Flywheel Design – Energy, Inertia and Rotating Stress Calculations

A flywheel is a rotating mass used to store kinetic energy and control variations in the flow of energy through a mechanical system. Flywheels are commonly used to smooth cyclic power delivery, supply short bursts of energy and limit fluctuations in rotational speed.

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This reference covers the basic energy and inertia relationships used in flywheel calculations together with the stresses produced in rotating discs and rings.

Introduction to Flywheels

When a mass is accelerated from rest to a rotational speed it acquires kinetic energy. Reducing its rotational speed releases some of this stored energy. A rotating mass designed specifically to store and exchange energy in this way is known as a flywheel.

Flywheels can be used to balance the energy flow between a relatively steady power source and a load whose energy demand varies during the operating cycle. Conversely, they can smooth the output from a machine or engine whose torque delivery is inherently cyclic.

Traditional flywheels are generally comparatively massive discs or rims rotating at moderate speed. Flywheel energy storage systems (FES) may instead use relatively lightweight rotors operating at very high rotational speeds. Such systems require consideration of rotor strength, bearing losses, aerodynamic losses, containment and rotor dynamics.

Energy Stored in a Flywheel

The rotational kinetic energy stored in a flywheel is:

E = \frac{1}{2} I ω^{2}

where:

E = stored kinetic energy (J)
I = mass moment of inertia about the axis of rotation (kg·m²)
ω = angular velocity (rad/s)

The moment of inertia may also be expressed in terms of the mass M and radius of gyration k:

I = M k^{2}

Flywheel geometry and rotational energy diagram

Typical Flywheel Applications

Punching, shearing and press machinery where a relatively large amount of energy is required during a short part of the operating cycle.
Internal-combustion and reciprocating steam engines where the flywheel reduces cyclic variations in rotational speed.
Mechanical energy-recovery systems used in road and rail transport.
Short-duration energy storage and uninterruptible power systems.
Power-quality and energy-balancing applications.

For conventional machinery in which the flywheel provides short-term energy balancing, the design calculations can often be based on energy variation, required speed fluctuation and mass moment of inertia.

High-speed flywheel energy-storage systems require additional engineering considerations including:

bearing friction and bearing technology;
aerodynamic losses, often reduced by operating the rotor in a vacuum;
rotor material strength at high peripheral speeds;
rotor dynamics and critical speeds;
containment in the event of rotor failure; and
energy conversion and control systems.

Nomenclature

Symbol	Description	SI Unit
t	Time	s
I	Mass moment of inertia about the axis of rotation	kg·m²
M	Mass	kg
k	Radius of gyration	m
E	Stored kinetic energy	J
r₁	Inside radius of flywheel ring or annular disc	m
r₂	Outside radius of flywheel ring or annular disc	m
T	Torque	N·m
θ	Angular displacement	rad
ω₁	Initial angular velocity	rad/s
ω₂	Final angular velocity	rad/s
n	Rotational speed	rev/min
α	Angular acceleration	rad/s²
ν	Poisson's ratio	–
ρ	Material density	kg/m³
σ_r	Radial stress	Pa
σ_t	Tangential or hoop stress	Pa

Basic Flywheel Equations

The following relationships are useful in determining flywheel energy, rotational speed, torque and acceleration.

Basic flywheel rotational energy equations

E = \frac{1}{2} I ω^{2}

ω = \frac{2 π n}{60}

I = M k^{2}

Flywheel torque, work and angular acceleration equations

T = I α

For constant torque acting through an angular displacement:

W = T θ

Flywheel angular motion and energy equations

For constant angular acceleration:

ω_{2} = ω_{1} + α t

ω_{2}^{2} = ω_{1}^{2} + 2 α θ

Flywheel Inertia for a Permitted Speed Variation

It is often necessary to determine the flywheel mass moment of inertia required to keep rotational speed within specified limits. If the flywheel speed varies between ω_min and ω_max, the corresponding change in stored energy is:

Flywheel energy variation and required moment of inertia equations

Δ E = E_{\max} - E_{\min}

Δ E = \frac{1}{2} I (ω_{\max}^{2} - ω_{\min}^{2})

Therefore:

I = \frac{2 Δ E}{ω_{\max}^{2} - ω_{\min}^{2}}

Stresses in Rotating Flywheel Rings and Discs

A flywheel rim or annular disc may be analysed as a rotating disc subject to radial and tangential stresses produced by its own inertia. The equations below apply to a thin annular disc under the assumptions used in the RoyMech Rotating Disks and Cylinders reference.

The principal assumptions are:

the disc thickness is small compared with its diameter;
the thickness is constant;
the material is homogeneous and isotropic;
the rotor rotates at constant angular velocity;
the inner and outer cylindrical surfaces are free from radial traction; and
axial stress is neglected for the thin-disc analysis.

Rotating annular disc showing inner radius, outer radius and radial position

Radial and Tangential Stress

For an annular disc with inside radius r₁, outside radius r₂, density ρ, Poisson's ratio ν and angular velocity ω, the radial and tangential stresses at radius r are:

Equations for radial and tangential stresses in a rotating annular disc

σ_{r} = \frac{(3 + ν) ρ ω^{2}}{8} (r_{1}^{2} + r_{2}^{2} - \frac{r_{1}^{2} r_{2}^{2}}{r^{2}} - r^{2})

σ_{t} = \frac{ρ ω^{2}}{8} ((3 + ν) (r_{1}^{2} + r_{2}^{2} + \frac{r_{1}^{2} r_{2}^{2}}{r^{2}}) - (1 + 3 ν) r^{2})

Maximum Tangential Stress

The maximum tangential stress occurs at the inside surface, r = r₁.

Maximum tangential stress equation for a rotating annular disc

σ_{t, \max} = \frac{ρ ω^{2}}{4} ((1 - ν) r_{1}^{2} + (3 + ν) r_{2}^{2})

Maximum Radial Stress

The maximum radial stress occurs at:

r = \sqrt{r_{1} r_{2}}

Maximum radial stress equation for a rotating annular disc

σ_{r, \max} = \frac{(3 + ν) ρ ω^{2}}{8} {(r_{2} - r_{1})}^{2}

High-speed flywheel design: The equations above provide an idealised elastic stress analysis for a constant-thickness rotating disc. High-speed energy-storage rotors require additional consideration of material behaviour, fatigue, rotor dynamics, stress concentrations, manufacturing defects and safe containment.

Advantages and Limitations of Flywheels

Advantages

Rapid absorption and release of mechanical energy.
Useful for smoothing short-duration power fluctuations.
High cycle life when appropriately designed.
Direct mechanical energy storage is possible.
Can provide high power for short periods.

Limitations

Stored energy is lost through bearing and aerodynamic losses.
Rotor stress increases strongly with rotational speed.
High-speed rotors require careful containment and safety design.
Advanced systems may require vacuum enclosures and specialised bearings.
Flywheels are generally more suited to short-duration than long-duration energy storage.

Related RoyMech References

External References and Further Reading

Flywheel Energy Storage – Wikipedia – General background information on flywheel energy storage.
HowStuffWorks – Car Clutch – An example of the application of a flywheel in an automotive drivetrain.
Flywheel Basics Tutorial – Introductory notes comparing flywheel energy storage with battery storage.
Flywheel Energy and Power Storage Systems – More detailed information on flywheel energy and power storage systems.

Flywheel Design – Energy, Inertia and Rotating Stresses