Flywheels in Machines

What is a flywheel?

• Flywheels are used in machines to act as an energy reservoir, by storing and supplying energy when required.
• They consist of a heavy metal disc or wheel that can rotate rapidly and so has a large moment of inertia. This means they have a high mass and a large radius.
• Once a flywheel starts spinning, it is difficult to make it stop. The key part is that the flywheel has stored rotational energy, which it can transfer for some time after there is no input.
• An example is a treadle (pedal) sewing machine: a big flywheel is connected to a small wheel by a rope which drives the sewing machine. When the pedal is not pressed, the smaller wheel will still rotate for some time due to the energy stored in the flywheel.

Smoothing torque and speed

• Flywheels are used to smooth out fluctuations in rotational speed, torque, and power (such as in vehicle engines).
• Power in an engine is not produced continuously. It is only produced in the "power stroke" or "combustion" part of an engine cycle, so energy is released in bursts. This causes the engine to produce a torque that fluctuates.
• If the torque is uneven, it will cause a jerky motion and unwanted vibrations. This is a waste of energy and uncomfortable for the passengers.
• The flywheel added will speed up or slow down over a period of time because of its inertia and, as a result, the sharp fluctuations in torque are "smoothed". The greater the moment of inertia of the flywheel, the smaller the fluctuation in speed.

Regenerative braking in vehicles

• In conventional braking, the kinetic energy store of the vehicle is transferred as waste through to the thermal energy store.
• Instead, when regenerative brakesA braking system that engages a flywheel to store the kinetic energy lost during braking. This stored energy is then used to accelerate the vehicle later. Also known as KERS (kinetic energy recovery systems). are applied, a flywheel is engaged and will "charge up" by using the energy lost by braking.
• When the vehicle needs to accelerate later, the energy stored by the flywheel is used to do this. These systems are sometimes called "KERS" (kinetic energy recovery systems).

Production processes

• An electric motor in industrial machines can be used along with a flywheel. The motor charges up the flywheel, which can then transfer short bursts of energy (such as needing to connect two materials together in a riveting machine).
• This prevents the motor from stalling, and a less powerful motor can be used.
• A flywheel transfers just enough power to a wheel to overcome frictional torque as it rotates. When power is needed to the rest of the engine, the flywheel can reduce its speed and transfer some power.

Factors affecting the energy storage capacity

• The mass of the flywheel: since the moment of inertia $I$ is directly proportional to the mass $m$, as mass increases the moment of inertia also increases. The rotational kinetic energy is directly proportional to $I$, so this also increases.
• The angular speed of the flywheel: the rotational kinetic energy is proportional to the square of the angular speed. If the angular speed increases, the rotational kinetic energy stored also increases.
• Friction: although they are very efficient, flywheels can still lose some stored energy as friction and air resistance between the wheels and its bearings. Friction can be reduced by lubricating bearings, using bearings made of superconductors (so the flywheel can levitate and have no contact), or using the flywheel in a vacuum or sealed container to reduce air resistance.

Flywheel shape

• The shape of the flywheel determines how much energy it can store. Crucially, a hoop (wheel)-shaped flywheel ($I = mr^2$) is preferred over a disc-shaped one ($I = \frac{1}{2}mr^2$) because the hoop has a greater moment of inertia for the same mass and radius.
• For a solid disc of radius $R$, thickness $t$, mass $M$ and density $\rho$: $$M = \rho V = \pi R^2 t \rho$$
• The moment of inertia about the axis of rotation for a disc is: $$I = \frac{1}{2}MR^2 = \frac{1}{2}(\pi R^2 t \rho)R^2 = \frac{1}{2}(\pi t \rho)R^4$$
• The rotational kinetic energy is therefore: $$E_k = \frac{1}{2}I\omega^2 = \frac{1}{2}\left(\frac{1}{2}(\pi t \rho)R^4\right)\omega^2$$
• Since $t$ and $\rho$ are constant: $$E_k \propto R^4\omega^2$$
• The rotational kinetic energy stored therefore depends on the moment of inertia, determined by the shape of the flywheel.

Worked example: combined flywheels

• A flywheel of mass $M$ and radius $R$ rotates at a constant angular velocity $\omega$ about an axis through its centre. Its rotational kinetic energy is $E_K$ and its moment of inertia is $I = \frac{1}{2}MR^2$.
• A second flywheel of mass $\frac{1}{2}M$ and radius $\frac{1}{2}R$ is placed on top of the first flywheel. The new angular velocity of the combined flywheels is $\frac{2}{3}\omega$.
• The kinetic energy of the first flywheel alone: $$E_K = \frac{1}{2}I\omega^2 = \frac{1}{2}\left(\frac{1}{2}MR^2\right)\omega^2 = \frac{1}{4}MR^2\omega^2$$
• The combined moment of inertia: $$I_{\text{new}} = I_1 + I_2 = \frac{1}{2}MR^2 + \frac{1}{2}\left(\frac{1}{2}M\right)\left(\frac{1}{2}R\right)^2 = \frac{9}{16}MR^2$$
• The new kinetic energy: $$E_{K\text{ new}} = \frac{1}{2}I_{\text{new}}\omega_{\text{new}}^2 = \frac{1}{2}\left(\frac{9}{16}MR^2\right)\left(\frac{2}{3}\omega\right)^2 = \frac{1}{2}\left(\frac{1}{4}MR^2\omega^2\right) = \frac{1}{2}E_K$$