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 An industrial flywheel. A flywheel mounted at the end of an automobile engine crankshaft. A Landini tractor with exposed flywheel. Sorry, your browser either has JavaScript disabled or does not have any supported player. You can download the clip or download a player to play the clip in your browser. A flywheel with variable moment of inertia, conceived by Leonardo da Vinci.

A flywheel is a rotating mechanical device that is used to store rotational energy. Flywheels have a significant moment of inertia and thus resist changes in rotational speed. The amount of energy stored in a flywheel is proportional to the square of its rotational speed. Energy is transferred to a flywheel by applying torque to it, thereby increasing its rotational speed, and hence its stored energy. Conversely, a flywheel releases stored energy by applying torque to a mechanical load, thereby decreasing its rotational speed.

Three common uses of a flywheel include:

• They provide continuous energy when the energy source is discontinuous. For example, flywheels are used in reciprocating engines because the energy source, torque from the engine, is intermittent.
• They deliver energy at rates beyond the ability of a continuous energy source. This is achieved by collecting energy in the flywheel over time and then releasing the energy quickly, at rates that exceed the abilities of the energy source.
• They control the orientation of a mechanical system. In such applications, the angular momentum of a flywheel is purposely transferred to a load when energy is transferred to or from the flywheel.

Flywheels are typically made of steel and rotate on conventional bearings; these are generally limited to a revolution rate of a few thousand RPM.[1] Some modern flywheels are made of carbon fiber materials and employ magnetic bearings, enabling them to revolve at speeds up to 60,000 RPM.[2]

## Applications

Flywheels are often used to provide continuous energy in systems where the energy source is not continuous. In such cases, the flywheel stores energy when torque is applied by the energy source, and it releases stored energy when the energy source is not applying torque to it. For example, a flywheel is used to maintain constant angular velocity of the crankshaft in a reciprocating engine. In this case, the flywheel—which is mounted on the crankshaft—stores energy when torque is exerted on it by a firing piston, and it releases energy to its mechanical loads when no piston is exerting torque on it. Other examples of this are friction motors, which use flywheel energy to power devices such as toy cars.

A flywheel may also be used to supply intermittent pulses of energy at transfer rates that exceed the abilities of its energy source, or when such pulses would disrupt the energy supply (e.g., public electric network). This is achieved by accumulating stored energy in the flywheel over a period of time, at a rate that is compatible with the energy source, and then releasing that energy at a much higher rate over a relatively short time. For example, flywheels are used in punching machines and riveting machines, where they store energy from the motor and release it during the punching or riveting operation.

The phenomenon of precession has to be considered when using flywheels in vehicles. A rotating flywheel responds to any momentum that tends to change the direction of its axis of rotation by a resulting precession rotation. A vehicle with a vertical-axis flywheel would experience a lateral momentum when passing the top of a hill or the bottom of a valley (roll momentum in response to a pitch change). Two counter-rotating flywheels may be needed to eliminate this effect. This effect is leveraged in reaction wheels, a type of flywheel employed in satellites in which the flywheel is used to orient the satellite's instruments without thruster rockets.

## History

The principle of the flywheel is found in the Neolithic spindle and the potter's wheel.[3]

The Andalusian agronomist Ibn Bassal (fl 1038–1075), in his Kitab al-Filaha, describes the flywheel effect employed in a water wheel machine, the saqiya.[4][unreliable source?]

The flywheel as a general mechanical device for equalizing the speed of rotation is, according to the American medievalist Lynn White, recorded in the De diversibus artibus (On various arts) of the German artisan Theophilus Presbyter (ca. 1070–1125) who records applying the device in several of his machines.[3][5]

In the Industrial Revolution, James Watt contributed to the development of the flywheel in the steam engine, and his contemporary James Pickard used a flywheel combined with a crank to transform reciprocating into rotary motion.

## Physics

A flywheel is a spinning wheel or disc with a fixed axle so that rotation is only about one axis. Energy is stored in the rotor as kinetic energy, or more specifically, rotational energy:

• $E_k=\frac{1}{2} I \omega^2$

Where:

• ω is the angular velocity, and
• $I$ is the moment of inertia of the mass about the center of rotation. The moment of inertia is the measure of resistance to torque applied on a spinning object (i.e. the higher the moment of inertia, the slower it will spin when a given force is applied).
• The moment of inertia for a solid cylinder is $I = \frac{1}{2} mr^2$,
• for a thin-walled empty cylinder is $I = m r^2$,
• and for a thick-walled empty cylinder is $I = \frac{1}{2} m({r_{external}}^2 + {r_{internal}}^2)$,[6]

Where m denotes mass, and r denotes a radius.

When calculating with SI units, the standards would be for mass, kilograms; for radius, meters; and for angular velocity, radians per second. The resulting answer would be in joules.

The amount of energy that can safely be stored in the rotor depends on the point at which the rotor will warp or shatter. The hoop stress on the rotor is a major consideration in the design of a flywheel energy storage system.

• $\sigma_t = \rho r^2 \omega^2 \$

Where:

• $\sigma_t$ is the tensile stress on the rim of the cylinder
• $\rho$ is the density of the cylinder
• $r$ is the radius of the cylinder, and
• $\omega$ is the angular velocity of the cylinder.

This formula can also be simplified using specific tensile strength and tangent velocity:

• $\frac{\sigma_t}{\rho} = v^2$

Where:

• $\frac{\sigma_t}{\rho}$ is the specific tensile strength of the material
• $v$ is the tangent velocity of the rim.

## Table of energy storage traits

Flywheel purpose, type Geometric Shape Factor (k)
(unitless - varies with shape)
Mass
(kg)
Diameter
(cm)
Angular velocity
(rpm)
Energy stored
(MJ)
Energy stored
(kWh)
Small battery 0.5 100 60 20,000 9.8 2.7
Regenerative braking in trains 0.5 3000 50 8,000 33.0 9.1
Electric power backup[7] 0.5 600 50 30,000 92.0 26.0

### High-energy materials

For a given flywheel design, the kinetic energy is proportional to the ratio of the hoop stress to the material density and to the mass:

• $E_k \varpropto \frac{\sigma_t}{\rho}m$

$\frac{\sigma_t}{\rho}$ could be called the specific tensile strength. The flywheel material with the highest specific tensile strength will yield the highest energy storage per unit mass. This is one reason why carbon fiber is a material of interest.

For a given design the stored energy is proportional to the hoop stress and the volume:

• $E_k \varpropto \sigma_tV$

### Rimmed

A rimmed flywheel has a rim, a hub, and spokes.[12] The structure of a rimmed flywheel is complex and, consequently, it may be difficult to compute its exact moment of inertia.[citation needed] A rimmed flywheel can be more easily analysed by applying various simplifications. For example:

• Assume the spokes, shaft and hub have zero moments of inertia, and the flywheel's moment of inertia is from the rim alone.
• The lumped moments of inertia of spokes, hub and shaft may be estimated as a percentage of the flywheel's moment of inertia, with the remainder from the rim, so that $I_{rim}=KI_{flywheel}$

For example, if the moments of inertia of hub, spokes and shaft are deemed negligible, and the rim's thickness is very small compared to its mean radius ($R$), the radius of rotation of the rim is equal to its mean radius and thus:

• $I_{rim}=M_{rim}R^2$

## References

1. ^ [1]; "Flywheels move from steam age technology to Formula 1"; Jon Stewart | 1 July 2012, retrieved 2012-07-03
2. ^ [2], "Breakthrough in Ricardo Kinergy ‘second generation’ high-speed flywheel technology"; Press release date: 22 August 2011. retrieved 2012-07-03
3. ^ a b Lynn White, Jr., "Theophilus Redivivus", Technology and Culture, Vol. 5, No. 2. (Spring, 1964), Review, pp. 224–233 (233)
4. ^ Ahmad Y Hassan. "Flywheel Effect for a Saqiya". History-science-technology.com. Retrieved 2010-11-30.
5. ^ Lynn White, Jr., "Medieval Engineering and the Sociology of Knowledge", The Pacific Historical Review, Vol. 44, No. 1. (Feb., 1975), pp. 1–21 (6)
6. ^ [3] (page 10, accessed 1 Dec 2011, Moment of inertia tutorial
7. ^ http://www.vyconenergy.com/pq/VDCtech.htm
8. ^ "Flywheel Energy Calculator". Botlanta.org. 2004-01-07. Retrieved 2010-11-30.
9. ^ "energy buffers". Home.hccnet.nl. Retrieved 2010-11-30.
10. ^ "Message from the Chair | Department of Physics | University of Prince Edward Island". Upei.ca. Retrieved 2010-11-30.
11. ^ "Density of Steel". Hypertextbook.com. 1998-01-20. Retrieved 2010-11-30.
12. ^ Flywheel Rotor And Containment Technology Development, FY83. Livermore, Calif: Lawrence Livermore National Laboratory , 1983. pp. 1-2
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