speed fall; because energy goes as the square of speed, allowing the default 1.10 kJ wheel to drop from 1,000 to 900 rpm releases `1 − 0.9² = 19` percent of its store, about 208 J, without stopping the machine.[^engsim-fly] A small motor with a large flywheel therefore does work no small motor could do directly, and the motor's rating is set by the average power, the flywheel's by the peak. Used purely as storage, the flywheel becomes the machine rather than a part of it. [[Flywheel_energy_storage|Flywheel energy storage]] systems spin a composite rotor in a vacuum on [[Magnetic_bearing|magnetic bearings]] to cut windage and friction, and deliver power through a motor-generator: the applications that suit them are those needing many cycles and high power for short times, such as ride-through in an [[Uninterruptible_power_supply|uninterruptible power supply]] and frequency regulation on a grid, rather than bulk [[Energy_storage|energy storage]] over hours. The [[Gyrobus|gyrobus]] of the 1950s carried a large steel flywheel charged from overhead posts at each stop, and [[Kinetic_energy_recovery_system|kinetic energy recovery systems]] built for motorsport around 2009 recovered braking energy into a small high-speed rotor, an approach related to electrical [[Regenerative_braking|regenerative braking]] but storing the energy mechanically.[^gyrobus] A last family uses neither the energy nor the smoothing but the [[Angular_momentum|angular momentum]] itself. A spacecraft's [[Reaction_wheel|reaction wheel]] is a flywheel that is deliberately accelerated so that the vehicle turns the other way, trading momentum with the wheel to point an instrument without spending propellant, and the same principle stabilises ship and camera gyros. In all these cases the equations of the sections above are unchanged; only the quantity being harvested — energy, steadiness or momentum — is different. ## See also - [[Moment_of_inertia]] - [[Rigid_body]] - [[Flywheel_energy_storage]] - [[Dual-mass_flywheel]] - [[Accumulator_(energy)]] - [[List_of_moments_of_inertia]] - [[Uninterruptible_power_supply]] - [[Clutch]] - [[Gear]] (section 9) - [[Four-stroke_engine]] (section 19) - [[Torsion_(mechanics)]] (section 6) ## References [^b080]: Idema, Timon. *Mechanics and Relativity*. Delft: TU Delft Open. Rotational motion, pp. 64–79: Table 5.1 of moments of inertia (disc `½mR²`, hoop `mR²`), the parallel-axis theorem, torque and angular acceleration, and rotational kinetic energy. Portal Book 080. (Edition year not pinned for this article.) [^b077rot]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. Fixed-axis rotation: moment of inertia and rotational kinetic energy, pp. 481–488, with the revolutions-per-minute to radians-per-second conversion at p. 487. https://openstax.org/details/books/university-physics-volume-1 (Portal Book 077). [^storage]: The framework's `energy.storage` flywheel functions — `flywheelEnergy`, `flywheelInertiaDisc`, `flywheelInertiaRim`, `flywheelBurstSpeed`, `flywheelBurstOmega` — cite Portal Book 115 (Yan, *Introduction to Engineering Thermodynamics*) and Portal Book 097 (Murphy) for the specific-energy form `E/m = Kσ/ρ`. The portal's source table records those pages as not yet pinned, so no page number is given here. [^shigley16]: Shigley's *Mechanical Engineering Design*, chapter 16, "Clutches, Brakes, Couplings and Flywheels": the coefficient of speed fluctuation, the energy exchange `E₂ − E₁ = Cs I ω̄²`, and the range of coefficients used for different classes of driven machine. Cited as a named textbook form; edition and page numbers are not pinned for this article, and the run's build report records the sim's energy balance as agreeing with this form. [^engsim-fly]: Engineering portal microsim spec `specs/sims/Flywheel.json` and its build report (job E-E, `reports_engrun_2026-09-18.md`, September 18, 2026). Inertia by `energy.storage.flywheelInertiaDisc` and `flywheelInertiaRim` (other shapes by `mech.rotation.inertia` with `parallelAxis`), energy by `flywheelEnergy`, coast time by `flywheelStopTime`, the torque balance integrated by `ode.integrators.rk4`. Hand check at the defaults (disc, 10 kg, r = 0.2 m, A = 1, 1,000 rpm, steel): `I` = 0.200 kg·m², ω = 104.72 rad/s, `E` = 1.10 kJ, `ΔE` = 40 J, `Cs` = 0.018 with a band of 991–1,009 rpm, coast time 1.05 s, and a disc burst speed of 351 m/s = 16,800 rpm. ILLUSTRATIVE, as labelled on the sim's sheets and in its sources note: the one-cylinder pulse shape `T = 20 N·m (1 + A sin θ)` and its 20 N·m mean; the strength-and-density table ("ILLUSTRATIVE typical: strength, density, burst speed"); and the drawing's clamped thickness and fixed cone and plate proportions. Recorded library gap: `flywheelBurstSpeed(σ, ρ, K)` returns `√(Kσ/ρ)`, the square root of the specific energy rather than the tip speed its comment claims — low by `√2` for a rim and by 2 for a disc — and `flywheelBurstOmega` inherits the same factor; the sim's hooks multiply by `√(2/k)` with `k = I/(mr²)` to recover the hoop-stress tip speeds `√(σ/ρ)` for a rim and `√(2.42σ/ρ)` for a disc, and this article quotes the corrected forms. [^engsim-rigid]: See-also variant `Rigid_body` of the same spec (job E-E, September 18, 2026): the default rim on a shaft 0.15 m off its centre, with readouts of 2.19 kJ rotational and 1.23 kJ translational kinetic energy for a total of 3.43 kJ. [^watt]: The pairing of a flywheel with a centrifugal governor on the rotative steam engines of the later eighteenth century is standard in histories of the steam engine and is associated above all with James Watt's rotative engines. No single source is pinned for this article; a history of the steam engine or the Watt patents would settle the detail. [^gyrobus]: The gyrobus, a flywheel-driven bus recharged from posts at its stops, ran in the 1950s; flywheel kinetic energy recovery systems were developed for motorsport around 2009. Both dates are as recalled and were not re-checked for this article; manufacturer and regulator records would settle them. [^theophilus]: Theophilus Presbyter (c. 1122). *De diversis artibus* ("On divers arts"). The attribution of the earliest clear description of a flywheel to this treatise is standard in histories of medieval technology; the date, the passage and the modern critical edition were not re-checked for this article, and a scholarly edition of the text would settle all three.  **Microsim — three.js (Wikitube framework):** *Flywheel* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Flywheel.html" data-title="Flywheel"></div> *Built from `MICROSIM_GUIDE/specs/sims/Flywheel.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Flywheel) : [Wikitube](https://en.wikitube.io/wiki/Flywheel) - skeleton pinned to revision 1369224288 (2026-09-18).