Camille J. Moore and Thomas G. Kraft

Glenn Research Center, Cleveland, Ohio

Metallic Rotor Sizing and Performance Model for 

Flywheel Systems

NASA/TM—2012-217808

December 2012
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Camille J. Moore and Thomas G. Kraft

Glenn Research Center, Cleveland, Ohio

Metallic Rotor Sizing and Performance Model for 

Flywheel Systems

NASA/TM—2012-217808

December 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

Ralph Jansen for providing funding to support this work.

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NASA/TM—2012-217808 1

Metallic Rotor Sizing and Performance Model for Flywheel Systems

Camille J. Moore1 and Thomas G. Kraft

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

The NASA Glenn Research Center (GRC) is developing flywheel system requirements and designs 

for terrestrial and spacecraft applications. Several generations of flywheels have been designed and tested 

at GRC using in-house expertise in motors, magnetic bearings, controls, materials and power electronics. 

The maturation of a flywheel system from the concept phase to the preliminary design phase is 

accompanied by maturation of the Integrated Systems Performance model, where estimating relationships 

are replaced by physics based analytical techniques. The modeling can incorporate results from 

engineering model testing and emerging detail from the design process.

Introduction

GRC is designing and building a new flywheel system for a terrestrial application. As part of this 

effort, an integrated performance model of a flywheel system is being developed to evaluate evolving 

concepts and to provide performance predictions of future flywheel designs. Of the subsystems in the 

flywheel, the rotor dictates much of the design. For this reason, developing a rotor sizing and performance 

prediction tool is of the most importance. 

Flywheel Design

For terrestrial applications of flywheel systems, maintaining low cost is typically the driver, whereas for 

space applications, reductions in system mass typically drive the design. Therefore, for the current terrestrial 

application, a metallic rotor was selected over composite designs to reduce cost. The vertically oriented 

rotor is supported radially and axially by an active magnetic bearing system. In the event of a magnetic 

bearing system fault, a set of rolling element mechanical (back-up) bearings are mounted on each shaft end. 

The entire flywheel system is housed within a vacuum chamber to minimize rotor drag losses. 

Rotor Analysis Code Development

The rotor analysis code is written in MATLAB (The Mathworks, Inc., Natick, MA) and organized in 

modular form, dividing the calculations of the rotor subsystem into more specific categories. The first 

modular section of the code displays a graphical user interface that allows the user to select a rotor 

material, specify the required energy and then choose between parametric or point design. A small 

material database was generated to allow the user to select from a group of potential metallic rotor 

materials. Material and fatigue properties were taken from Metallic Materials Properties Development 

and Standardization (MMPDS), sixth edition. The fatigue analysis also references the NASA Rotating 

Machinery Design Guide.

 1 Camille J. Moore, NASA Glenn Research Center, Cooperative Education (Co-op) student.
NASA/TM—2012-217808 2

For the flywheel design, the fatigue stress ratio was chosen to be 0.10. This means that the alternating, or 

minimum, stress of a stress cycle is 10 percent of the maximum stress. This ratio is then used in the 

fatigue equations given by MMPDS-06:

( ) ( ) ( ) ( ) N B B σeq log = 1 + 2 ⋅log (1)

max (1 ) (3) eq R B m σ = σ ⋅ − − (2)

Where

σeq Equivalent Stress, ksi

σmax Maximum stress, ksi

N Fatigue life, number of cycles to failure

m optimized constant obtained from MMPDS-06

B material constants obtained from MMPDS-06

Solving Equations (1) and (2) for σmax will warrant the maximum allowable stress for the rotor due to 

fatigue. 

Once the material is chosen, the user inputs a rotor inner diameter and outer diameter, or a range of 

outer diameters for the parametric case. For the selected rotor material, based on the user inputs, the 

maximum allowable stress for the rotor is calculated by comparing the ultimate and yield stress, with 

safety factors applied, to the maximum stress over a specified life due to fatigue. After each is calculated,

the most conservative stress value is selected as the maximum allowable stress for the rotor. For each 

outer diameter in the specified range, a maximum allowable rotational speed is calculated (based on 

allowable maximum stress).

Next, the model determines the necessary length of the rotor for the required energy output of the 

flywheel. This is calculated using relationships between the maximum allowable stress, angular velocity 

of the rotor, and required energy output. These relationships are different depending on if the user 

specifies a solid rotor (inner diameter = 0) versus a hollow rotor (inner diameter > 0). The basic equations 

for radial and hoop stress for a solid disk are: 

( ) ( ) 2 2 2

8

3 b r r ⋅ρ ⋅ω ⋅ − + ν

σ = (3)

( ) 2 2 ( ) 2 2

8

1 3

8

3 b ⋅ρ ⋅ω ⋅r

+ ⋅ ν ⋅ρ ⋅ω ⋅ − + ν

σθ = (4)

Where

σr = radial stress, ksi

σθ = hoop stress, ksi

ν = Poisson’s ratio

ρ = material density, lbm/in.3

ω = angular velocity, rad/s

b = maximum outer radius of rotor, in.

r = radius at which the stress is to be determined, in.
NASA/TM—2012-217808 3

Because the maximum radial and hoop stress occurs at the center of the disk, 

( ) 2 2

MAX MAX 8

3 b r ⋅ρ ⋅ω ⋅ + ν

σ = σθ = (5)

Since the maximum allowable stress has already been determined based on material properties, the

Equation (5) is solved for angular velocity, yielding the maximum rate at which the flywheel rotor can spin. 

( 3)

1 8 MAX

MAX

ρ ⋅ ν +

⋅σ

ω = θ

b (6)

From this, the maximum tip speed is calculated by multiplying Equation (6) by the rotor radius and is 

then used to determine the required length of the rotor.

The required energy output of the flywheel dictates the length of the rotor. For the purposes of this 

parametric flywheel model, any energy inefficiencies are calculated externally. This is because the model 

only takes into account the mass of the metallic rotor and not that of the motor laminates or other 

subsystems. As the Integrated Systems Performance Model is expanded, other energy inefficiencies can 

be introduced by each additional subsystem. The energy of a rotating body is defined as:

2

2

1 E = ⋅ I ⋅ω (7)

Where

I = Moment of inertia about the rotating axis, in.

4

By substituting equations for the moment of inertia (Eq. (8)) and mass of the rotor (Eq. (9)), (Eq. (7))

becomes (Eq. (10)):

2

2

1 I = ⋅m ⋅b (8)

m = ρ ⋅ π ⋅ b ⋅ L 2 (9)

( ) 2 2 2

2

1

2

1 ⋅ω 



 



 E = ⋅ ⋅ ρ ⋅ π ⋅b ⋅ L ⋅b (10)

Where

m = mass of the metallic rotor, lbm

L = length of the rotor, in.

Then solving for length and using required energy and maximum angular velocity yields the required 

rotor length:

2

4

MAX

4

req

req

π ⋅ ⋅ρ ⋅ω

⋅ = b

E

L (11)

If the user wishes to design a hollow rotor then Equations (6) and (11) become:
NASA/TM—2012-217808 4

( ) ( ) 



 



 ⋅ + ν

− ν ⋅ + 



 



 ν +

ρ ⋅

σ

ω = θ

2 2

MAX

MAX

3

1

4

3 b a

(12)

( 2 ) ( 2 ) MAX

4

MAX

4

req

req

4

π ⋅ ⋅ρ ⋅ω − π ⋅ ⋅ρ ⋅ω

⋅ =

a b

E

L (13)

Where

a = inner diameter, in.

The process outlined above for solid rotors calculations is the same for hollow rotors.

Graphical User Interfaces

An important aspect of the parametric model is ease of use. The model is being designed to be able to 

be run by personnel inexperienced with stress analysis, as well as provide customizable control for an 

experienced design engineer. To accommodate this need, several graphical user interfaces (GUIs) were 

developed. This allows for quick changes to governing variables as well as detailed control over the 

model. Designed using the MATLAB Guide feature, the main GUI (shown in Fig. 1) is executed like any 

other m-file from the MATLAB command window. In the main GUI, the user controls the entire model. 

The model can be used in parametric analysis over a specified range of outer diameters, or for a point 

design. This is regulated by two mutually exclusive radio buttons found in the upper right hand corner. By 

employing a drop-down menu, the user can select a material to use for the rotor from an established 

materials database. This materials database was created using data from the most recent version of the 

Metallic Materials Properties Development and Standardization (MMPDS-06). Within the database a 

more detailed description can be found for how to locate the data within MMPDS-06. If the user wishes 

to use a material not included in the database the GUI provides the option to manually input material 

properties by clicking on “Material Input.” Doing so opens another GUI, as shown in Figure 2.

If the user selects a material from the drop-down menu the “Input Material” button on Figure 1 

changes to read “View Properties.” Figure 2 will then display the material properties, of the selected 

material, from the database. 

The main GUI also features a button that when clicked, reveals information on how to change the 

fatigue properties of a materials. This was included because a design engineer may want to have more 

control over the equations, from MMPDS-06, used for fatigue analysis. 

Another feature of the main GUI is the ability to include cost analysis. When the GUI is opened, the 

cost analysis section default is to be disabled. Because cost of materials is constantly changing with the 

market, it is important for cost analysis to be easily controlled by the user. Enabling cost analysis will 

warrant a basic assessment of the material cost of the rotor; this does not include manufacturing or 

shipping costs. Also incorporated into the main GUI is a series of pop-up warnings to alert the user if a 

user has not entered proper values into each field.
NASA/TM—2012-217808 5

Figure 1.—Flywheel rotor design analysis.

Figure 2.—Input material properties.
NASA/TM—2012-217808 6

Sample Results

Depending on whether the user has selected parametric or point design, the results of the model are 

displayed in different forms. If the user specifies parametric design, the results are shown in multiple 

graphs (Fig. 3), plotted against the range of specified outer diameters. The plots are rotor length, 

rotational speed, cost per joule and inertia ratio. These are results that a design engineer would be most 

interested in. Important for rotor dynamic analysis, the inertia ratio is a design check related to the 

dynamic control of the rotor. If the user specifies point design, individual results are displayed in a pop-up 

GUI, as shown in Figure 4. A stress profile plot (Fig. 5), is also created for the point design case to 

highlight the stress distribution over the entire radius of the rotor.

Figure 3.—Parametric rotor analysis.
NASA/TM—2012-217808 7

Figure 4.—Results.

Figure 5.—Stress plot.
NASA/TM—2012-217808 8

Conclusions

With this parametric rotor design tool, the GRC Flywheel project will be able to quickly perform 

rotor sizing studies, for metallic flywheels. It is important for the project to assess the performance of new 

concepts in a timely manner. This model will be used as stepping stone for further development of an 

Integrated Systems Performance model for the GRC Flywheel system. Ultimately, a flywheel Integrated 

Systems Performance model could be used to size flywheels for both terrestrial and spacecraft 

applications.

References

1. Metallic Materials Properties Development and Standardization (MMPDS), 6th Edition, FAA, April 

2011.

2. “Fatigue Analysis,” Rotating Machinery Design Guide, Engineering Development Division, NASA, 

20 March 2012.

3. Cook, Robert; Young, Warren: Advanced Mechanics of Materials, Second Ed., Prentice Hall, 1998.

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Camille J. Moore, NASA Glenn Research Center, Cooperative Education (Co-op) student. 

14. ABSTRACT

The NASA Glenn Research Center (GRC) is developing flywheel system requirements and designs for terrestrial and spacecraft 

applications. Several generations of flywheels have been designed and tested at GRC using in-house expertise in motors, magnetic bearings, 

controls, materials and power electronics. The maturation of a flywheel system from the concept phase to the preliminary design phase is 

accompanied by maturation of the Integrated Systems Performance model, where estimating relationships are replaced by physics based 

analytical techniques. The modeling can incorporate results from engineering model testing and emerging detail from the design process.

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