on the optimization of composite flywheel rotors

The Pennsylvania State University

The Graduate School

College of Engineering

ON THE OPTIMIZATION OF COMPOSITE FLYWHEEL ROTORS

A Dissertation inEngineering Science and Mechanics

byJacob Wayne Ross

c© 2013 Jacob Wayne Ross

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

August 2013

The dissertation of Jacob Wayne Ross was reviewed and approved* by the following:

Charles E. BakisDistinguished Professor of Engineering Science and MechanicsDissertation Advisor, Committee Chair

Renata S. EngelProfessor of Engineering Science and Mechanics and Engineering DesignAssociate Dean for Undergraduate Studies and International Programs

George A. LesieutreProfessor of Aerospace EngineeringDepartment Head of Aerospace Engineering

Albert E. SegallProfessor of Engineering Science and Mechanics

Judith A. ToddProfessor of Engineering Science and MechanicsP. B. Breneman Department Head of Engineering Science and Mechanics

*Signatures are on file in the Graduate School.

Abstract

Energy storing flywheel rotor technology has yet to be fully optimized given the design

possibilities. There have been many design approaches that have been published over the

years, but no overall comparisons of the many manufacturable design options have been

put forth. This research increases understanding of how boundary constraints coupled with

optimization objective selection can affect optimized designs. This research also compares

different design options by searching for the global optimum for all cases investigated: (A)

varying the fiber/matrix ratio of each material ring in the composite rotor; (B) including

radial fibers in addition to the circumferential fibers for each material ring; (C) co-mingling

two fiber materials with a variable ratio in each material ring; (D) material ring press-

fitting; and (E) matrix ballasting, where high-density particles can infiltrate the matrix

to vary the density, stiffness, and thermal coefficient of expansion within each material

ring. The results show that fixing both the outer and inner rotor radii in an optimization

search produces a highly restrictive design constraint compared to when at least one of the

radii are allowed to vary. The combinations that hold the most promise without producing

overly restrictive, trivial, or degenerate solutions are as follows: (1) total stored energy,

fixed outer radius, (2) specific energy, fixed outer radius, with a minimum total stored

energy constraint. Concerning the design options, (B) has been shown to be a very poor

design choice and should not be used. Option (D) has shown to have great potential in

maximizing both total stored energy and specific energy, being the single best design option

investigated. However, if the inner radius is allowed to vary, the total stored energy can be

comparably maximized by combining options (C) and (E). Option (E) was found to be the

second best design option with consistent reliability in improving performance.

iii

Contents

List of Figures viii

List of Tables xiii

List of Symbols xvi

Acknowledgments xxi

1 Introduction 1

1.1 Energy Storage Technology Overview . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Automotive Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Electrical Utility Sector . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Organizational Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Literature Review 17

2.1 Research Work: Up To the 1990s . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Research Work: 1990s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Research Work: 2000s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Research Work: 2010 To The Present . . . . . . . . . . . . . . . . . . . . . 45

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Objectives 52

3.1 Phase I: Objective/Constraint Investigation . . . . . . . . . . . . . . . . . . 52

iv

3.2 Phase II: Design Option Investigation . . . . . . . . . . . . . . . . . . . . . 54

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Methodology 56



4.3 Single Ring Rotor Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Multiple Ring Rotor Optimization Program . . . . . . . . . . . . . . . . . . 61

4.4.1 Analytic Formulation & Material Models . . . . . . . . . . . . . . . 61

4.4.2 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.3 Penalty Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.4 Program Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Results & Discussion 72


5.1.1 Fixed Inner & Outer Radii . . . . . . . . . . . . . . . . . . . . . . . 74

5.1.2 2–B: Enclosed Volume Energy Density, Fixed Inner Radius . . . . . 74

5.1.3 2–C: Enclosed Volume Energy Density, Fixed Outer Radius . . . . . 75

5.1.4 1–D: Total Stored Energy, Constraint-Free Radii . . . . . . . . . . . 76

5.1.5 2–D: Enclosed Volume Energy Density, Constraint-Free Radii . . . . 78

5.1.6 1–B: Total Stored Energy, Fixed Inner Radius . . . . . . . . . . . . . 79

5.1.7 4–C: Specific Energy, Fixed Outer Radius . . . . . . . . . . . . . . . 80

5.1.8 4–B: Specific Energy, Fixed Inner Radius . . . . . . . . . . . . . . . 86

5.1.9 4–D: Specific Energy, Constraint-Free Radii . . . . . . . . . . . . . . 87

5.1.10 3–C: Material Volume Energy Density, Fixed Outer Radius . . . . . 89

5.1.11 3–B: Material Volume Energy Density, Fixed Inner Radius . . . . . 90

5.1.12 3–D: Material Volume Energy Density, Constraint-Free Radii . . . . 92

5.1.13 The Remaining Combinations . . . . . . . . . . . . . . . . . . . . . . 93

v


5.2.1 Variable Fiber/Matrix Option . . . . . . . . . . . . . . . . . . . . . . 94

5.2.2 Radial/Hoop Fiber Option . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.3 Two-Fiber Co-mingling Option . . . . . . . . . . . . . . . . . . . . . 97

5.2.4 Press-fitting Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.5 Matrix Ballasting Option . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.6 Comparing the Valid Objective/Constraint Combinations . . . . . . 102

5.2.7 Best Rotor Design Option Combinations . . . . . . . . . . . . . . . . 102

6 Conclusions & Recommendations 104



6.2.1 Design Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2.2 Material Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.3 Objective Constraint Combinations . . . . . . . . . . . . . . . . . . . 109

6.2.4 Convergence Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.3 Avenues for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography 112

Appendix A. Analytic Formulations 118

A.1 Single Ring Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A.2 Multiple Ring Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A.2.1 Stress Calculation Procedure . . . . . . . . . . . . . . . . . . . . . . 129

Appendix B. Composite Material Properties 131

Appendix C. Composite Material Models 135

C.1 Preliminary Calculation Models . . . . . . . . . . . . . . . . . . . . . . . . . 135

C.2 Ballast Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

C.3 Single Fiber Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.4 Two-Fiber Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

vi

C.5 Two-Ply Laminate Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 141

Appendix D. Multiple Ring Rotor Optimization Program Results 143

D.1 Total Stored Energy, Fixed Radii . . . . . . . . . . . . . . . . . . . . . . . . 144

D.2 Total Stored Energy, Fixed Inner Radius . . . . . . . . . . . . . . . . . . . . 150

D.3 Total Stored Energy, Fixed Outer Radius . . . . . . . . . . . . . . . . . . . 156

D.4 Specific Energy, Fixed Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Appendix E. Nontechnical Abstract 168

vii

List of Figures

1.1 Components necessary for inserting a high-voltage battery to a vehicle for

propulsion. Engine and transmission for a hybrid powertrain are not shown. 4

1.2 Components necessary for inserting a capacitor bank into a vehicle for propul-

sion. Engine and transmission for a hybrid powertrain are not shown. . . . 7

1.3 Components necessary for inserting a flywheel system based on electrical

energy transmission into a vehicle for propulsion. Engine and transmission

for a hybrid powertrain are not shown. . . . . . . . . . . . . . . . . . . . . . 10

1.4 Components necessary for inserting a flywheel system based on mechanical

energy transmission into a vehicle for propulsion. Engine and transmission

for a hybrid powertrain are not shown. . . . . . . . . . . . . . . . . . . . . . 11

2.1 One rotor design technique for multiple ring flywheel rotors is to incorporate

a hyperelastic interlayer between the composite rings to reduce peak radial

stresses throughout the entire rotor. . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Axial profiles and shape factors for flywheels using isotropic materials [16]. 22

2.3 Flywheel design developed by General Electric Company [16, 4]. Left is a

picture of the flywheel. Right is a drawing of the design. . . . . . . . . . . . 23

2.4 Flywheel design by Garrett AiResearch [16]. Left is a picture of the flywheel.

Right is a drawing of the design. . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Flywheel design by the AVCO Corporation [16]. Left is a drawing of the

continuous bidirectionally woven ply. Right is a drawing of the assembled

flywheel with a nylon hub. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 A sample of two of the flywheel designs investigated by Ikegami et al. [17]. 26

viii

2.7 Ring disks of complementary varying thickness corresponding to radial and

hoop stress distributions proposed by Miyata [19]. . . . . . . . . . . . . . . 27

2.8 Schematic of chord winding around a filament-wound composite ring. The

pole radius is denoted rp, inner ring radius a, outer ring radius b, chord

thickness hc, and ring thickness is denoted H [20]. . . . . . . . . . . . . . . 28

2.9 Multi-ring flywheel design that was analyzed and tested by Gabrys and Bakis

[22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.10 Multi-direction composite flywheel design by Gowayed et al. [31]. . . . . . . 37

2.11 Cases investigated by Portnov et al. with (a) showing in-plane translation

and (b) showing out-of-plane rotation [34]. . . . . . . . . . . . . . . . . . . . 40

2.12 Example of varying the ply angle for the radial plies of the single ring flywheel

[37]. Hoop plies are not shown. . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.13 Split-type hub design first investigated by Ha et al. [46], then by Krack et

al. [45]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.14 Three rotor design cases with four material rings of co-mingled fiber with

carbon and glass tows, and press fitting in two of the cases [47]. . . . . . . . 49

3.1 Illustrative description of the five rotor design options to be investigated. . 54

4.1 The Schwefel test function in two dimensions. This test function is multi-

modal with a deceptive optimum at the corner of the search space. . . . . . 63

4.2 An illustration of how the CMA-ES uses a displacement vector and covari-

ance matrix to move, scale, shape, and orient the next generation statistical

distribution [62]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Step size control illustration showing a step size that is too large (left), too

small (right), and appropriate (middle) [62]. . . . . . . . . . . . . . . . . . . 65

4.4 Graphical illustration of hard and soft penalty factors applied in one dimension. 69

4.5 Flowchart of decision and calculation procedure when incorporating penalties

into the objective calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Notational illustration of the optimization case coding scheme used for the

figures and tables in Appendix D. . . . . . . . . . . . . . . . . . . . . . . . . 73

ix

5.2 Notional illustration of radial and hoop stress distributions in a spinning,

single-material ring with zero, compressive, and tensile inner boundary radial

stresses P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 For a fixed inner and outer radius, optimal solutions for K, Kev, and Kmv

are identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 For a fixed outer radius, the optimal K search converges on a physically

realistic solution identical to the optimal Kev. . . . . . . . . . . . . . . . . . 76

5.5 The maximum K for a single-material ring of fixed outer radius as a function

of inner radius with a series of curves corresponding to P values of –0.8, 0,

and +0.8 of YT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6 Optimized K for a single-material ring as a function of outer radius with

various P values ranging from –0.8 to +0.8 of YT with both ri and ro allowed

to vary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.7 For objective/constraint combination 1–D, the optimal K search produces

an infinitely large ring with the same λ factors as that of 1–C. . . . . . . . . 79

5.8 Optimized Kev for a single-material ring various P values ranging from –0.8

to +0.8 of YT with both ri and ro allowed to vary. . . . . . . . . . . . . . . 80

5.9 The maximumKm for a single-material ring of fixed outer radius as a function

of inner radius with a series of curves corresponding to P values of –0.8, 0,

and +0.8 of YT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.10 Optimized Km for a single-material ring with various P values ranging from

–0.8 to 0 times the value YT with both ri and ro allowed to vary. . . . . . . 85

5.11 For objective/constraint combinations 3–B, 3–C, 4–B, 4–C, the optimal Km

or Kmv search produces an infinitesimally thin ring that is scale invariant. . 87

5.12 The maximum Kmv for a single-material ring of fixed outer radius as a func-

tion of inner radius with a series of curves corresponding to P values of –0.8,

0, and +0.8 of YT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.13 Optimized Kmv for a single-material ring with various P values ranging from

–0.8 to 0 times the value YT with both ri and ro allowed to vary. . . . . . . 92

x

5.14 Hoop and radial ply stresses for the radial direction under processing tem-

perature change loading of ∆T = −110 ◦C without any centrifugal loading

corresponding to the optimal case E-O-B-110 with one ring. . . . . . . . . . 97

5.15 Notational illustration of the optimization behavior of design option (B),

showing that a step change in the optimization as the last fiber in a direction

is removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

D.1 Energy objective with both radii fixed at 100 mm and 250 mm. Includes zero

and one design option selected. . . . . . . . . . . . . . . . . . . . . . . . . . 145

D.2 Energy objective with both radii fixed at 100 mm and 250 mm. Includes two

design options selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

D.3 Energy objective with both radii fixed at 100 mm and 250 mm. Includes

three design options selected. . . . . . . . . . . . . . . . . . . . . . . . . . . 147

D.4 Energy objective with both radii fixed at 100 mm and 250 mm. Includes four

and all five design options selected. . . . . . . . . . . . . . . . . . . . . . . . 148

D.5 Energy objective with both radii fixed at 100 mm and 250 mm. Includes all

the design options except for design options (B) and (D). . . . . . . . . . . 149

D.6 Energy objective with the inner radius fixed at 100 mm. Includes zero and

one design option selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

D.7 Energy objective with the inner radius fixed at 100 mm. Includes two design

options selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

D.8 Energy objective with the inner radius fixed at 100 mm. Includes three design


D.9 Energy objective with the inner radius fixed at 100 mm. Includes four and

all five design options selected. . . . . . . . . . . . . . . . . . . . . . . . . . 154

D.10 Energy objective with the inner radius fixed at 100 mm. Includes all the

design options except for design options (B) and (D). . . . . . . . . . . . . . 155

D.11 Energy objective with the outer radius fixed at 250 mm. Includes zero and

one design option selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

xi

D.12 Energy objective with the outer radius fixed at 250 mm. Includes two design


D.13 Energy objective with the outer radius fixed at 250 mm. Includes three design


D.14 Energy objective with the outer radius fixed at 250 mm. Includes four and

all five design options selected. . . . . . . . . . . . . . . . . . . . . . . . . . 160

D.15 Energy objective with the outer radius fixed at 250 mm. Includes all the

design options except for design options (B) and (D). . . . . . . . . . . . . . 161

D.16 Specific energy objective with both radii fixed at 100 mm and 250 mm.

Includes zero and one design option selected. Units are in W·h/kg. . . . . . 163


Includes two design options selected. . . . . . . . . . . . . . . . . . . . . . . 164


Includes three design options selected. . . . . . . . . . . . . . . . . . . . . . 165


Includes four and all five design options selected. . . . . . . . . . . . . . . . 166


Includes all the design options except for design options (B) and (D). . . . . 167

xii

List of Tables

1.1 Specific energy and power for automotive energy storage technologies [1,

2, 3, 4, 5]. * Entire flywheel system specific power, including all auxiliary

components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 Optimization objective validity and equivalency for different radial constraint

combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 Design option combinations investigated for a given objective/constraint com-

bination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1 Flywheel single ring rotor results using the multiple ring rotor program with

design option (A) selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Converged results for combination 4–C with the default design options as

well as (A), (C), and (E). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Converged results for combination 4–B with the default design options as

well as (A), (C), and (E). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88


well as (A), (C), and (E). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


well as (A), (C), and (E). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.6 Best designs with (A) for three rings on each of the three valid objec-

tive/constraint combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.7 Best designs with (C) for three rings on each of the three valid objective/constraint

combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xiii

5.8 Best designs with (D) for three rings on each of the three valid objec-

tive/constraint combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.9 Best designs with (E) for three rings on each of the three valid objective/constraint

combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.10 Best designs for three rings on each of the three valid objective/constraint

combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.1 Optimization objective validity and equivalency for different radius constraint

combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.1 Material properties of selected unidirectional fiber composite materials for

the single material ring rotor optimization program [25, 66]. . . . . . . . . . 132

B.2 Densities, composite volume fractions, elastic moduli, and Poisson’s ratios

for the multiple ring rotor program composite materials database. . . . . . 132

B.3 Longitudinal and transverse elastic moduli and coefficients of thermal expan-

sion for the multiple ring rotor program composite materials database. . . . 133

B.4 Longitudinal and transverse tensile and compressive strengths for the multi-

ple ring rotor program composite materials database. . . . . . . . . . . . . . 133

B.5 Epoxy matrix density, Poisson’s ratios, elastic moduli, and coefficients of

thermal expansion for the multiple ring rotor program composite materials

database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

B.6 Ballast material density, Poisson’s Ratios, elastic moduli, and coefficients of

thermal expansion for the multiple ring rotor program composite materials

database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

D.1 Energy objective with both radii fixed at 100 mm and 250 mm. Best results

after 75,000 optimization runs. Gray shading indicates convergence failure.

Units are in kW·h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

D.2 Energy objective with the inner radius fixed at 100 mm. Best results after

35,000 optimization runs. Gray shading indicates convergence failure. Units

are in kW·h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

xiv

D.3 Energy objective with the outer radius fixed at 250 mm. Best results after

75,000 optimization runs. Gray shading indicates convergence failure. Units

are in kW·h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

D.4 Specific energy objective with both radii fixed at 100 mm and 250 mm. Best

results after 75,000 optimization runs. Gray shading indicates convergence

failure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

xv

List of Symbols

a Inner ring radius of flywheel

b Outer ring radius of flywheel

c Vector for integration constants

f Force loading vector

fsoft Exponential decay soft penalty factor

g Matrix for radial coordinate integration constant multiplication

h Axial length or thickness

hc Chord axial thickness of chord flywheel

iiGA Island Injection Genetic Algorithm

k Single material ring stiffness matrix

m Mass

r Radius, radial coordinate

rp Pole radius of chord flywheel

sGA Simple Genetic Algorithm

u Displacement

v Velocity

x constraint parameter

xo constraint parameter limit

A Ballasting shear modulus factor

B Ballasting shear modulus factor

C Ballasting shear modulus factor

xvi

Cp Integration constant

E Young’s modulus

F Global laoding vectors

FEM Finite Element Modeling

G Shear Modulus

GPS Generalized Plane Strain

H Total axial length of chord flywheel

I∗ A -1, 1 diagonal matrix

K Kinetic energy, total stored energy, global stiffness matrix, bulk modulus

Km Specific energy

Kmv Material volume energy density

Kev Enclosed volume energy density

L Dimensionless radial coordinate factor

LLNL Lawrence Livermore National Laboratory

MGPS Modified Generalized Plane Strain

MLC Magnetically Loaded Composite

Qpq Element of the stiffness matrix

P Inner radial boundary stress, indivdual variable penalty

PGA Parallel Genetic Algorithm

PIM Power Inverter Module

PS Plane Stress

R Tsai-Wu failure factor

Spq Element of the compliance matrix

SA Simulated Annealing

SOC State Of Charge

U Global displacement vector

UPS Uninterrupted Power Supply

V Volume fraction

W Outer radial boundary stress

xvii

X Hoop/longitudinal direction strength

Y Radial/transverse direction strength

Z Tsai-Wu Failure vector and matrix

α Coefficient of thermal expansion

β Coefficient of moisture expansion

γ Shear strain

δ Interference Displacement

ε Normal strain

ε0 Constant strain for a GPS assumption

ε1 Strain coefficient for linearly varying strain of a MGPS assumption

ζ Material ring inner-to-outer radius ratio factors

η Halpin-Tsai factor, ballasting shear modulus factor

κ Dimensionless principal stiffness element factor

λ Rotor inner-to-outer radius ratio

µ Dimensionless hoop/radial elastic modulus factor

ν Poisson’s ratio

ξ Halpin-Tsai empirical factor

ρ Mass density

σ Normal stress

τ Shear stress

φp Material property notational simplification factor

χ Radius variable to outer radius ratio

ω Angular velocity

∆C Moisture change

∆T Temperature change

Φ Material property notational simplification matrix

Subscripts:

xviii

1 Longitudinal composite direction

2 Transverse composite direction

b Boundary-related factor

h Hoop direction

hard Hard Penalty variable

i Inner radial boundary

o Outer radial boundary

r Radial coordinate, radial direction, radial ply

soft Soft penalty variable

z Axial coordinate

C Compressive stress

T Tensile Stress, Halpin-Tsai parameter

ε Strain-related factor

δ Interference-related factor

θ Angular or circumferential coordinate

σ Stress-related factor

ω Centrifugal loading

∆T Temperature change

5 θz shear component

6 θr shear component

Superscripts:

1 Co-mingling composite material number

2 Co-mingling composite material number

2 Transverse composite direction

b ballast material property

e epoxy material property

f Fiber-related property

m Matrix-related property

xix

n Ring number, increasing number with increasing radius

N Number of material rings

T Transpose

xx

Acknowledgments

I would like to thank Dr. Joel R. Anstrom, the Thomas D. Larson Pennsylvania Trans-

portation Institute, and the Graduate Automotive Technology Education (GATE) Center

for providing funding, energy conversion educational resources, and hands-on research op-

portunities in energy-related issues. This work was also supported in part through instru-

mentation funded by the National Science Foundation through grant OCI–0821527.

xxi

Chapter 1

Introduction

Throughout history, flywheels have been used as energy storage and load-leveling devices

for a variety of purposes that include speed control of a potter’s wheel and inertial loading

of engines. However, only within the last century have flywheels begun to approach the

power and energy densities required for industrial and transportation applications. Much

of this progress is due to the invention of fiber-reinforced composites. Based on the classical

kinetic energy equation, the kinetic energy of a system varies linearly with the mass but

quadratically with the velocity, which is shown in Equation 1.1.

K =1

2mv2 =

1

2mr2ω2 (1.1)

where:K: Kinetic energym: Massr: Radiusv: Velocityω: Angular velocity

Since the energy stored grows faster with velocity than with mass, it is preferred to

have a light, high-speed flywheel than a heavy, low-speed device. For centrifugal loading,

1

2

the stresses produced are also related to the velocity as shown in Equation 1.2.

σ = ρv2 = ρr2ω2 (1.2)

where:ρ: Mass Densityσ: Normal stress

Like the energy, the stresses produced in a flywheel from centrifugal loading are pro-

portional to the square of the velocity. Therefore, a popular method to produce a high-

performance flywheel is to build it out of a material that has high strength characteristics

to obtain high angular speeds. Many types of carbon fibers have stiffness, strength, and

mass density characteristics in the fiber direction that are superior to those of metal alloys,

making them the good candidates to improve flywheel performance. However, flywheel

technology has become a minor contributor compared to other energy storage technologies.

Research for chemical batteries has been popular, and high power density ultracapacitors

have been garnering significant research interest. A brief review of various energy storage

technologies can provide justification for how flywheels can contribute to energy storage

technology demands.

1.1 Energy Storage Technology Overview

In society today, there are two primary commercial power technology sectors that can

benefit from energy storage technologies. These are the transportation and electrical power

utility sectors. In both sectors, kinetic energy storage devices have attractive characteristics

compared to both chemical batteries and ultracapacitors. For the transportation sector,

there is a need to provide high power in a very compact, lightweight package that is reliable

given a range of environmental conditions. For the power utilities industry, there is a need

to have technologies that have a very long life with low maintenance requirements.

3

1.1.1 Automotive Sector

For automotive applications, there are a number of requirements that must be satisfied in

order to produce an energy storage system that can be placed into a vehicle. Regardless of

the energy storage form, the energy must be delivered as kinetic energy to move the vehicle.

For a chemical battery, conversion technology is required such as a motor/generator and

a Power Inverter Module (PIM) to transform high voltage D/C power to three-phase A/C

electrical power that the motor generator can efficiently accept. For a capacitor bank, there

is an additional D/C to D/C converter module necessary to maintain a relatively constant

high voltage line to the PIM. For compressed air, a turbine/compressor would need to be

attached to the drive line. An alternative would be to use the compressed air to increase

the combustion pressure for an internal combustion engine. For flywheels, there can exist

another PIM for the motor/generator attached to the flywheel itself for an electromechanical

transmission. An alternative to the mechanical/electrical conversion of energy for a flywheel

would be a direct mechanical system.

Some of the latest industry technology product data can provide a rough comparison of

energy storage characteristics, which is given in Table 1.1. However, most of these values

do not incorporate the auxiliary component requirements, dynamic behaviors, or environ-

mental effects on these technologies. As indicated in Table 1.1, the specific power of the

compressed air is based on the turbine system used to recover the energy. Given an auto-

motive application, there are advantages and disadvantages for each of these technologies.

Table 1.1: Specific energy and power for automotive energy storage technologies [1, 2, 3, 4,5]. * Entire flywheel system specific power, including all auxiliary components.

Li-ionUltracapacitors

CompressedFlywheels

Batteries Air

Specific Energy131 4.0 28 72.8

(W·h/kg)

Specific Power2.4 5.6 Variable 2.64*

(kW/kg)

4

Chemical Batteries

The advantages chemical batteries have over the alternatives can be significant and have

led to their dominance as an energy storage technology. Batteries provide power directly in

an electrical form that maintains a constant D/C voltage throughout most of their energy

storage range. Such a capability simplifies energy transmission and reduces the required rec-

tifier component to transform the energy into a usable form for the motor/generator, which

is shown in Figure 1.1. Some Li-ion battery chemistry technology has become established

for large-scale use.

Figure 1.1: Components necessary for inserting a high-voltage battery to a vehicle forpropulsion. Engine and transmission for a hybrid powertrain are not shown.

With further research into improving this technology, continuous evolution is fully ex-

5

pected. Because batteries transform energy electro-chemically, they possess high efficiencies

that exceed those of heat engines for their operating range. Also, the latest Li-ion batteries

have very good performance characteristics. These characteristics make for a very compact

energy storage device. Nonetheless, there are many disadvantages with chemical batteries.

To ensure safe, reliable operation, many limitations are imposed on a chemical battery

system. Rechargeable batteries used in vehicles do not operate at their full State-Of-Charge

(SOC) range. To limit battery aging and degradation so that it can last throughout the

entire life of the vehicle, the maximum operation energy range is reduced. For the weather

conditions that most vehicles are designed to operate in, battery performance can degrade

or be limited significantly. Colder temperatures lower ion transport and increase the inter-

nal resistance of the battery, reducing the power, energy, and efficiency of the battery. High

temperatures increase the performance of the battery, but there are temperature-dependent

safety limitations for the materials used, requiring that performance be lowered. Because of

the thermal management required to keep the battery operating within a relatively narrow

temperature range, active auxiliary temperature control equipment must be added. The

thermal management may include metal fins and air blowers or coolant pumps for cool-

ing, and resistive heating elements or heat pumps for heating. Li-ion battery technologies

possess energy densities that are a significant fraction of what is normally considered for

explosive materials. In the case of an emergency where the battery is damaged and a short

circuit results, the battery can easily ignite. Poisonous gases from this fire must be directed

away from the vehicle through ducting, and the battery container must be sealed to pre-

vent infiltration into the passenger compartment. Another limitation is discharge speed.

Batteries become less efficient the faster they are loaded and can be damaged if loaded

too quickly. Battery management systems provide high-power constraints and limitations

related to the SOC, time away from the equilibrium, and charge/discharge rate. Once

the maximum charge for the high-power operation has been reached, the battery manage-

ment system will force the battery to operate at the lower discharge rate. Incorporating

6

nanoscale structure into batteries may increase performance, but it also has a tendency

to increase manufacturing complexity, and thus cost. This complexity adds to the already

cost-intensive manufacturing and materials procurement necessary to produce Li-ion bat-

teries. Li-ion batteries can have cycle life limitations due to battery aging. Li-ion batteries

are also much more sensitive to damage when being charged compared to being discharged,

meaning that the powers for charging may be less than for discharging. This factor limits

Li-ion battery regenerative braking energy absorption capabilities.

Ultracapacitors

The primary advantage of ultracapacitors is their high specific power [2]. Standard capaci-

tors store their energy strictly as an electrostatic potential. This energy storage mechanism

allows for fast charge or discharge for a given change in voltage. Electric double-layer capac-

itors, or ultracapacitors, have different characteristics than standard capacitors relating to

their use of an electrolyte for ion transport. In addition to higher specific powers, they have

longer cycle lives compared to chemical batteries that could last the life of an automotive

vehicle. Because there is ion transport for a very short distance without the burden of an

electro-chemical reaction, these devices are highly efficient with low resistive losses. As a

result, cooling requirements are significantly reduced, reducing system size and cost. Stan-

dard materials combined with established, simple manufacturing methods can also lower

device cost. Because performance characteristics are based on material properties of the

electrode, there is potential in improving the technology with better materials and fabrica-

tion techniques. Despite these advantages, there are issues that must be addressed when

using these devices.

Ultracapacitor energy storage is based on voltage; as the device releases its energy, the

voltage of the device changes. In order to ensure a reasonable voltage operation, a converter

capable of transforming large electrical power from a variable DC signal to a limited-voltage

range DC bus line, which is shown in Figure 1.2, would add significant weight to the system.

7

An ultracapacitor system would require some kind of containment in case there is dielectric

breakdown and the ultracapacitors explode. Electrolytic ultracapacitors have some of the

same problems as batteries. While power densities for capacitors are significantly higher

than the best chemical batteries, energy densities can be more than an order of magnitude

lower. This limitation restricts a capacitor bank that would be of comparable size to an

electric powertrain battery pack to only be able to satisfy pulse power requirements for a

hybrid powertrain.

Figure 1.2: Components necessary for inserting a capacitor bank into a vehicle for propul-sion. Engine and transmission for a hybrid powertrain are not shown.

Also, ultracapacitors have high self-discharge rates and cannot be used for long-term

energy storage. Having an electric powertrain based solely on ultracapacitors is not yet

8

feasible for personal vehicles. If a voltage difference is maintained for an extended period of

time, dielectric absorption can occur, preventing the ultracapacitors from fully discharging.

Because the electrolyte is a liquid, there are temperatures at which the electrolyte solidifies.

Even before then, ion transport is lowered by cold temperatures, increasing the internal

resistance of the ultracapacitors. Like batteries, ultracapacitors do not operate through

their entire energy range.

Compressed Air

Compressed air has an advantage in that the medium for storage is readily abundant. Direct

mechanical linkages to the drive line are possible. Different configurations can provide

different benefits. A configuration where there is a direct mechanical connection from the

compressor/turbine to the drive line can provide direct kinetic energy transmission. Another

configuration uses the compressed air to increase the compression pressures of the internal

combustion engine, using the released energy to improve engine performance. The cycle

life of the storage tank can match that of the vehicle. Nonetheless, there are disadvantages

with this technology.

The tank must be confined to a pressure vessel geometry for optimal energy storage and

may be awkward and space inefficient to place into a vehicle. Because the gas is heated as

it is compressed and cooled upon release, an active temperature management system may

be required for both heating and cooling of the gas. Self-discharge rates are dependent on

the tank sealing technology. Power capabilities and efficiencies are dependent on the size of

the turbine/compressor system.

Flywheels

There are many advantages to incorporating flywheels into a hybrid vehicle system. The

charge/discharge capability of flywheels that lead to high specific powers make for a practical

load-leveling assist to vehicles [5, 6]. Flywheels have a long cycle life that can match the life

9

of a vehicle. Even with contact bearings introducing friction into the system, short-term

efficiencies can be very high [5]. Also, an active temperature management system may

not be required for some of the flywheel systems, significantly reducing weight, space, and

cost requirements [5]. Many of the research developments of modern flywheel systems for

automobile use that have come within the last decade have made the difference between a

novel academic research topic and a competitive energy storage system worthy of industry

attention. Two of these advances are now discussed.

For the electrical power transmission shown in Figure 1.3, a significant innovation that

has been applied to flywheel rotors is the Magnetically Loaded Composite (MLC) [7]. This

technology incorporates ferromagnetic particles into the epoxy matrix of a composite, al-

lowing the composite matrix serve as a permanent magnet rotor for electrical energy trans-

mission. By using higher strength and stiffness materials such as glass/epoxy composites,

designers can produce higher strength rotors that are not handicapped by weaker, heavier

permanent magnet materials. Using this higher strength, the composite material increases

the specific energy of the rotor. Also, because the ferromagnetic particles are not in elec-

trical contact with each other and the composite material is not electrically conductive,

magnetically-induced eddy currents cannot form. Normally, the energy from the eddy cur-

rent losses is converted into thermal energy which would need to be removed from the

flywheel containment. The MLC rotor reduces motor losses due to eddy currents, increas-

ing the efficiency of the system. Also, a large temperature management system would have

to be employed to remove the heat generated from eddy current losses. Reduction of the

eddy current losses limits the need for a temperature management system, further reducing

the required weight, space, and cost of the flywheel system while increasing the efficiency.

Mechanical transmission of flywheel energy has been made more attractive by relatively

recent developments in vacuum sealing and continuously variable transmission technology

[5, 8, 9]. These technologies allow the flywheel to maintain a vacuum seal while producing

direct shaft work that allows for continuous conversion of speed and torque suitable to

10

Figure 1.3: Components necessary for inserting a flywheel system based on electrical energytransmission into a vehicle for propulsion. Engine and transmission for a hybrid powertrainare not shown.

the flywheel. There are a number of advantages to the system shown in Figure 1.4. The

direct mechanical transmission removes the need for a magnetic rotor and stator within

the flywheel system containment, allowing the rotor and hub to be further optimized for

energy storage while substantially reducing the weight and cost at the same time. The motor

and PIMs that would be required for electrical to mechanical transmission are also removed,

further reducing weight, space, and costs requirements while increasing efficiency. Although

these technological developments are relatively recent, automotive manufacturers have taken

11

notice, with Volvo announcing development and testing of this system in an automobile

[6]. Testing has shown promising results, and with the development of more sophisticated

control methods and techniques, further improvements are expected. Nonetheless, there are

disadvantages to both flywheel designs.

Figure 1.4: Components necessary for inserting a flywheel system based on mechanicalenergy transmission into a vehicle for propulsion. Engine and transmission for a hybridpowertrain are not shown.

Composite flywheels have limitations relating to the maximum strength of the flywheel.

The maximum strength of the composite can be reduced when factoring in loading fa-

tigue. For automotive applications, solid contact bearings are required which produce higher

amounts of friction compared to magnetic bearings. Use of contact bearings both decreases

the efficiency and increases the self-discharge rate of the flywheel. For electro-mechanical

transmission, two PIMs are required to transfer energy. These added components have the

12

negative effect of additional cost, weight, and space requirements compared to a chemical

battery while inducing switching losses for the high-power insulated-gate bipolar transistors.

For the mechanical transmission, there are additional losses at the vacuum lip seals from the

hermetic sealing fluid, further decreasing efficiency and increasing the self-discharge rate.

1.1.2 Electrical Utility Sector

The primary goals for producing viable energy storage technologies suited for utility grid

applications are to reduce the capital and maintenance costs. One reason to use energy

storage technologies for these applications is to provide an Uninterrupted Power Supply

(UPS) for emergency situations. Those institutions that require emergency backup systems

typically have emergency diesel generators for a longer term power outage, but require an

intermediate power solution for the time period between the start of the outage and the

time the generators have been brought online. Certain energy storage technologies can fill

in this gap.

Another reason why utilities may want to use energy storage technologies would be for

load-leveling purposes. Renewable energy sources provide power in an unreliable fashion.

Energy storage systems working in concert with renewable sources can serve as a buffer to

even out the power supplied to the grid. Even when using more predictable and reliable

power technologies, consumer demand can fluctuate significantly throughout the course of a

single day. Energy storage systems can provide load-leveling in this case as well, absorbing

energy during periods of low demand and dispensing energy during peak demand periods.

Given these scenarios, energy storage technologies must possess characteristics that are

different from that of automotive applications.

Chemical Batteries

For utility applications, capital cost can be the most important factor. Although higher

energy densities typically translate into smaller systems which reduce cost, the fact that

13

high performance battery technologies are currently expensive to produce tends to negate

this advantage. Utility companies also favor the reliability of established technologies. As

such, utilities typically choose lead-acid batteries for much of their energy storage needs.

Nonetheless, there are issues relating to chemical batteries. Temperatures for the battery

pack must be maintained within certain limits. The primary concern is colder temperatures

degrading battery performance. To provide a solution, waste heat from power plants can

provide adequate heating. If the battery system needs to be placed in a more isolated

setting, underground storage is an option. Regardless, both solutions incur higher cost.

For daily load-leveling use, life cycle issues for batteries are a limiting factor, with capital

cost per energy storage capability being a secondary limitation. Needing to replace the

battery pack every few years incurs high maintenance costs for utilities and makes batteries

unattractive for any kind of routine load-leveling function. Overall, inexpensive lead-acid

batteries are suitable for UPS applications, but chemical batteries are not suited to large-

scale load leveling in general.

Ultracapacitors

Ultracapacitor technology is not currently used as a energy storage technology for utility

applications in a large-scale manner. While the large cycle life limits are attractive, high self-

discharge rates along with dielectric absorption issues make this technology not suited for

the larger energy storage time scales that utilities have compared to automotive applications.

Also, the energy storage limitations are not justified given the high capital cost to construct

these devices. However, material advances can come in leaps and bounds. Therefore, the

viability of this technology is highly susceptible to change.

Compressed Air

Compressed air storage is competitive for large-scale load-leveling applications, but not well

suited for UPS applications [10]. Large-scale load-leveling is cost-effective due to the fact

14

that the storage tank is a large cavern that has already existed due to natural geological

phenomena or mining. Therefore, the large volume for storage is available at very low cost.

Also, the fact that the air storage is underground and has such a high volume eliminates

the need for temperature management of the storage tank. The availability of these caverns

limits the location of the storage unit, but an entire independent power plant can be run

for a suitably large cavern. Current applications include using the compressed air as a

combustion enhancer for natural gas power plants [11]. Applications being investigated

are coupling this technology with wind turbine power production. Compressed air energy

storage is not well suited for small scale UPS applications because the system must be

constructed at a fixed location, such as a hospital or existing power plant. These sites may

not have access to an existing large volume cavern. Producing high-pressure storage tanks

of the volume necessary for UPS applications significantly increases capital cost. Also, a

temperature management system may be required for the storage tanks, further reducing

cost-effectiveness.

Water Storage

Water storage has been a popular existing load-leveling technology for use in both the

utility and water management industries. The large reservoirs upstream of hydroelectric

dams serve as massive energy storage systems. Pumped water can be cost-effective if the

reservoirs can be constructed out of existing high-altitude locations where water can be

pumped up into the reservoirs for storage and released during periods of peak demand

[10, 11]. Water towers serve as a load-leveling mechanism for water management systems

and provide emergency water pressure in the case of electrical power outages. However,

hydro storage is not suited to UPS applications due to the fact that storing water in the

necessary quantities of mass and elevation would not be cost-effective.

15

Flywheels

For both emergency UPS and small-scale load-leveling applications, flywheel energy stor-

age has many attractive features. First, the high cycle life of flywheels translates into

lower maintenance costs. Flywheels are less temperature sensitive compared to batteries,

which improves reliability. Because utility applications require energy storage systems to

be stationary, flywheel system designs can incorporate different design features to improve

performance. The bearings used for stationary flywheel applications are typically magnetic

in nature. Removing all points of contact between the rotor and the rest of the system

significantly reduces frictional losses and related maintenance needs. Because the location

of the flywheels is somewhat arbitrary, these devices can be placed underground to minimize

the size of the containment for the flywheels; the earth would supply most of the contain-

ment. One utility equipment provider, Beacon Power Corporation, produces commercial

flywheels that it advertises requires maintenance once every twenty years [12]. Nonetheless,

flywheels are not as competitive as compressed air or water storage for large load-leveling

applications [10].

1.2 Conclusions

For both automotive and utility applications, there are multiple factors that must be ac-

counted for beyond the theoretical performance values given in research articles. There are

many ancillary system components beyond the raw energy storage medium that must be

accounted for in terms of added weight, space, and cost as well as parasitic power losses.

Additional factors such as long-term application in an environment subject to changing con-

ditions can also lower performance characteristics. Given these factors, flywheel systems are

competitive for certain applications and conditions. While not capable of providing long-

term energy storage for automotive applications, flywheel systems with mechanical energy

transmissions can be a competitive technology for hybrid vehicle applications where a pulse

16

power energy assists are useful. For utility application, flywheel systems can be competitive

for UPS systems and small-scale load-leveling assistance.

1.3 Organizational Structure

After this introduction, an overview of existing energy storing flywheel rotor research is

presented in Chapter 2. The time period spans from the 1970’s to the present time and

includes a variety of research projects with a many different goals and analysis methods.

From this information, the objectives of this research to add to the understanding of flywheel

optimization are given in Chapter 3. Next, the analysis methodology is provided in Chapter

4 and Appendices A, B, and C. The results along with their discussion are given in Chapter 5

and Appendix D. Last, the research conclusions and recommendations are given in Chapter

6.

Chapter 2

Literature Review

Relatively modern research into flywheels started in the 1960s and has progressed since then.

This literature review identifies significant research into rotor optimization throughout the

decades up to the present day. Different optimization objectives are considered with both

theoretical and experimental work. Various design ideas along with analytic formulations

are seen. The analytic formulations vary widely in both the simplifying assumptions used

and the manufacturing calculations made. For some research investigations, no manufactur-

ing factors are incorporated. These factors can include thermo-elastic effects from processing

temperature change due to epoxy curing, stresses from filament winding, autoclave pres-

sure loading, and material ring stress states from the press-fitting process depending on the

design. The material ring press-fitting process includes ensuring the material rings survive

manufacturing, then press-fitting, then zero load and centrifugal loading states. The various

phenomena associated with manufacturing are denoted as manufacturing method effects.

2.1 Research Work: Up To the 1990s

Before the advent of high-strength fibers in the late 1960s, flywheels were typically con-

structed out of various kinds of steel to provide load-leveling on a small scale, such as

engine flywheels. There were a few attempts to produce flywheels that could be used on

17

18

trains and buses, but the energy storage capabilities of the flywheels using metals were

still too heavy to be viable for transportation purposes [13]. However, with the advent of

aramid, carbon, and glass fibers, researchers began to contemplate the use of these fibers

in flywheels. The oil crisis of 1973 to 1974 caused many in the US to search for alternatives

to oil and ways to reduce the need fossil fuels altogether. The US Department of Defense

and Energy Research & Development Administration began funding multiple energy-related

research projects and national laboratories. Starting in the mid-to-late 1970s, researchers

began intensively investigating fiber reinforced composites for energy storage applications.

Some of the results of work by US national laboratory researchers was produced by

Christensen and Wu [14]. In their analysis, they chose a 2-D plane stress formulation with

a maximum strain failure criterion. The stress-strain relations are defined according to

Equation 2.1.

σθ

σr

=

Qθθ Qθr

Qrθ Qrr

εθ

εr

(2.1)

where:Qij : Element of the stiffness matrixε: Normal strain

subscripts:r: Radial coordinateθ: Angular or circumferential coordinate

Because they wanted to optimize the rotor design, they confined the optimal limit to be

a uniform, constant strain field throughout the rotor. They required that the flywheel fail

at every point simultaneously by achieving this uniform maximum strain throughout the

flywheel. They incorporated an axial thickness function into the stress equilibrium equa-

tion. All analysis was made in a strictly theoretical sense, with no material requirements

given except for the assumption that was cylindrically orthotropic. Given this formulation,

19

the authors demonstrated that the optimization of the flywheel in terms of specific energy

required only optimizing the axial thickness variation. The elastic property solution to

the optimal specific energy allows for variation in the radial and circumferential directions,

meaning that the fiber reinforcement can lie strictly in the radial direction, the circumfer-

ential direction, or anything in between and still produce an optimal design. Cases where

the Young’s moduli in the radial and circumferential directions were allowed to vary were

investigated in order to understand the axial thickness shape function that was evaluated.

A solution was found for the isotropic case becoming a solid disk , and rings—where the

thickness is zero at the center—for the radial and circumferential reinforcement cases. The

authors noted that for the same invariant stiffness properties, the solid isotropic disk was

preferred due to more efficient space use and no need for a hub. This analysis was theo-

retical only; no properties were inserted for definitive performance values. No checks for

processing temperature or manufacturing-related residual stresses were introduced.

Multi-ring flywheel optimization attempts were presented by Danfelt et al. around the

same time in 1977 [15]. In their research, the authors produced a formulation based on a 2-D

plane stress assumption. Their formulation was geared toward a finite element solution that

accommodated ring elements of different anisotropic materials and constant axial thickness.

The method for optimization was a trial-and-error procedure that started with attempting

to maximize the energy of one ring at a time, with each additional ring designed to produce

a distribution that satisfied the maximum stress failure criterion and was as uniform as

possible. Hyperelastic rubber acted as the interface between composite rings in order to

reduce the radial stresses, which is shown in Figure 2.1. For this numerical study, six

composite rings with five interlayers were used. Parametric studies were performed to

optimize the rotor design. The outer diameter was set to 914 mm and the inner diameter

was 50.8 mm. The material used was an aramid/epoxy composite for the large rings and

a rubber material for the interlayers. A baseline case of constant radial thickness and

properties for the composite and elastomeric rings were performed first. After this baseline,

20

the authors attempted to change the density with ballasting elements in the matrix material

of the composite rings to develop more uniform failure limits for the rings, but found that

it was not possible to do with density changes alone.

Figure 2.1: One rotor design technique for multiple ring flywheel rotors is to incorporate ahyperelastic interlayer between the composite rings to reduce peak radial stresses throughoutthe entire rotor.

Another variation case included changing the stiffness of the elastomeric interlayers.

The authors noted that changing the axial thickness of the rings would have only small

benefits since the in-plane stresses are fairly independent of axial thickness for multi-ring

flywheels with elastomeric interlayers. The last case study included varying both the density

and radial thickness of the composite rings. The highest specific energy obtained was 81.4

W·h/kg and came from varying the stiffness of the interlayers. The highest energy density

21

was 160 kW·h/m3, and was obtained by varying both the density and radial thickness of

the composite rings. The authors noted that this analysis was simply an overview of the

possible variations in flywheel design to provide optimization and did not provide a definitive

optimization strategy or solution. Curing temperature changes and manufacturing method

effects on the inherent stresses of the rotor were not included in the analysis.

During the period of 1975 to 1983, a program to develop high-performance flywheel

energy storage systems by various industry leaders was sponsored by the Department of

Energy and administered by Lawrence Livermore National Laboratory (LLNL) [16]. Devel-

opment research accelerated in 1978 as the laboratory partnered with the General Electric

Company to research flywheel system integration for automobiles. During these years,

LLNL performed comprehensive flywheel research that included material and manufactur-

ing process options surveys along with theoretical analysis into flywheel limits and rotor

dynamic stability. The shape profiles analyzed were based on an isotropic flywheel material

subjected to a maximum stress failure criterion. The optimal shape profile exhibits failure

at all points simultaneously and maximizes the specific energy, which is shown as the top

shape in Figure 2.2. Other profiles were given a shape factor that was defined by the ratio

of the given shape specific energy to the optimal shape specific energy.

In addition to this analysis, LLNL sponsored a competition between industry leaders to

design and develop flywheel designs for testing. After the initial design and test phase for

six competing flywheels were finished, three candidates were selected for further develop-

ment and testing. These candidate designs came from General Electric Company, Garrett

AiResearch, and the AVCO Corporation [16]. The General Electric flywheel design con-

sisted of an outer ring of filament-wound carbon/epoxy composite with a thick, inner ring

consisting of a glass fiber/epoxy prepreg with a quasi-isotropic layup. This design is shown

in Figure 2.3.

The Garrett AiResearch flywheel design consisted of fifteen aramid fiber rings that were

not bonded together. Layers of the rotor were separated by Teflon tape, which is shown in

22

Figure 2.2: Axial profiles and shape factors for flywheels using isotropic materials [16].

Figure 2.4. The hub was composed of graphite/epoxy struts bonded to aluminum sheets.

The AVCO Corporation flywheel consisted of a ring that contained a bidirectional weave

where fibers were placed in the hoop and radial directions. The goal of this weave was to

attempt to produce a constant stress profile where with radial fibers would interact with the

hoop fibers to transmit the outer hoop stresses to the inner hoop fibers, which is shown in

Figure 2.5. Manufacturing the flywheel rotor involved using a mold, but it did not specify

23

Figure 2.3: Flywheel design developed by General Electric Company [16, 4]. Left is a pictureof the flywheel. Right is a drawing of the design.

whether the disks were press consolidated, resin transfer molded, or whether a wet lay-up

was used. The hub was constructed of nylon with an aluminum shaft.

Given these three flywheel designs, the Garrett flywheel performed the best with the

highest recorded burst specific energy of 72.8 W·h/kg, and the General Electric flywheel

came in second with 68.0 W·h/kg [4]. The AVCO design had significant manufacturing

issues where they were only able to produce a single flywheel for testing. They were sub-

sequently dropped from the competition after testing of the single flywheel. For this com-

petition, the performance parameter of interest was specific energy and ancillary flywheel

system components were not factored into the design or economic analysis.

At the same time the LLNL final report was published, researchers in Japan were also

investigating design ideas for flywheels. The design by Ikegami et al. included a laminated

quasi-isotropic glass/epoxy composite with a carbon/epoxy outer ring that was filament

wound [17]. This design was highly similar to General Electric’s design and the design

parameters that were investigated included the hub radial and axial thicknesses as well

as ring radial and axial thicknesses. In this analysis, a 3-D axisymmetric finite element

24

Figure 2.4: Flywheel design by Garrett AiResearch [16]. Left is a picture of the flywheel.Right is a drawing of the design.

method was stated to be used. The failure criterion used was an interactive stress criterion

called Hoffman’s condition [17]. The flywheel designs shown include glass cloth rings that

had filament-wound reinforcement on the side as well as flywheels that had filament-wound

carbon fiber rings attached to a glass cloth inner ring that consisted of two thickness sections,

which is shown in Figure 2.6. For all cases, the inner diameter was set at 30 mm and the

outer diameter to 130 mm. Analysis was performed before experimental testing and included

six design test cases. The maximum angular speed for the best design was 62.2 krpm for

a tip speed of 847 m/s. The highest specific energy calculated for this design was 60.8

W·h/kg. Curing temperature changes and manufacturing method effects on the inherent

stresses of the rotor were not included in the analysis.

In 1984, Genta proposed a different design technique based on the 2-D plane stress

analysis that included stiffness and density tuning [18]. In addition to requiring a constant

hoop stress profile to maximize energy storage, Genta also chose to impose a zero radial

stress condition and a constant hoop-radial Poisson’s ratio throughout the flywheel rotor.

Based on these assumptions, equations for the hoop stiffness Eθ and composite density ρ

became functions of the radial location in the disk. He also noted that the radius-dependent

25

Figure 2.5: Flywheel design by the AVCO Corporation [16]. Left is a drawing of thecontinuous bidirectionally woven ply. Right is a drawing of the assembled flywheel with anylon hub.

axial thickness h(r) and radial stiffness Er fell out of the equations and were not required

to define the system. Also, a minimum inner-to-outer radius ratio λ for the flywheel ring

could be analytically determined. For his numerical discrete ring stress analysis, the author

chose to use a λ ratio of 0.4. Genta proposed that, based on this analysis, an optimum

design could be achieved by continuously varying the fiber content as a function of radius

to tune hoop stiffness as required. In order to continuously change the density with radial

location, Genta proposed incorporating a ballasting material such as lead particles into the

polymer matrix. After this optimization, the author noted that the axial thickness could

be varied separately to further optimize the design in the case that the condition of zero

radial stresses could not be completely met. If this condition could be met, the axial shape

profile would be irrelevant. Although he did not produce specific energy numbers, he did

propose that, for the case of approximated steps for ballasting and tuning the axial profile,

a shape factor of 0.365 could be achieved relative to an isotropic constant stress optimal

design. He admitted that the comparison was a stretch considering the material properties

were not constant for his design. No other optimal design information was given. For this

26

Figure 2.6: A sample of two of the flywheel designs investigated by Ikegami et al. [17].

analysis, no effects relating to induced stresses from temperature changes due to curing or

manufacturing methods were included.

Miyata proposed that, using a 2-D plane stress analysis that incorporated an axial

thickness parameter, it was possible to develop a solution that required the radial and

circumferential elastic moduli to be tuned so the stresses could match the strength limits

of the rotor [19]. The optimization parameter he used was energy density; however, the

volume included the vacant space lying within the hole of the flywheel ring. He proposed to

accomplish this tuning by placing one hoop fiber ply of variable thickness on top of a radial

fiber ply of complementary thickness so that when the plies come together, they produce a

layer that has a constant thickness. This is shown in Figure 2.7.

Carbon/epoxy composite material was used. For the numerical calculations, a λ value

of 0.2 was used. The failure criterion was maximum stress. Other than estimated shape

27

Figure 2.7: Ring disks of complementary varying thickness corresponding to radial andhoop stress distributions proposed by Miyata [19].

factors, no definitive performance data was given. For this analysis, no effects relating to

induced stresses from temperature changes due to curing or manufacturing methods were

included.

In 1988, Portnov and Kustova attempted to optimize chorded flywheel designs with

parametric studies [20]. The parameters chosen to be varied were the pole radius rp, of

which the chords were wrapped around, and the ratio of chord axial thickness hc to the axial

thickness of the filament-wound ring H. This is shown in Figure 2.8. The failure criterion

was maximum stress. The ring material was glass/epoxy, and the chord material was

carbon/epoxy. With these parameters selected, the filament-wound ring radial thickness was

tuned to produce an optimal design that corresponded to the flywheel failing simultaneously

due to radial stresses in the ring and the shear/tensile stresses acting on the chord fibers.

With this parametric study, the authors observed that as the ratio hc/H approaches unity,

the energy density reaches a maximum. This observation also corresponds to having a ring

with a radial-to-hoop fiber ratio of unity. Also, a maximum energy density was achieved by

reducing the pole radius rp to zero. This achievement corresponds to orienting the chord

28

fibers as close to the radial direction as possible. The authors’ parametric study achieved a

maximum energy density of 9.44 kW·h/m3 for the case where rp = 0.1 and hc/H = 1. No

other performance information was given. For this analysis, no effects relating to induced

stresses from temperature changes due to curing or manufacturing methods were included.

Figure 2.8: Schematic of chord winding around a filament-wound composite ring. The poleradius is denoted rp, inner ring radius a, outer ring radius b, chord thickness hc, and ringthickness is denoted H [20].

2.2 Research Work: 1990s

In 1995, Curtiss et al. performed a case study analysis of a flywheel energy storage system

that included composite flywheel optimization for a single ring containing electrical current

conducting materials in the ring for pulsed power applications [21]. For the single ring

design, the authors use the formulation derived by Danfelt et al. of a 2-D plane stress

assumption and a maximum stress failure criterion [15]. In their formulation, the authors

29

set λ to 0.5. Optimization was performed by tuning the ratio of hoop stiffness to radial

stiffness Eθ/Er. To be able to physically tune this parameter, the authors stacked different

plies of radial and hoop fiber orientations. This design was similar to Miyata’s as shown

in Figure 2.7 except that the ply thickness was constant throughout the radial thickness.

Carbon/epoxy composite material was used in the ring. Once the composite design was

optimized, the design was compared to that of isotropic materials such as high-strength

titanium and steel alloys. For this comparison, an aluminum conductor was placed in each of

the options as a uniform, zero-stiffness parasitic mass. For this analysis, the composite ring

was shown to be superior. However, the authors determined that a conductive aluminum

sheet was inferior to a copper Litz wire ply for electrical high-current applications. Since

the wires must be placed in some sort of non-conductive matrix, the composite ring was

produced. The optimized design incorporated a hoop/radial ply thickness ratio of 0.904

with a copper wire/composite ply thickness ratio of 0.23. This design analysis indicated

that a tip speed of 650 m/s can be achieved for a specific energy of 36.7 W·h/kg. For this

analysis, no effects relating to induced stresses from temperature changes due to curing

or manufacturing methods were included. Also, no hub stresses were incorporated into

the analysis. However, spin tests to failure were conducted on composite rings that had

fiber orientations that approximated the analytic designs. In spin testing, the first flywheel

failed by hub-ring debonding. The second flywheel failed from burst. In both cases, failure

indicated that analytic design limits could be attained.

In 1997, Gabrys and Bakis published a case study in design analysis and testing of

composite flywheel rings [22]. Equations for stresses were obtained from Genta [13] and

Lekhnitskii [23]. Equations given in Genta pertain to a 2-D plane stress analysis [13]. A

maximum stress failure criterion was used. These equations were used to analyze the three

flywheel designs given in the article. The first design included elastomeric interlayers to

minimize radial stress transmission between selected composite rings. For the interfaces

that do not have elastomeric urethane interfaces, the strategy was to place a ring with a

30

lower hoop stiffness/density ratio on the inside of a ring with a higher hoop stiffness/density

ratio. This is shown in Figure 2.9. The ring thickness selection and placement methodology

centered on the fact that the high-strength and stiffness T1000 Carbon fiber should be used

only where appropriate due to its high cost. The manufacturing methodology was discussed

in detail. A press-fitting technique was used to compress the rings in the radial direction

and counter the tensile radial stresses developed during centrifugal loading. The analysis

included residual stresses from both the press-fitting and processing temperature changes.

Figure 2.9: Multi-ring flywheel design that was analyzed and tested by Gabrys and Bakis[22].

Analysis of this first flywheel indicated that the failure would occur due to radial stresses

in the outermost ring. Theoretical analysis estimated the maximum ring speed to be 1715

m/s for a specific energy of 204 W·h/kg. Spin tests were unable to achieve sufficient speed

for failure. The first test failed due to mechanical separation of the flywheel hub and the

spin arbor. The second test was limited by the inability to achieve a sufficient vacuum in

the test rig. Air drag losses prevented the achievement of tip speeds beyond 1101 m/s.

The second design included an aluminum hub with a filament-wound S-2 glass fiber

composite thick ring with an elastomeric urethane matrix. The third design was identical

to the second except that the fiber used was AS4C carbon fiber. The advantage of the two

elastomeric matrix flywheels was that the matrix was highly compliant to the radial stress,

31

allowing for radial expansion as required during loading. When such a matrix material

was used, very thick rings could be manufactured for high centrifugal loading. Theoretical

analysis predicted a maximum tip speed of the second flywheel as 1025 m/s for a specific

energy of 74 W·hr/kg. For the third flywheel, a maximum tip speed of 1265 m/s was

predicted for a specific energy of 113 W·h/kg. For the experimental analysis, only the

second flywheel was tested. This flywheel had resonance vibration problems that were

being analyzed, but could not be resolved at the time of publishing.

In 1998, Ha et al. published a strictly theoretical optimization analysis that included

a 2-D plane strain formulation instead of plane stress [24]. The assumption corresponded

to a thick flywheel with multiple filament-wound rings and included press-fitting the rings

together. Press-fits were modeled using ring interface interference displacements. All ten

ring thicknesses were kept constant at 5 mm and the press-fit interference between the

rings were the variables to be optimized. Another set of variables the authors attempted to

optimize was the axial/hoop filament winding angles. Shear stresses and strains associated

with axial/hoop interactions are shown in Equation 2.2.

σθσrσzτθz

=

Qθθ Qθr Qθz Qθ5Qrθ Qrr Qrz Qr5Qzθ Qzr Qzz Qz5Q5θ Q5r Q5z Q55

εθεr

εz = 0γθz = 0

(2.2)

where:γ: Shear strainτ : Shear stress

Subscripts:z: Axial coordinate5: θz shear component

As can be seen, the plane strain condition of zero axial strain was applied. Also, the θz

shear stresses were incorporated into the analysis since the ply angles for each ring could

vary in the θz plane. The 3-D Tsai-Wu failure criterion was used to determine stress limits

32

[25]. For this analysis, no effects relating to induced stresses from temperature changes

due to curing were included. The optimization strategy included the use of a sequential

linear programming method to search the solution space and perform sensitivity studies.

Three case studies were performed. The first was the baseline case where the ply angles

were set to the hoop direction only and the interferences were set to zero. The second

case set the interferences to zero, but allowed the ply angles of the rings to vary freely.

The third case allowed the interferences to vary freely, but all ply angles were set to the

hoop direction only. The second case showed that a 29% improvement in specific energy

could be obtained over the baseline case of 29.3 W·h/kg by appropriately varying the θz

ply angles. The advantage comes from tuning the inner rings with large ply angles to

incur higher axial stresses, allowing more expansion in the radial direction, and thus lower

radial stresses. The third case showed that a 145% improvement in specific energy could

be obtained by varying the interferences between the rings. The third case produced an

optimum specific energy of 72.0 W·h/kg and a tip speed of 910 m/s. Sensitivity studies

show that optimizing interferences is substantially more important than ply angles. For this

analysis, no effects relating to induced stresses from temperature changes due to curing or

manufacturing methods were included.

In 1999, another paper by Ha et al. investigated the requirement of placing permanent

magnets along the inner radius of the flywheel to incorporate a motor/generator function in

the rotor. This magnet material placed a compressive stress condition along the inner radius

of the multi-ring rotor [26]. This strictly theoretical analysis used a plane strain assumption

with fibers wound in the hoop direction only, producing the stress-strain relations given in

Equation 2.3.

σθσrσz

=

Qθθ Qθr QθzQrθ Qrr QrzQzθ Qzr Qzz

εθεr

εz = 0

(2.3)

The optimization parameters for this flywheel setup were the material choices for each

33

ring, the inner and outer radii of the permanent magnet material, and the composite ring

thicknesses. The criteria used for acceptability were the Tsai-Wu 3-D stress failure criterion

and the minimum rotor voltage requirements generated by the magnets [25]. The minimum

voltage requirement criterion added a constraint on the geometry of the permanent magnet.

The compressive stresses of the permanent magnet on the composite ring during loading

constrained the outer radius of the flywheel ring. Overall, the inner and outer radii were

variables, but were constrained by the magnetic material conditions. The parameter for

optimization was the total stored energy of the flywheel. The authors started by investi-

gating the material sequence for the composite rings. The five composite materials selected

used glass, aramid, and three grades of carbon fiber as the choices for the rings. After

investigating the possible permutations, the authors concluded that having materials that

increase the stiffness/density ratio as the radius was increased provided the optimum se-

quence. After the sequence investigation, the authors chose to investigate magnet radii and

ring thicknesses in four sections. Section 1 consisted of three tests where only one compos-

ite ring was analyzed. Section 2 contained four cases where two rings were used unless the

program saw fit to impose a zero thickness on one of the rings. This scenario occurred in

one of these cases. Section 3 contained six cases where three rings were tested with two

cases where one of the rings was eliminated and one case where two rings were eliminated.

Section 4 contained five cases where four rings were permuted, with two cases of single ring

elimination and one case of double elimination. The best design found included four rings

in the sequence of glass, aramid, low-grade carbon, and high-grade carbon fiber. It was

predicted to attain a maximum total stored energy of 2010 W·h for a design with an inner

magnet radius of 29.6 mm and an outer radius of 151 mm. This result would indicate that

the more rings the flywheel contains, with the proper order, the higher the total stored en-

ergy can be. Also, the cases that had one or more rings eliminated occurred due to sequence

violation of the stiffness/density ratio observation. Usually, a higher stiffness/density ratio

fiber would be placed in an inner ring and would eliminate any lower stiffness/density ratio

34

ring outside of it. For this analysis, no effects relating to induced stresses from temperature

changes due to curing or manufacturing methods were included.

At roughly the same time, Eby et al. used the axial thickness parameter h(r) of isotropic

material flywheels as an optimization problem that could be used to test two types of ge-

netic algorithms [27]. First, a simple Genetic Algorithm (sGA) was used to determine the

optimum shape using one isotropic material with a plane stress assumption. This result

matched that of the well-known constant-stress profile. Next, the authors used this algo-

rithm to recalculate the design with a homogeneous outer boundary condition to eliminate

the requirement of a force on the outer radius of the flywheel. After this, an optimized

isotropic ring was determined as well given a set inner radius to provide results for a later

ring optimization test. These 2-D plane stress results were next used as the starting search

basis for more complex 3-D modeling. With the sGA determined valid for the plane stress

analysis, the authors chose to mix and match sGA with a modified Simulated Annealing

(SA) approach and an island injection Genetic Algorithm (iiGA). In the results of using

these algorithms separately or in conjunction with each other, the hybrid approach of using

SA along with a local search iiGA was proven to be the fastest. In this approach, the plane

stress sGA results were incorporated into a 3-D Finite Element Modeling (FEM) package

for higher refinement. Last, the problem complexity was increased with a higher number

of rings and multiple material selections. Also, a Parallel Genetic Algorithm (PGA) was

tested to compare to the various types of iiGA concepts tested previously. This test showed

that both the standalone and hybrid iiGAs produced superior results in a fraction of the

computational time compared to the PGAs. Because the flywheel optimization was not

the focus of the research, optimum specific energy results were not given. Also, residual

stresses due to processing temperatures or manufacturing methods were not included in this

analysis.

35

2.3 Research Work: 2000s

In 2000, an article was published by Emerson and Bakis that presented concerns about

press-fit relaxation by producing a viscoelastic formulation and experimental results [28].

The formulation included a linear viscoelastic model with plane stress assumptions and

constant centrifugal loading. Residual stresses due to press-fitting were included; however,

thermal processing effects were not. Multiple flywheel rotor designs were investigated and

included three material options with aluminum, glass/epoxy, and carbon/epoxy composites.

Flywheels designs also included both high-pressure and low-pressure press-fitting for a total

of six designs. When compared to experimental results, the models were very close for

the composite-only designs. However, the designs that included an aluminum hub were

significantly under-predicting the experimental results. After hundreds of hours of testing,

pressure losses of 3-5% were measured.

In 2001, Tzeng investigated various mechanical concerns for a press-fit rotor design

subject to centrifugal and electromagnetic loading [29]. The design consisted of outer and

inner composite rings with an aluminum coil cylinder set radially between the two composite

rings. Centrifugal and electromagnetic loadings were analyzed to determine the value of

the press-fitting. The author discussed stress relaxation of the press-fit preloading over

time. The author displayed test results of press-fit relaxation at elevated temperatures

for an extend period of time. The elevated temperatures used were 93 and 75 ◦C. Last,

the author discussed how compressive stresses resulting from press-fitting can increase the

fatigue failure cycle life from shear stresses generated from the electromagnetic loading.

In another article by Ha et al., an optimum flywheel design was investigated using a Mod-

ified Generalized Plane Strain (MGPS) assumption [30]. This assumption was compared to

a plane stress assumption and 3-D FEM solutions. Also, this article incorporated residual

stresses associated with processing temperature changes. The MGPS is an expansion of the

plane strain assumption. The plane strain assumption sets the axial strains to zero. The

Generalized Plane Strain (GPS) assumption sets the axial strain to a constant as follows:

36

εz = ε0. The MGPS assumption incorporates a radially-dependent linear variation to the

axial strain as follows: εz = ε0 + ε1r. The constants were solved by requiring the axial force

and moment resultant to vanish. The limiting design was determined by the Tsia-Wu failure

criterion [25]. When compared to the 3-D FEM and plane stress assumptions, the MGPS

assumption was determined to be more accurate compared to the plane stress assumption

for axial stresses and strength ratios. The higher accuracy of the axial stress distribution

translates into a higher peak axial stress for the MGPS assumption. However, the plane

stress assumption is more conservative in determining limiting strengths. These results were

consistent in the cases of processing temperature changes that were 0 and −100 ◦C. Once

the accuracy of the MGPS assumption was confirmed, it was used to perform a case study

analysis similar to the previous paper by Ha et al. on multi-ring hoop-wound rotors [26].

However, contrary to the previous article, there was no permanent magnet rotor segment

to account for. The three material rings given were a glass fiber composite, a low-grade

carbon fiber composite, and a high-grade carbon fiber composite. The inner radius was

set to 50 mm, and the spin speed was set to 60,000 rpm. The only free variables were the

three thicknesses of the rings. To solve the optimization problem, the ”modified method

of feasible directions for constrained minimization” strategy was used with total stored en-

ergy as the objective. The best case presented with a zero processing temperature included

all three rings in a sequence relating to an increasing stiffness/density ratio as the radius

increased. The total stored energy of this case was 1141 W·h with a tip speed of 1070

m/s. For the case of residual stresses originating from a processing temperature change of

−100 ◦C, the maximum total stored energy was 794 W·h with a tip speed of 975 m/s. This

corresponded to a 30.4% reduction in energy storage capability and proves that, in certain

cases, the processing temperature changes can decrease flywheel performance significantly.

In 2002, Gowayed et al. performed a flywheel optimization design analysis for a single

ring, hub, and rotor mount [31]. The novelty of this analysis lies in the fact that both the

hub and the ring contained fibers oriented in the hoop and radial directions for a multi-

37

directional composite. Much like Miyata [19], radial and hoop plies ware stacked axially to

produce the composite. Only one ring was analyzed and the ply thicknesses were constant

throughout the radial thickness of the ring, which is shown in Figure 2.10. Two analysis

methods were used. The first included a plane stress assumption, and the second included

a FEM tool. For both cases, the Tsai-Hill failure criterion was used. The tool used to

perform the 3-D stress analysis was a package called ANSYSTM. The flywheel hub consisted

of an S-2 glass fiber/epoxy composite. The ring was constructed of a T1000 carbon/epoxy

composite. For all cases, the ring outer diameter and axial thickness were set. The variables

for all the cases were the radial location of the interface between the ring and hub, the hub

axial thickness, and the radial/hoop ply thickness ratios for both the hub and ring. The

optimization was performed based on a sequential quadratic programming procedure. For

the case that included the rotor mount, hub, and ring; a tip speed of 1211 m/s was achieved

for a specific energy of 93 W·h/kg. No residual stresses concerning processing temperature

changes or manufacturing methods were factored into this analysis.

Figure 2.10: Multi-direction composite flywheel design by Gowayed et al. [31].

38

In 2003, Emerson and Bakis published a continuation of their previous work concerning

press-fit relaxation [32]. This model is the same as previously stated [28], except that

it includes time-varying loads and residual stresses from temperature changes. Analysis

included a design without press-fitting and constant loading after manufacturing, a press-fit

design with discrete step loading and temperature changes, and a press-fit design with cyclic

loading and temperature changes. Based on this analysis, the authors predicted a decrease

in energy storage capacity of roughly 4% after 10 years, suggesting minimal viscoelastic

performance degradation effects. Nonetheless, changes in the radial strains were predicted

to be large.

In 2004, Corbin et al. presented a case study in flywheel design, manufacturing, and

testing [33]. The stress analysis tool for both the 2-D and 3-D modeling was ANSYSTM. The

flywheel used a multi-ring design with a steel shaft as the base; an elastomeric polyurethane

hub; and three rings consisting of an E-glass fiber, a low-grade carbon fiber, and a high-

grade carbon fiber. The Tsai-Wu failure criterion was used to determine flywheel limits

[25]. The flywheel had been optimized, but the method for performing this optimization

was not given. The manufacturing of the three composite rings was performed with filament

winding. The liquid polyurethane hub was cast around the steel shaft and allowed to cure.

To fit the hub and shaft onto the composite ring, the hub/shaft assembly was cooled in

a freezer and the large coefficient of thermal expansion of the polyurethane allowed the

assembly to shrink. During testing of the flywheel design, two failure modes were seen.

The first observed failure came from the hub debonding from the steel shaft. The second

occurred in the composite rings, indicating radial stress failure. The authors note that they

were not able to adequately predict the radial strengths due to uncertainties such as curing

temperature, fiber volume content, and filament winding tension. Analysis estimated the

maximum angular speed to be 7120 rad/s for a total stored energy of 30.5 W·h. No other

geometric or performance data were given.

Also in 2004, Portnov et al. performed both a theoretical formulation and numerical

39

calculations to investigate of the behavior of elastomeric interlayers [34]. In this paper, the

authors provided a formulation for the stress state of an interlayer by assuming a plane stress

state for the composite inner and outer layers, and a plane strain state for the interior regions

of the elastomeric interlayer. Formulations were generated for rotation about the axis of

symmetry, translation in the plane normal to the axis of symmetry, and rotation outside the

plane of symmetry. The latter two loading states are shown in Figure 2.11. Concerning the

symmetrical axis rotation, the proposed method of analysis was compared to typical plane

stress analysis and 3-D axisymmetric FEM analysis. The proposed formulation was shown

to be valid for radially thin interlayers at points distant from the axial ends, whereas the

plane stress formulation was more accurate as the interlayer became thicker. The proposed

formulation deviated from the FEM results in regions near the axial ends of the flywheel.

Last, it was noted that it was possible to eliminate the radial stresses imposed on the inner

edge of the outer ring by tuning the thickness of the elastomeric interlayer. No residual

stresses relating to manufacturing methods were addressed. Because the purpose of this

article was to test the accuracy of a formulation and understand elastomeric interlayer

behavior, no specific design or flywheel performance information was given.

Another paper published by Portnov et al. in 2005 discusses the prospect of varying

the filament winding tension to produce an appropriately pre-stressed flywheel that could

compete with press-fit pre-stressing [35]. Both methods generate compressive radial stresses

in the flywheel to counter the tensile radial stresses produced by centrifugal loading. The

authors constructed a formulation of plane strain similar to that by Ha et al. [30] and

used literature references and numerical calculations to verify its validity for a hybrid steel

and fiber/epoxy composite flywheel. This formulation included residual stresses relating

to processing temperature changes. It was noted that for compound cylinders using the

plane stress assumption, the radial stresses were higher and hoop stresses were lower com-

pared to the plane strain assumption. Radial stresses were 50 to 60% higher for the plane

stress cases. After this analysis, the authors move on to display the formulation for vari-

40

Figure 2.11: Cases investigated by Portnov et al. with (a) showing in-plane translation and(b) showing out-of-plane rotation [34].

able filament-winding tensioning. This formulation was based on a linear elastic material

model which completely neglected any viscoelastic effects during the filament winding pro-

cess. This formulation also neglected axial stresses and assumed there were no non-circular

deformations in the filament winding process, such as what would arise from gravitational

body forces. The authors cite a single source in the literature concerning experimental evi-

dence of the retention of linear elastic behavior being maintained throughout and after the

curing process. However, the statistical distribution for these experimental data points was

quite large. The authors further discussed the potential viability of their variable tensioning

model, especially with in situ curing, but did not provide their own experimental evidence

in this paper.

In 2006, Arvin and Bakis published a paper that investigated optimization of filament-

wound multi-ring composite flywheels that had been press-fitted together and possessed an

attached interior permanent magnet ring [36]. The formulation used was referenced from

41

Gabrys and Bakis [22], which obtained plane stress solutions from Genta and Lekhniskii

[13, 23]. The authors also incorporated residual stresses relating to processing temperature

changes. These residual stresses were tested for failures in the case of an unloaded situation

for each ring before press-fitting the rings together and in the cases of both unloaded

and centrifugally loaded situations for the entire rotor after the press-fitting procedure.

The failure criterion used was Tsai-Wu [25]. For the optimization, they used a simulated

annealing algorithm. For all cases, the three composite materials available for use were three

grades of carbon fiber. This optimization method was first tested without respect to stresses

in the individual rings prior to press-fitting. The second test set did incorporate the residual

stresses in the rings prior to press-fitting. Once the simulated annealing tests were validated

across multiple runs, the second test set incorporated a matrix of scenarios. The scenarios

included cases for four, five, six, seven, and eight rings; and stress free temperatures of 112

and 176 ◦C. A total of 10 cases were evaluated. Results showed how calculating residual

stresses for all the rings manufactured before press-fitting and the completed ring after

press-fitting can significantly lower the specific energy of the optimal design. The results

listed a potential drop of 40% from 48 to 28 W·h/kg. Results also indicated that lowering

the processing temperature reduced the number of rings needed to achieve a set specific

energy. However, as the number of rings is increased, both processing temperature cases

appeared to move toward the same specific energy upper limit. Increasing the number of

rings can compensate for higher processing temperatures. This increase would be expected

as thinner rings have lower residual stresses. Also, the number of press-fit interferences and

ring thickness parameters allow the flywheel ring to be more finely tuned and optimized.

The best case of 112 ◦C processing temperature with 8 rings, the highest specific energy

achieved was roughly 50 W·h/kg with a tip speed of 860 m/s. However, the authors felt

that a more reasonably cost-effective solution was with 5 rings for a specific energy of 46

W·h/kg and a tip speed of 840 m/s.

In 2007, Fabien performed an investigation concerning using different failure criteria

42

and a single ring that contained both hoop and radial plies stacked axially [37]. This design

optimization was highly similar to Miyata [19] except that instead of varying the radial

and hoop ply thickness, Fabien chose to vary the ply angle of the radial fiber plies. Fabien

selected a 2-D plane stress formulation. Because the ply angle was allowed to vary, the

in-plane shear stress needed to be taken into account. This is shown in Equation 2.4.

σθσrτθr

=

Qθθ Qθr Qθ6Qrθ Qrr Qr6Q6θ Q6r Q66

εθεrγθr

(2.4)

where:6: θr shear component

The alteration in the ply angle for the radial plies is shown in Figure 2.12. It is again

noted that the ply angle alteration concerns the radial plies only, with the hoop plies having

all the fibers oriented strictly in the hoop direction. For the analysis, the radial ply was

discretized into a series of small rings stacked together. Each thin ring has its own ply

angle and corresponding stiffness matrix. For this analysis, three different failure criteria

were used to compare to each other: maximum stress, maximum strain, and Tsai-Wu [25].

For all cases, the inner radius was set to 25.4 mm and the outer radius to 168.9 mm. Four

designs were generated for comparison of the three failure criteria. The first design set all

the radial fibers strictly in the radial direction and served as a benchmark for the optimized

designs. The second, third, and fourth designs were optimized based on each of the three

failure criteria. The optimization method for these three designs included using sequential

quadratic programming.

For the results, all four designs were tested against each of the failure criteria for specific

energy. Although the maximum stress and strain cases possessed different design results,

they produced identical specific energy results. The result for these cases was 168 W·h/kg.

The most conservative case was based using the Tsai-Wu failure criterion for a specific

energy of 156 W·h/kg. No velocity information was given in this article. The author

43

Figure 2.12: Example of varying the ply angle for the radial plies of the single ring flywheel[37]. Hoop plies are not shown.

acknowledges the manufacturing difficulty entailed with trying to vary the radial ply angle

in a nearly continuous manner. For this analysis, residual stresses relating to manufacturing

were not included.

Also in the same year, Strasik et al. published results on constructing a complete

flywheel energy storage system [38]. For this experiment, the flywheel rotor consisted of

four carbon/epoxy composite rings with three elastomeric interlayers. The entire system

had an energy storage capacity of 5 kW·h. The hub and rotor weighed 164 kg for a specific

energy of 30.5 W·h/kg with the hub included. No geometrical information was included,

but angular speeds of 2480 rad/s were experimentally achieved. This flywheel system also

included a high-temperature superconducting bearing to levitate the flywheel.

In 2008, Arslan analyzed to isotropic flywheel designs by performing FEM analysis on

44

variations of the iso-stress disk in order to explore other geometries [39]. Cases investigated

vary the geometry of an AISI 1006 steel alloy disk. Geometries included 6 cases with a con-

stant thickness disk, constant thickness ring, parabolic tapering, linear tapering, truncated

iso-stress exponential decay tapering, and a modified iso-stress case. The last case truncates

the iso-stress case, however, it places a small torus at the end of the taper to increase the

specific energy and develop a radial stress at the end of the taper. As expected, the constant

thickness ring was the worst performer. The highest performer was the modified iso-stress

profile with a specific energy of 8.977 W·h/kg, which was 48.7% higher than the constant

thickness disk.

In the same year, Ha et al. investigated how geometric scaling of flywheels affects

the energy storage capability, press-fit interferences, and rotational speed [40]. For all

cases, the curing temperature remained the same. A plane stress formulation was used.

The geometric scaling was made such that the specific energy was kept constant. Two

flywheels were investigated. The first flywheel was a double ring that had an inner layer

of a glass/epoxy composite and an outer layer of carbon/epoxy composite. The second

flywheel consisted of four composite rings: two inner rings made of glass/epoxy, two outer

rings made of carbon/epoxy. The double ring held a specific energy 64.8 W·h/kg with a

tip speed of 890 m/s, and the quadruple ring had 81.2 W·h/kg with a tip speed of 1013

m/s. The authors developed special scaling factors that increased to the cubic power of the

energy storage capacity. Press-fits were determined to scale linearly, and rotational speed

scaled inversely with the factors. Two flywheels of the double ring design were made to

report the manufacturing requirements of these designs.

Also in the same year, Ha et al. published an analysis similar to that of Arvin and Bakis

[36] concerning the residual stresses due to processing temperature changes and stress states

before and after press-fitting but before centrifugal loading [41]. A plane stress formulation

was used as was the the Tsai-Wu failure criterion [25]. For all analyzed cases, the goal was

to maximize the specific energy of the flywheels. Also, the inner radius constraint was set to

45

100 mm with the outer radius allowed to vary to maximize specific energy, but a maximum

angular speed limit was set to 30 krpm. For each design, two processing temperature change

cases of 0 and −50 ◦C were analyzed. Multiple cases were analyzed with up to five rings

used with the number of glass and carbon/epoxy composite rings varied. In all cases, the

glass/epoxy composite rings were placed inside the carbon/epoxy composite rings. For all

cases, the processing temperature change of −50 ◦C was only slightly inferior to the 0 ◦C

change. This change was displayed for the optimal case of two glass/epoxy inner rings and

three outer carbon/epoxy rings. The 0 ◦C case gave a specific energy of 131.0 W·h/kg with

a tip speed of 1372 m/s, and the −50 ◦C case gave a specific energy of 130.7 W·h/kg with

a tip speed of 1370 m/s. After this, the authors discuss the press-fitting procedure for two

flywheels of 0.5 and 5.0 kW·h.

2.4 Research Work: 2010 To The Present

In 2010, Krack et al. published an investigation into energy/cost optimization of a rotor

with two material rings [42]. The inner ring material was set to a glass/epoxy composite

with a carbon/epoxy composite for the outer ring. The inner radius of the rotor was set to

120 mm and the outer radius was set to 240 mm. A press-fit interference of 0.5 mm was set

in between the material rings. The only two variables in this optimization search were the

middle radius of the interface between the two material rings and the rotational speed. This

highly constrained problem included a plane stress formulation highly similar to Ha et al.

[24], which was verified by the FEM package ANSYSTM. The maximum stress, maximum

strain, and Tsai-Wu failure criteria were used individually and separately. To analyze the

cost of the flywheel rotor, a cost ratio between the carbon/epoxy and glass/epoxy was

created. This ratio was multiplied by the mass of the carbon material as a weighting

factor. This ratio was manually varied from zero to an arbitrarily high number such that

optimized behavior could be observed as this ratio is changed. For the optimization, the

Coliny evolutionary algorithm was employed along with a nonlinear interior-point method

46

to produce a more robust optimization search. It is important to note that only the material

cost of the rotor was analyzed. Manufacturing costs for the rotor as well as material and

manufacturing costs for the auxiliary flywheel system components were not included. The

authors were able to show that there were step jumps/drops in the optimized solutions as

this ratio was changed, with four distinct regions found. The author claim that highest

specific energy for the best cost ratio range was 2.48 kW·h/kg with a total stored energy of

3.46kW·h and angular speed of 804 rad/s. Because the specific energy was roughly a factor

of ten higher than that found in other research results, it is considered to be in error.

In the same year, Lin et al. published an article proposing that incorporating carbon

nanotubes into the epoxy matrix of a carbon/epoxy composite could increase the per-

formance of flywheels [43]. After a basic theoretical analysis was performed for various

materials, the authors used FEM analysis to show that the stress distribution of a centrifu-

gally loaded flywheel ring can benefit from incorporating carbon nanotubes in a functional

gradient. This gradient would serve to stiffen the matrix where it was needed most. The

authors did not provide any further theoretical analysis of how significant the performance

improvement would be. The authors also did not explain how such a gradient could be

physically produced.

In 2011, Prez-Aparicio and Ripoll produced a theoretical formulation that included

physical effects such as residual stresses from temperature and moisture changes as well as

shear stresses from rotational acceleration [44]. The full accounting of these effects is shown

in Equation 2.5.

εθεrεzγθr

=

Sθθ Sθr Sθz 0Srθ Srr Srz 0Szθ Szr Szz 00 0 0 S66

σθσrσzτθr

+

αθαrαz0

∆T +

βθβrβz0

∆C (2.5)

where:Sij : Element of the compliance matrix∆T : Temperature change

47

∆C: Moisture changeα: Coefficient of thermal expansionβ: Coefficient of moisture expansion

These relations serve as a basis for both the plane stress and plane strain formulation,

where the simplifying assumptions were incorporated later into the analysis as numerical

factors that could be changed to suit the assumptions without needing to change the under-

lying equations. The article discussed the need for factoring in shear stresses to account for

energy transfer. The article extended the discussion into the benefits of producing a gra-

dient cure temperature and moisture saturation in order to improve flywheel performance.

No manufacturing details of how these benefits could be achieved were discussed. Also there

was no accounting for the fact that high speed composite flywheels need to operate in a

vacuum, where no moisture is present. Last, hub effects were lightly investigated, displaying

the benefits of using an elastomeric urethane hub to mitigate interface stresses.

Also in 2011, Krack et al. expanded on their previous research to include a hub in

their cost optimization analysis [45]. The included hub was of a split-type design previously

investigated by Ha et al. [46]. This hub design is shown in Figure 2.13. In this analysis, the

authors compared and contrasted the performance of combining the analytic and numerical

calculations using different methodologies to minimize computational time. The authors

determined that the surrogate strategy tested was the most efficacious. Once this method

had been determined, four additional analyses were performed, which were divided into two

sets. The first set included an optimization search that was nearly identical to that of the

2010 article except that the cost ratio was set to 4, no press-fit interferences were used, and a

solid hub was included with a set thickness of 10 mm. Both the split type and solid ring hub

designs were compared with that of a no-hub scenario. The peak energy/cost ratio for the

solid ring hub was 80.3% of the no-hub scenario, and the peak ratio for the split-type hub

was 99.7%. Two additional cases were made where the hub thickness was allowed to vary.

When this variation was allowed, the solid ring case produced a vanishing hub thickness,

48

and the split-type hub produced a hub thickness of 3.80 mm for an energy/cost ratio that

was 103.7% of the no-hub scenario, indicating a slight improvement when using this hub

design. It is important to note that the energy/cost analysis included neither the added

kinetic energy nor the manufacturing/material cost of the hub. The purpose of the hub was

strictly to change the stress distribution of the rotor, thus changing the energy/cost of the

rotor itself.

Figure 2.13: Split-type hub design first investigated by Ha et al. [46], then by Krack et al.[45].

In 2012, Ha et al. presented research concerning their investigation of combining fiber

co-mingling, where two or more fibers are used at the same time for one material ring,

and press-fitting [47]. The three cases that were presented are shown in Figure 2.14. As

is shown, the innermost material rings of the cases have mainly glass fibers, transitioning

to mostly carbon fiber content in the outermost ring. In these cases, the inner and outer

radii of the rotors were fixed at 0.27 and 0.45 m. The three radii in between the inner and

outer rotor radii were variables in an optimization search. The carbon/glass fiber ratios of

the four material rings were also variables in the optimization search. Last, for the cases

that include press-fitting, the interferences were also variables. In the stress analysis, a

49

plane stress assumption was used with the Tsai-Wu failure criterion. The procedure for the

stress analysis was first to calculate the residual stresses for the cured rings. If there was

press-fitting, then the rings were analyzed according to the press-fit partitions individually.

The rings were then analyzed at each point in the press-fitting procedure where they must

be able to exist without assistance. Once the rotor was fully assembled while surviving

the stationary residual stress associated with manufacturing, then it was analyzed with

spin-loading. The parameters to be optimized were the strength ratios Rr = σr/YT and

Rθ = σθ/XT , given a set angular speed of ωmax = 15 krpm. From this analysis, the authors

believed that while Case B with three press-fit interferences produced the best design, Case

C, with only one interference, was the most cost effective.

Figure 2.14: Three rotor design cases with four material rings of co-mingled fiber withcarbon and glass tows, and press fitting in two of the cases [47].

In 2013, van Rensburg et al. reviewed three previous articles from the standpoint of

the shape factor in order to further optimize them [48]. The authors reviewed in detail the

shape factor concept given by Genta [13] and applied it to three articles concerning flywheel

50

rotor optimization. These articles were Krack et al. [42], Arvin and Bakis [36], and Ha et

al. [30]. The authors believed that using the shape factor in a multi-ring composite rotor

could help in increasing the specific energy of the rotor. The shape factor for these articles

were calculated to be 0.39, 0.22, and 0.18; all of which were much lower that the theoretical

limit of 0.5 for a composite hoop-wound rotor. The authors then attempted to reanalyze the

rotors according to this criterion. Only two-ring rotors were analyzed with the plane stress

assumption and the Tsai-Wu failure criterion. It is not clear what optimization method was

used although particle swarm and genetic algorithms were mentioned. The only variables

in the optimization were the overall thickness ratio of the rotor λ, the radial locations of the

interfaces for the material rings within the rotor, and the angular speed. The authors showed

improvements in all three cases; however, the authors did not analyze the rotor in a manner

similar to that of the previous authors. Krack et al. performed a cost optimization, not a

specific energy or energy density optimization [42]. Arvin and Bakis analyzed their rotor

with respect to inner magnetic material for power transmission, which was not included in

this analysis [36]. Last, Ha et al. chose to optimize on total stored energy. In performing

this analysis, the authors believed that enclosed volume energy density was a more preferred

optimization objective than specific energy.

2.5 Summary

As has been listed in the referenced work, there are many avenues and options for improving

the performance of flywheels. Early research focused on producing optimized solutions by

using formulations that provided purely analytic optimization results. The analytic results

required that the flywheels have certain properties that continuously vary based on location.

Some more recent work also investigated the continuous variation of flywheel properties

to optimize performance, but did not provide a strictly analytic formulation result to do

so. Later work expresses greater interest in a more discrete approach to optimization

with multi-ring flywheels or suggested a discretized change in properties to reflect a more

51

easily producible rotor design. Later work also expressed an interest in using more complex

optimization tools such as sequential linear and quadratic programming, genetic algorithms,

and simulated annealing. Throughout the entire research history, case studies have been

performed to analyze, manufacture, and test flywheel designs and ideas. With this research

work, a few different optimization goals have been presented. By far, the optimization

parameter of greatest interest has been specific energy. However, other parameters such as

the total stored energy, usable stored energy, energy per unit cost, and energy density have

been used.

Given all of these options and possibilities, there is a clear need to narrow the research

range to the most relevant analyses. Theoretically optimized solutions for rotors with con-

tinuously varying material or geometric properties can serve as benchmarks against new

composite design ideas. However, manufacturing difficulties associated with continuously

varying the flywheel properties can make these designs highly uneconomical. Discrete vari-

ation (multi-ring rotors) is preferred from a manufacturing standpoint, but there are many

options for discrete changes such as density, press-fit stresses, material selections, elas-

tomeric interlayers, and fiber selection and orientation.

What optimization objective and constraints should be used to ensure the analysis

results most closely match the needs of a consumer of this technology? Which combination

of design options allows the highest performance given the ease of manufacturing associated

with those design options? What search methods/tools should be used to determine this

optimal design? Last, what is the most realistic way to model these designs? These are

issues and questions worthy of investigation in order to define an appropriate research path.

The next chapter presents a discussion and underlying relevant approach to be used in

improving the performance of flywheels.

Chapter 3

Objectives

With all of the possible flywheel rotor model formulations and design options, it is important

to clearly define the objectives of this research to advance technological understanding in

both regards. It was the original goal of this research to only focus on five multi-ring

composite flywheel rotor design options. However, upon investigating the methodology

for constructing the optimization problem, this author found it necessary to also pursue

research identifying useful, independent objective and geometric constraint combinations.

3.1 Phase I: Objective/Constraint Investigation

In the preceding publications reviewed, several optimization objectives were established:

total stored energy K, specific energy Km, material volume energy density Kmv, enclosed

volume energy density Kev, and energy per unit cost. Since attempting to characterize both

material and manufacturing costs of the flywheel rotor and auxiliary system components is

a highly complex endeavor that requires substantial economic data and is subject to large

variations depending on technology application and environmental factors, energy per unit

cost is excluded from this investigation. However, when including simplifying assumptions

and restrictions, the other options become feasible to analyze. The first simplifying factor

is the 2-D plane stress assumption. The second factor is an axisymmetric rotor design

52

53

restriction. The third is the exclusion of shear stresses associated with power-transferring

torques and bearing-to-rotor gravitational loadings. The fourth is the restriction to polar

orthotropic materials where the fibers are oriented in either the hoop or radial directions.

The fifth is the exclusion of a hub. This fifth factor is required because realistic hubs are

often neither axisymmetric nor 2-D in their design. With these factors, the objectives to be

investigated are K, Km, Kmv, and Kev. With these objectives selected, the next issues to

address are the geometric constraints.

When making an accounting of all the variables that are included with the flywheel

design, an issue arises as to how to handle the rotor radii. Should the outer rotor radius

be fixed, or should it be allowed to vary? Should the inner rotor radius be fixed, or should

it be allowed to vary? Is it reasonable to allow both inner and outer rotor radii to vary?

These questions are multiplied for each objective to produce an objective/constraint decision

matrix, which is shown in Table 3.1.

Table 3.1: Optimization objective validity and equivalency for different radial constraintcombinations.

(A) (B) (C) (D)

Fixed RadiiFixed Inner Fixed Outer Constraint-

Radius Radius Free Radii

1. Total Stored? ? ? ?

Energy K

2. Encl. Vol. Energy? ? ? ?

Density Kev

3. Mat. Vol. Energy? ? ? ?

Density Kmv

4. Specific? ? ? ?

Energy Km

This table summarizes the first major goal of this research: identify all valid and in-

dependent objective/constraint combinations as well as any combination that is not valid

or useful through equivalencies, non-physical limiting behavior, or unbounded optimization

54

behavior. In doing so, it is possible to summarize the information and complete Table 3.1.

Note that this table contains numbers corresponding to the objectives and capital letters

corresponding to the geometric constraints. This table is designed to act as a code for future

use. An example would that 2–C would correspond to enclosed volume energy density with

a fixed outer radius. This portion of the research will be denoted as Phase I. Only when

this portion is finished and the valid objective/constraint combinations have be selected for

use can the second phase begin.

3.2 Phase II: Design Option Investigation

There are five design options to be investigated in a portion of the research denoted as

Phase II. A notational overview of the options is given in Figure 3.1 and a description is

provided below:

Figure 3.1: Illustrative description of the five rotor design options to be investigated.

(A) Variable Fiber/Matrix Option: allowing the the fiber/matrix ratio of each material

ring in the multi-ring composite rotor to each vary.

(B) Radial/Hoop Fiber Option: allowing variable proportions of radial and hoop fibers in

55

each material ring.

(C) Two-Fiber Co-mingling Option: combining two fiber materials in a variable ratio in

each material ring.

(D) Press-fitting Option: taking the material rings and press-fitting them together with

set overlapping interferences to produce compressive radial stress in each material

ring.

(E) Matrix Ballasting Option: infiltrating the composite matrix with high-density parti-

cles to vary density, stiffness, and coefficient of thermal expansion within the matrix

of each material ring.

These options are investigated individually and in combination with each other and

compared with the zero design option scenario. All of these combinations are investigated

with respect to the valid, independent objective/constraint combinations identified in Phase

I. Phase I allows for the comparison of design options with the objective/constraint combi-

nations.

3.3 Summary

The first goal is to characterize many popular objective/constraint combinations in terms

of their validity and usefulness. With these combinations identified, the design options

are investigated to characterize their potential to benefit or hinder the optimization of the

desired objective. All this information can be highly useful to flywheel rotor designers to

better understand how they can avoid pitfalls and trivial optimization traps in setting up

their optimization search. It also can direct them to either further investigate or avoid the

design options investigated here.

Chapter 4

Methodology

This chapter describes the structure of the two investigative phases outlined in the previous

chapter. It goes into the assumptions, constraints, and formulations necessary to perform

the research. It also describes the tools and techniques used to generate the results, such

as the computer codes written. Subsequently, it goes into detail in sequence of calculations

made and how the optimization algorithms work.


In a number of the objective/constraint combinations given in Table 3.1, simple, straight-

forward dimensional analysis can be performed to identify equivalencies in the optimization

goals. As a result of this fact, these simple degenerate combinations are analyzed first. Four

of the sixteen cases are analyzed using this method. There are two cases that require only

analysis from the single ring rotor program. Since this analysis is slightly more complex

than the previous four combinations, it is presented next. One combination requires the

multiple ring rotor optimization program to demonstrate its conditional validity and is pre-

sented as the seventh case. Three cases concerning specific energy Km require use of both

the multiple ring rotor program and the single ring rotor program to robustly investigate

and validate the issues regarding their result. Three cases concerning the material volume

56

57

energy density Kmv largely mimic the cases concerning specific energy with some notable

exceptions. Last the remaining cases are considered valid and are used in the Phase II

investigation with the design options.

It is important to note that, in most cases, definitive proofs are not made. Closed-

form analytic solution are intractable if not impossible to produce. The reason for this

intractability can be provided by counting the number of search dimensions for a 3-ring

problem with all the design options checked: 27-D search space. Other factors that could

change results include the following:

• Changing the material properties/models

• Changing from a 2-D plane stress analysis to another formulation such as a 2-D planestrain, 2-D MGPS, 3-D FEM, etc.

• Including a designed hub that may not be 2-D or axisymmetric

• Adding other design options such as fiber pretensioning, axial thickness variation,orienting fibers more toward the axial direction, etc.

• Introducing design-specific constraints not listed here

Therefore, these results are only presented as a guide for flywheel rotor designers to

assist them in choosing how they want to construct their rotor optimization problem. For

the objective/constraint combinations where the multiple ring program is used, plot lines

are shown to demonstrate the behavior of using (1) none of the design options, (2) each of

the five options individually, (3) four of the design options grouped together, (4) and then

all five design options grouped together for a total of eight lines. Each line contains three

data points corresponding to one, two, and three material ring rotors.

When using the multiple ring rotor program, a many of optimization runs made pro-

duced results that indicate non-physical behavior. For the conditionally valid objective/

constraint combination 1–B; 35,000 optimization runs were made for each data point shown

in Table D.2. For the convergence criteria to be satisfied, the best ten results for each data

point must be identical to within 3 significant figures. If not, the best value is used and

the corresponding data point given in Table D.2 is shaded gray. For combinations 3–C and

58

4–C, 30,000 runs were made for the design results shown, and all of those results converged

to satisfaction.


The design option investigation only includes results from the three valid objective/constraint

combinations: total stored energy, fixed radii; total stored energy, fixed outer radius; and

specific energy, fixed radii. There are a total of 32 design option combinations which are

shown as four independent graphs as listed in Table 4.1. In all of these cases, the residual

stresses due to an epoxy cure cooling temperature change of ∆T = −110 ◦C was applied to

the stress analysis. Each design option is measured with three ring options of one, two, and

three material rings per rotor. This results in a total of 288 plot points.

Table 4.1: Design option combinations investigated for a given objective/constraint combi-nation.

Graph 1 Graph 2 Graph 3 Graph 4

0 AB ABC ABCDE

A AC ABD ABCD

B AD ABE ABCE

C AE ACD ABDE

D BC ACE ACDE

E BD ADE BCDE

BE BCD

CD BCE

CE BDE

DE CDE

Each design option adds to the number of search space dimensions that already exist.

Since there can be multiple rings, the number of added search space dimensions is subject

to change. The list below identifies all of these search dimensions that include and exclude

the design options. In this list, N is the number of materials rings in the flywheel rotor.

• Default Search Space Dimensions

59

◦ N or N − 1 ring radii selections depending on if one or both rotor radii are fixed

◦ N Fiber #1 material selections

◦ 1 Angular speed selection

• A: Variable Fiber/Matrix Option

◦ N fiber-matrix ratios

• B: Radial/Hoop Fiber Option

◦ N radial-to-hoop fiber ratios

• C: Two-Fiber Co-mingling Option

◦ N Fiber #2 material selections

◦ N (Fiber #1 / Fiber #2) ratios

• D: Press-fitting Option

◦ N − 1 press-fit interferences

• E: Matrix Ballasting Option

◦ N ballast material selections

◦ N particle-to-matrix volume ratios

As can be seen, the number of search space dimensions is highly variable depending

on both the number of material rings and design option combinations. The most simple

case would be a single material ring rotor with none of the options selected and both rotor

radii fixed. This situation would correspond to a 2-D search space associated with the

composite material selection and the peak angular speed allowed. The most complex case

would include fixing only one radii while incorporating three material rings with all five

design options. This situation would correspond to a 27-D search space.

For the completely valid objective constraint conditions 1–A, 1–C, and 4–A; 75,000

optimization runs were made for each data point. For the convergence criteria to be satisfied,

the best ten results for each data point must be identical to within 3 significant figures. If

not, the best value is used and each corresponding data point that failed to converge given

in Tables D.1, D.3, and D.4 is shaded gray.

60

4.3 Single Ring Rotor Program

The analysis performed here is highly similar to that of Gabrys & Bakis [22], which is based

on that of Genta [13] and Lekhnitskii [23]. This formulation is based on the plane stress

assumption for an axisymmetric, uniform-thickness ring geometry with polar orthotropic

materials in the form of hoop-wound fiber composite materials. This formulation is given

in Section A.1. With this information and formulation, it is possible to develop a search

algorithm to determine the optimum inner radius for a set of assigned material, bound-

ary stresses, and outer radius conditions with an optimization objective chosen. Both the

straight forward calculations and the optimized results are produced. For this analysis, two

popular hoop-wound composite materials are chosen: E-glass/epoxy and high strength car-

bon/epoxy. The formulation and calculation procedure are listed in Section A.1. Properties

of the unidirectionally reinforced hoop-wound composites are listed in Table B.1.

With these equations, it is a simple matter to calculate the corresponding maximum

K, Kmv, Kev, and Km for a given rim material, inner and outer radii, and inner and

outer boundary stresses. With this information and formulation, it is possible to develop

a search algorithm to determine the optimum inner radius for a set of assigned material,

boundary stresses, and outer radius conditions with an optimization objective chosen by

simply scanning the 1-D angular speed variable until the peak is found. Graphs are produced

to demonstrate this as necessary. Both the straight forward calculations and the optimized

results are produced.

To validate the single ring rotor program and analysis, each major calculation step

made by the single ring program code was verified by an identical calculation made in the

numerical analysis package Maxima. Also, the stress distributions and objective calculations

were verified by comparing them to the multiple ring rotor program, which uses a different

stress formulation as given in Section A.2. These calculations were also plotted to look for

any unusual or non-physical behavior.

61

4.4 Multiple Ring Rotor Optimization Program

In this section, all aspects of the multiple ring rotor program are discussed. First, the an-

alytic thermo-elastic formulation is presented with all accompanying approximations and

simplifications. Also, the composite material models are presented along with the accom-

panying material model database. This database includes composite, matrix, and ballast

material properties. Next, an overview of the optimization algorithm used is provided

along with justification of its use over other optimization algorithms. The penalty factors

are listed and explanation of how they affect search space boundary behavior is provided.

Last, an outline of the program is given along with a flowchart of that illustrates how the

objective calculation is made.

4.4.1 Analytic Formulation & Material Models

The full formulation information is given in Section A.2. This formulation is different from

the single ring formulation due to the fact that design option (B) requires individual ply

stress calculations, which require strain information. This fact is indicated in Section A.2.1.

Also, using two different formulations allowed for more robust program cross-checking. It

uses a 2-D plane stress assumption [23, 22, 49]. This assumption is used due to the fact

that it produces a more conservative approximation for the radial stresses [30]. Also, no

shear stresses relating to power transferring torques or hub-rotor gravitational shear loading

are calculated. It is well known that a superposition of these shear stress fields under the

plane stress and axisymmetric assumptions can affect certain 3-D interactive stress failure

criteria. However, because the power transferring torques and the axial length effects of

the hub or rotor are not the subject of this investigation, they are not included. This

formulation assumes only thermo-elastic behavior. No moisture strains are included due to

the flywheel being made in factory-controlled conditions while being forced to operate in a

vacuum to avoid aerodynamic drag losses. Last, neither a flywheel hub nor its boundary

stresses are included in this analysis. This is due to the fact that hubs are typically neither

62

axisymmetric nor 2-D in their design and the inner boundary stresses imposed on the rotor

reflect this. After the formulation is presented, Subsection A.2.1 contains the calculation

procedure performed by the program.

Concerning the calculation of the material properties, the stock composite, accompa-

nying epoxy, and ballast material database information given in Appendix B is used. The

basic micromechanical model property factors are first back calculated from this material

database. Once these volume-fraction-independent factors are calculated, then the actual

material properties are calculated. If the matrix ballasting design option (E) is used, then

the epoxy/ballast material property calculations are made. Assigned fiber volume fractions

of 65% are used if the variable fiber/matrix design option (A) is not used. If (A) is used,

the the optimization algorithm defines the fiber/matrix ratio to be used. If the fiber co-

mingling design option (C) is used, then the default material properties are modified to

include the additional fiber material in the composite. Last, if there are radial fibers as well

as hoop fibers for design option (B), then multi-ply composite properties are calculated as

well. Equations for these material models are given in Appendix C. All of the models used

are referenced from books by Daniel & Ishai [25], Christensen [50], and Tsai & Hahn [51],

as well as an article by Christensen [52].

4.4.2 Optimization Algorithm

As given in the literature, there are many optimization methods used. These include sequen-

tial linear and quadratic programming, simulated annealing, and various genetic algorithms.

For the highly complex search space where the most complicated test case has 27 degrees

of freedom, a robust and efficient search algorithm is required. The sequential program-

ming techniques are acceptable for a search space that is not multi-modal, or many-peaked.

These algorithms climb the optimization hill on which they are placed and will not attempt

to robustly search for other peak solutions. Such algorithms are not suited to highly multi-

modal or deceptive search spaces. An example of both characteristics is given with the 2-D

63

Schwefel test function that is shown in Figure 4.1. Simulated annealing is an optimization

algorithm where the members of the search population do not attempt to learn from each

other. It is possible that the various genetic algorithms presented can search for the optimal

solution in a robust and efficient fashion; however, the effort needed to properly tune all of

the possible algorithm parameters is beyond the scope of the proposed research. Therefore,

what is needed is an algorithm that is capable of providing a robust search with a minimum

number of search parameters to be tuned.

Figure 4.1: The Schwefel test function in two dimensions. This test function is multi-modalwith a deceptive optimum at the corner of the search space.

With these issues in mind, an investigation into a number of different evolutionary

strategies and algorithms were conducted. The more notable ones include the following:

• Particle Swarm Optimization (PSO) [53]

64

• Differential Evolution (DE) [54]

• Compact Genetic Algorithm (cGA) [55, 56]

• Generalized Generation Gap Parent-centric recombination Genetic Algorithm (G3-PCX GA) [57]

• Covariant Matrix Adaptation – Evolutionary Strategy (CMA–ES) [58]

Of all the options presented, the most promising is the CMA-ES algorithm. This strategy

reduces the number of parameters to manipulate down to the optional population size. All

other parameters that would normally need to be tuned by other algorithms are self-tuning

for CMA-ES. It is also a robust algorithm that has been compared to others with favorable

results [57, 59, 60, 61]. It has been determined to be a suitable algorithm for the purposes of

this research. The only parameter truly required to be set for this algorithm is the limit on

the number of functional evaluations. CMA-ES works by taking sample distribution results

and learning from them by adapting a displacement vector and covariance matrix to move,

scale, shape, and orient a new statistical distribution. These steps are shown in Figure

4.2. As can be seen in the figure, a default symmetrical statistical distribution is generated

and sample points calculated. Next, the best points are used to move, scale, shape, and

orient a new covariance matrix. With the new search location and statistical distribution

information acquired, a new set of sample points can be generated.

Figure 4.2: An illustration of how the CMA-ES uses a displacement vector and covariancematrix to move, scale, shape, and orient the next generation statistical distribution [62].

Other notable features in this algorithm include covariance matrix history weighting

and step size control. It is important that the new covariance matrix constructed does

65

not forget what came before it by only relying on the new population information. Often,

the previous generation’s covariance information is still useful. Therefore, the algorithm

incorporates weighted covariance history information that also helps to create the new

matrix. This history weighting provides search inertia so that unreasonable leaps and

jumps do not occur. The step size control is needed to scale the length of the displacement

vector such that it is appropriate for efficient search. An example of picking the appropriate

step size is shown in Figure 4.3. Further details on the algorithm is provided in the tutorial

produced by its co-creator as well as the first publication to outline its design [62, 58].

Figure 4.3: Step size control illustration showing a step size that is too large (left), toosmall (right), and appropriate (middle) [62].

4.4.3 Penalty Factors

The chosen optimization algorithm includes many self-parameterizing mechanisms. How-

ever, it does not account for the search behavior at the boundaries of the search space.

For all places in the search space, the algorithm needs to know where to go. If there is

a stress failure, or other constraint that limits the objective, simply setting the objective

to zero in all limiting cases does not inform the algorithm where to go. Therefore, it is

necessary to inform the algorithm that this space leads to unnacceptable results while also

informing it where to go in order to avoid/leave this part of the search space. The first of

66

these mechanisms is called the soft penalty factor. This factor is calculated according to

Equation 4.1.

Psoft = exp

(−fsoft

∣∣∣∣x− xoxo

∣∣∣∣) (4.1)

where:Psoft: Individual variable soft penaltyfsoft: Exponential decay soft penalty factorx: Constraint parameter of interestxo: Constraint parameter limit

With this equation, the full objective calculations are made, then the exponential decay

penalty is applied. If there are multiple penalties for multiple constraints, then they are

multiplied together with the objective. In trying to find the right exponential decay factor,

a great deal of testing has been performed. If the factor is too aggressive, the objective may

return zero due to a numerical precision limit and no information is obtained. In this case,

the valid search space is not found. If the factor is too lenient, the constraint limits may

be overshot and non-valid spaces might be searched. The soft penalty factor was chosen to

be fsoft = 25. In addition to the stress failure constraint, this soft penalty equation is used

for the following non-stress related constraints:

Valid ring thickness: [0.5mm,∞]

Valid press-fit ring thickness: [5% · ri,∞]

Valid ballast fraction: [0, 0.25]

Valid fiber fraction: [0.05, 0.65]

Valid press-fit interference:

[0, 0.2% · ri + ro

2

]

Note that the soft penalty corresponds to impossible or reasonable limits, but attempting

the objective calculation does not break the code. The constraints relating to stress failure

67

are applied according to Equation 4.1. For the multiple ring program, both the maximum

stress and 2-D Tsai-Wu failure criteria are applied. Whichever is the most conservative

(limiting) is used in the stress-failure soft penalty. Concerning the stress-failure calculation,

a number of scenarios are analyzed depending on the design options.

In designing a flywheel rotor, it is important to factor in manufacturing and environmen-

tal concerns. For a flywheel system, the following situations must be tested for a multi-ring

flywheel:

• Manufactured rotor, temperature change due to cooling, no load

• Manufactured rotor, temperature change due to cooling, maximum spin load

A temperature change relative to manufacturing temperatures must be included to de-

termine residual stress restrictions. Therefore, the limiting case of a nominal environmental

temperature change is used. For press-fit interferences, additional manufacturing situations

must be taken into account:

• Individual rotor ring, temperature change due to cooling, no load

Before press-fitting of multiple material rings can be made, it is necessary to construct

the individual rings first, then allow them to cure and cool down [36]. Only then can

press-fitting be attempted. However, if one of these individual rings should fail before press

fitting, then the rotor has failed before it could even be constructed and therefore should not

be considered manufacturable and should not be a valid flywheel rotor design. Therefore,

stress distributions are calculated at each of these stages to ensure manufacturability as

well as performance.

In calculating the boundary penalties, there is a second mechanism which bypasses the

objective calculation altogether. The reason for this is that if the variables were allowed

to represent such unrealistic values, their introduction may break the code. These values

68

include negative ring thicknesses, volume fractions less than zero or greater than one, or

negative interferences (producing a gap instead of an overlap). The variables that must

stay within the hard penalty range are given below:

Valid ring thickness: [0,∞]

Valid ballast fraction: [0, 1]

Valid fiber fraction: [0, 1]

Valid hoop ply fraction: [0, 1]

Valid press-fit interference: [0,∞]

When these hard penalty boundaries are exceeded, a negative linear difference is calculated

for each objective variable. This simple difference for one variable is shown in Equation 4.2.

Phard = − |x− xo| (4.2)

where:Phard: Individual variable hard penalty

Using this method of hard penalty calculation, the algorithm is always pointed toward a

valid region of the search space. This is shown in Figure 4.4 for one search dimension. With

these penalties determined, the calculation procedure for the objective is given in Figure

4.5.

4.4.4 Program Structure

An outline for the structure of the multiple ring computer program that was written is

provided here. It includes all of the major subroutine and do-loops associated with the

relevant calculation procedures of the program. After each listing, a description of the

subroutine is given.

69

Figure 4.4: Graphical illustration of hard and soft penalty factors applied in one dimension.

• Preliminary Calculations: Calculates preliminary material property informationfrom Appendix B

• Case Do Loop: Performs the required number of optimization search runs for eachcase

◦ CMA-ES Run: Samples search space by either calling the Function Evalua-tion subroutine or making a fake hard penalty calculation for each sample point;and then moves, scales, shapes, and orients the distribution using a calculateddisplacement vector and covariance matrix

— Function Evaluation: Performs a functional evaluation of the objectiveparameter based on sample variables and penalty factor weighting

∗ Composite Material Calculation: Calculates the material propertiesfor each ring

· Ballast Material Calculation: Calculates the ballasted matrix ma-terial

· Fiber/Matrix Calculation: Calculates changes in the single-fiberunidirectional material properties due to volume fraction changes

· Fiber Co-mingling Calculation: Calculates the material propertiesfor the situation of using two fibers in the composite material

· Stiffness Matrix Calculation: Calculates the stiffness matrices andthermal expansion coefficient vectors based on hoop and possibly ra-dial fiber plies

70

Figure 4.5: Flowchart of decision and calculation procedure when incorporating penaltiesinto the objective calculation.

∗ Stress Calculation: Calculates the stress vectors for speed loaded,unloaded, and possibly unloaded individual material rings if there ispress-fitting

∗ Stress Failure Calculation: Calculates the stress failure criteria tofind worst case for rotor

∗ Penalty Calculations: Calculates and combines all soft penalty factorsby multiplying them together

∗ Objective Calculation: Calculates the objective parameter with penaltyfactors

4.4.5 Validation

To validate the multiple ring rotor program, each major calculation step made by the single

ring program code was verified by an identical calculation made in the numerical analysis

package Maxima. For the material models, plots were made to verify appropriate behavior

across volume fractions. The stress distributions and objective calculations were verified

by comparing them to the single ring rotor program as well as another multiple ring rotor

71

program provided for verification [63], both of which use a different stress formulation to

the multiple ring rotor program used in this analysis. These calculations were also plotted

to look for any unusual or non-physical behavior. Last, the optimization algorithm was

tested against four different standard test functions with up to 50 dimensions: Rosenbrock,

Ackley, Rastrigin, and Schwefel.

4.5 Summary

In this section, the overall investigation methods and program tools have been discussed.

Section 4.1 discussed the strategies for proving equivalencies and non-physical behavior as

well as the assumptions and simplifications. Section 4.2 listed the conditions of the plot

points to be presented and described the degrees of freedom possible. Section 4.3 described

both the formulation and the function of the single ring rotor program. Last, Section 4.4

provided a full overview of the analytic formulation, the optimization algorithm, boundary

behavior, and program structure of the multiple ring rotor program.

Chapter 5

Results & Discussion

This chapter is divided into two phases. Phase I presents the results and discussion of

the objective/constraint investigation. For this section, the invalid, conditionally valid,

and equivalent objective/constraint combinations are discussed in detail. Both the single

material ring rotor program and the multiple ring rotor program are used for some of the

cases. In Phase II, three objective/constraint combinations are investigated along with all

of the design option combinations. For this analysis, only the multiple ring rotor program

is used. For some of the program results, the coding scheme is given in Figure 5.1. This

notation is used extensively in Appendix D for both the figures as well as the tables. Details

of the design options are given in Section 3.2.


In this section, thirteen objective/constraint combinations are explicitly discussed, with the

remainder considered valid and worthy of further detailed investigation. Objective/constraint

combinations in this section are identified according to the code presented in Table 3.1. As

can be seen in this table, grid location 1–A corresponds to Total Stored Energy, Fixed Radii.

This coding scheme is used in order to simplify the descriptions and avoid confusion. Please

refer to Table 3.1 to identify the proper notation when it is given in this chapter.

72

73

Figure 5.1: Notational illustration of the optimization case coding scheme used for thefigures and tables in Appendix D.

For the results and discussion, it is important to start with a notational illustration of

the generic stress distribution for loading a single-ring rotor centrifugally as well as applying

an inner boundary stress. Figure 5.2 illustrates the stress distributions in a single spinning

ring loaded with three different inner boundary radial stresses P : P = 0, P > 0 (tensile

loading), and P < 0 (compressive loading). For the purpose of this analysis, the outer

boundary radial stress is zero, W = 0, corresponding to conditions of the outer material

ring. Note that the notation used is the same as that in Section A.1. Increasing the tensile

inner boundary radial stress P reduces the magnitude of hoop stress and increases the

magnitude of radial tensile stress throughout the ring. While the effects of the boundary

conditions on the hoop and radial stresses of the ring are well known, these effects are

important for understanding the outcome of specific optimization cases later. Thus they

are shown for reference.

74

Figure 5.2: Notional illustration of radial and hoop stress distributions in a spinning, single-material ring with zero, compressive, and tensile inner boundary radial stresses P .

5.1.1 Fixed Inner & Outer Radii

Figure 5.3 demonstrates possible behavior relating to constraining both the inner and outer

radius for a multi-ring optimization. In these cases, all optimization objectives are con-

sidered to produce physically realistic, yet unknown, solutions and are considered valid.

However, certain objectives produce equivalent optimized design solutions. This fact can

be proven through basic dimensional analysis. For Kmv; if the inner and outer radius is

fixed so that the material volume is constant, the optimize stored energy value K is divided

by the constant volume. The same analysis applies to the Kev; if the outer radius is fixed

so that the enclosed volume is constant, K is again divided by the constant volume. This

observation is valid for both multi-ring and single ring rotors. Therefore, the optimized

solutions for combinations 3–A, and 4–A are equivalent to 2–A.

5.1.2 2–B: Enclosed Volume Energy Density, Fixed Inner Radius

Objective/constraint combination 2–B produces an equivalent optimized design solution

with that of 3–B. This fact can be proven through basic dimensional analysis. Because

75

Figure 5.3: For a fixed inner and outer radius, optimal solutions for K, Kev, and Kmv areidentical.

the inner radius is set, the open space, or hole, in the center that is included with the

enclosed volume is set to a constant value. Subtracting off this constant value from the

material volume does not change the optimization objectives. This observation is valid for

both multiple ring and single ring rotors. Therefore, the optimized solutions for option 2–B

are equivalent to that of option 3–B. Further information on combination 3–B is given in

section 5.1.11.

5.1.3 2–C: Enclosed Volume Energy Density, Fixed Outer Radius

Figure 5.4 illustrates two optimization cases with a variable inner radius and a fixed outer

radius. When optimizing on either K or Kev, the inner radius moves inward to maximize the

mass and volume of the rotor while the angular speed is maximized. These two parameters

interact to produce realistic, yet unknown, solutions and are considered valid. Therefore,

2–C is considered a valid option. However, this objective/constraint combination produces

equivalent optimized design solutions to 1–C. This fact can be proven through basic dimen-

sional analysis. For Kev, if the outer radius is fixed so that the enclosed volume is constant,

the optimized stored energy value K is divided by the constant volume. This observation

is valid for both multiple ring and single ring rotors. Therefore, the optimized solutions for

option 2–C are equivalent to option 1–C.

76

Figure 5.4: For a fixed outer radius, the optimal K search converges on a physically realisticsolution identical to the optimal Kev.

5.1.4 1–D: Total Stored Energy, Constraint-Free Radii

Figure 5.5 shows the maximum K for single rings having a fixed outer radius and variable

inner radius. It is important to note that for this figure, the shapes of the curves are

independent of geometric scaling. This figure is generated with P set to zero, –0.8 of YT ,

and +0.8 of YT . An example of how to read these graphs would be to note that for the E-

glass composite with a set outer radius of 0.1 m and P = 0, a peak energy storage capability

exists for the inner radius selection of roughly 0.072 m. It is noted that this is the location

where the limiting angular speeds due to hoop and radial strength limitations match. This

means that the overall shape of each graph is the same as long as the inner and outer radii

are both multiplied by the same factor, i.e. λ.

Figure 5.5 shows that the highest energy storage capability may not necessarily be

achieved with the highest hoop strength alone. The interaction between hoop and radial

elastic characteristics as well as mass density are highly relevant in determining the peak

energy storage capability. Because the outer radius is fixed and thus the enclosed volume is

fixed, Figure 5.5 also displays the same overall behavior as that of Kev, as given in Section

5.1.3.

Figure 5.6 shows how the K is affected by a constraint-free condition. Figure 5.6 high-

lights the fact that as the outer radius is allowed to grow, the optimized K value also grows.

The optimized thickness ratio λ is always the same for a given material and the P value and

is independent of outer radius selection. Once identified, the radii can be scaled upward

77

Figure 5.5: The maximum K for a single-material ring of fixed outer radius as a functionof inner radius with a series of curves corresponding to P values of –0.8, 0, and +0.8 of YT .

with the same λ to further increase K.

Figure 5.7 illustrates the cases where both the inner and outer radii are variable. For

the constraint-free option with the optimization of K, the geometry would align in a specific

manner for a single ring as shown in Figure 5.5, and then continue to scale upward without

bounds as shown in Figure 5.6. This statement can also be expanded to a multiple ring

scenario because no matter what optimized multiple ring rotor shape can be generated

to maximize energy storage capability, that geometry can be scaled up to produce a new

design with greater K compared to the original. For both a single ring and a multiple ring

rotor, combination 1–D produces an unbounded total stored energy solution. Therefore, no

realistic optimized total stored energy solution can be realized for option 1–D.

78

Figure 5.6: Optimized K for a single-material ring as a function of outer radius with variousP values ranging from –0.8 to +0.8 of YT with both ri and ro allowed to vary.

5.1.5 2–D: Enclosed Volume Energy Density, Constraint-Free Radii

In the previous section, it was identified that the radii can be scaled upward with the

same λ to further increase K. However, Kev does not increase with increased scaling once

the optimal λ has been identified. Figure 5.8 is displayed as a bar chart because Kev is

independent of outer radius selection. Even when adding a constant tensile or compressive

P , the scaling still does not change with geometry. Because the optimal solution is scale

invariant, combination 2–D produces an identical optimal solution to 2–C, and is thus

equivalent to 1–C as well.

79

Figure 5.7: For objective/constraint combination 1–D, the optimal K search produces aninfinitely large ring with the same λ factors as that of 1–C.

5.1.6 1–B: Total Stored Energy, Fixed Inner Radius

There is no simple single ring analysis that could be performed to show that 1–B is not a

valid objective/constraint combination. The only problems that were encountered occurred

with the multiple ring rotor program as shown in Table D.2 and Figures D.6, through D.10.

Design option (A) stood out as strongly affecting optimized results such that a logarithmic

scale needed to be used for the energy axis. More insight can be obtained by looking at

the optimized results as shown in Table 5.1. As can be seen, the outer radius is 4 orders

of magnitude larger than the inner radius. This design is simply a solid disk rotor with a

small hole in the center of it. Because the size of the rotor is so large, the angular speed

is very low. The optimization is attempting to obtain the largest scaling possible. The

method for achieving this optimization lies in the fiber volume fraction, which is virtually

non-existent. As can be seen in Figures D.6, through D.10, any design option combination

that includes (A) has its energy optimization significantly increased. Since nearly all of

the fibers have been eliminated from the optimization, the material properties of the rotor

are close to being isotropic. This situation has been analyzed before by both Christensen

in [14] and Mohr & Walter [16]. When isotropic materials are used, the most optimized

solutions come in the form of a solid disk, not a ring. Also, the highest shape factor for

isotropic material, as shown in Figure 2.2, is the isostress solid disk. Even a uniform disk

has a higher shape factor than a thin ring. For a fixed inner radius with a variable outer

80

Figure 5.8: Optimized Kev for a single-material ring various P values ranging from –0.8 to+0.8 of YT with both ri and ro allowed to vary.

radius, variable material design options that can lead to an isotropic material produce only

a conditional validity for combination 1–B. However, if the composite material can remain

anisotropic, there may be a valid solution space that can be searched.

5.1.7 4–C: Specific Energy, Fixed Outer Radius

Upon initial multi-ring analysis, it was identified that many of the combination 4–C opti-

mized solutions were converging to the minimum radial thickness required by the multiple

ring rotor program. A few of the converged solutions after 30,000 optimization runs were

made are shown in Table 5.2. Design option (B) was not used due to manufacturing failure,

which is discussed in Section 5.2.2. Design option (D) is also not presented in Table 5.2 due

81

Table 5.1: Flywheel single ring rotor results using the multiple ring rotor program withdesign option (A) selected.

E–I–0–1101 Ring

Energy K (W·h) 5.13E+08

Angular Speed ω (rad/s) 9.02E-02

Ring Radius # 1 (m) 1.00E-01

Ring Radius # 2 (m) 1.03E+03

Composite Material # 5

Fiber Volume Fraction V f 2.98E-07

to the fact that the solution did not converge, with a maximum Km for the 3-Ring rotor

being 175 W·h/kg. This information poses an interesting insight: the optimization is mov-

ing toward an infinitesimally thin ring, with the 0.5 mm limit set by the code reached every

time. With this result, this objective/constraint combination required further investigation.

The discussion that follows shows that combination 4–C produces non-physical solutions

for the rotor as shown in Figure 5.9, and indicated in Table 5.2. It is important to note

that Figure 5.9 displays infinitesimally thin, non-physical optimal ring solutions for zero

and tensile P values.

For the situation of P = 0, the maximum Km for a fixed outer radius occurs at a

vanishing rotor thickness. Since the material is much stronger in hoop tension than radial

tension, a higher Km can be obtained when all of the material is uniaxially stressed in

the hoop direction, i.e. an infinitesimally thin ring. It is also noted that for Km, a slope

discontinuity occurs that corresponds to the peak K value and a transition between radial

tensile and hoop tensile failure modes. The limiting behavior of Km for P = 0 as the inner

and outer ring radii become equal can be analytically determined. Referring to Equation

A.7 in Appendix A, as ri → ro, then λ→ 1. Therefore, the angular speed for P = 0 can be

simplified according to Equation 5.1.

82

Table 5.2: Converged results for combination 4–C with the default design options as wellas (A), (C), and (E).

S-O-0-110 S-O-A-110 S-O-C-110 S-O-E-1103 Ring 3 Ring 3 Ring 3 Ring

Specific Energy Km 2.11E+02 2.11E+02 2.11E+02 2.11E+02(W·h/kg)

Angular Speed ω (rad/s) 4.94E+03 4.94E+03 4.95E+03 4.94E+03

Ring Radius 1 (m) 2.485E-01 2.485E-01 2.485E-01 2.485E-01




Fiber 1 Material 1 # 1 1 1 1



Fiber 2 Material 1 # 5



Fiber Fraction 1 V f 6.50E-01 6.50E-01 6.50E-01 6.50E-01



Fiber 1 Fraction 1 V f1 1.82E-01



Ballast Material 1 # 3



Ballast Fraction 1 V b · V m 3.00E-06



limλ→1

(ωh) =

√XT

ρr2o

(5.1)

For a vanishing thickness, the peak radial stress for P = 0 also goes to zero, allowing

the radial strength maximum angular speed to increase asymptotically. Therefore, ωh is

limiting when compared to ωr. The corresponding limit for Km is shown in Equation 5.2.

83

Figure 5.9: The maximum Km for a single-material ring of fixed outer radius as a functionof inner radius with a series of curves corresponding to P values of –0.8, 0, and +0.8 of YT .

limri→ro

(Km) =XT

2ρ(5.2)

The first situation to consider is the standalone single ring case with P = W = 0. It is

shown in Figure 5.9 that for the single free ring case with a fixed outer radius boundary, the

highest Km corresponds to an infinitesimally thin ring. If the angular speed and density

are held constant for a single ring, any rotor material placed inside the thin ring (smaller

radius) will lower Km. This is always the case for a single ring. However, this statement

does not cover a multi-ring rotor scenario. To investigate further, Km for a single thin

ring with P = W = 0 is analyzed first by taking one step back from the limit analysis of

84

Equation 5.2 to produce Equation 5.3 as shown.

limri→ro

(Km) =1

4

XT

ρ

[1 +

(riro

)2]

(5.3)

Note that the thin ring limit is used because it is the optimal case compared to a thick

ring case found by using Equations A.7 and A.8. Whenever material is placed at a radius ri

lower than boundary limit ro, there is always a penalty incurred because ri/ro < ro/ro = 1.

The issue in question is how to compensate beyond this penalty with an increase with

angular speed.

The first method to compensate beyond this penalty is with a multi-ring rotor that allows

the inner ring material to pull inward on the outer ring material. To compensate accordingly,

an inner ring material must have a higher strength-density ratio XT /ρ as the radius r is

lowered to be able to withstand the stresses of both itself and the outer ring material with

the lower XT /ρ. However, if this strategy is to be used, then the optimal solution would

instead be to simply place a material with the maximum XT /ρ at the outermost location.

A multi-ring material cannot improve Km with a superior XT /ρ material along the inner

radius if the best XT /ρ material is already being used as a single ring material located at

the outermost radius.

The second method to affect Km for a multi-ring rotor is to impose an inner boundary

radial stress P and somehow increase the maximum angular speed. This method is shown

in Figures 5.9 and 5.10 with the boundary conditions imposed. There are three possibilities:

(1) P = 0, (2) P > 0, (3) and P < 0.

1. The no-loading condition for a multi-ring rotor is a situation where each material

ring in the rotor does not impose radial stress on the other rings at the material

boundaries. This situation can at most match the limit of Figure 5.9; however, the

interior radius penalty previously stated prevents the addition of material to the inner

radial regions. So a multi-ring no-loading scenario can at best match the single ring

85

Figure 5.10: Optimized Km for a single-material ring with various P values ranging from–0.8 to 0 times the value YT with both ri and ro allowed to vary.

case with an infinitesimally thin ring as shown in Figure 5.9.

2. Constructing a multi-ring rotor with an inner ring applying a compressive P value on

the outermost ring decreases the optimized Km compared to a no-loading condition

as shown in Figure 5.10. Adding more hoop stress on the outer ring detracts from

the peak angular speed and thus P . This penalty can be combined with the interior

material radius penalty. Therefore, the optimal Km solutions with compressive P

value for multi-ring rotors are inferior to the single, infinitesimally thin ring case.

3. The tensile case is very interesting because the asymptotic increase in the Km shown in

Figure 5.9 reveals a possibility for an increase in the angular speed that can offset the

86

inner radial location penalty. As indicated in Figure 5.9, the reason this is happening

is because the centrifugal hoop stresses are being countered by the tensile P value. As

the radial thickness of the outer ring goes to zero, the angular speed can go to infinity

with a ring mass going to zero and P remaining constant. The problem with this

situation lies in the design of the inner ring structure relative to the outermost ring.

These inner rings have to pull inward on the outermost ring, meaning that they have

to withstand the loading from their own body force stresses in addition to the body

force stresses of the outermost ring. To be able to withstand the additional outermost

ring loading, the inner ring structure next to the outermost ring must have a higher

strength-density ratio XT /ρ than the outermost ring. If a multiple ring rotor is to be

designed with one or more inner rings having a higher XT /ρ than the outermost ring,

then it is inferior to a single-ring rotor that uses this higher XT /ρ material because

the single, infinitesimally thin ring avoids the inner ring radial location penalty.

The above analysis has reviewed and discredited all possible options for which a mul-

tiple ring rotor could produce a superior Km than a single, infinitesimally thin ring rotor.

Therefore, it can be stated that a multi-ring rotor cannot be physically constructed that is

superior in XT /ρ to a single, infinitesimally thin ring using the highest XT /ρ material. This

analysis combined with the multiple ring rotor program analysis allows for the following

statement: option 4–C produces a non-physical solution, as shown in Figure 5.11.

5.1.8 4–B: Specific Energy, Fixed Inner Radius

Initial multiple ring analysis had also revealed that many of the combination 4–B optimized

solutions were converging to the minimum radial thickness required by the multiple ring

rotor program. A few of the converged solutions after 30,000 optimization runs were made

are shown in Table 5.3. Again, design option (B) was not used due to manufacturing failure,

which is discussed in Section 5.2.2. Design option (D) is also not presented in Table 5.3

due to the fact that the solution did not converge, with a maximum Km for the 3-Ring

87

Figure 5.11: For objective/constraint combinations 3–B, 3–C, 4–B, 4–C, the optimal Km

or Kmv search produces an infinitesimally thin ring that is scale invariant.

rotor being 182 W·h/kg. This information poses the same interesting result as with 4–C:

the optimization is moving toward an infinitesimally thin ring, with the 0.5 mm limit set

by the code reached every time. Because this behavior is so similar to that of Section 5.1.7,

the Figures used there apply here as well.

Figure 5.9 shows the peak Km as ri is changed compared to the ro. However, this

graph can be used for any outer radius, regardless of scaling. The reason for this is that

optimizing on Km is scale-invariant. That is why Figure 5.10 is a column graph instead of

a plot of behavior as the radius changes. This scale-invariant attribute of the Km forces the

optimization to an infinitesimally thin ring given by Equation 5.2 regardless of whether it is

the ri or ro that is fixed. So long as one of the radii can move with no other constraints, a

non-physical thin ring solution results. Therefore, the optimized result of combination 4–C

is non-physical and thus not valid, as shown in Figure 5.11.

5.1.9 4–D: Specific Energy, Constraint-Free Radii

As discussed in Sections 5.1.8 and 5.1.7, Km is a scale-invariant objective. If one of the

radii can vary, the objective approaches the solution given in Equation 5.2. When both

radii are allowed to vary, the optimized solution does not change. Therefore, combination

88

Table 5.3: Converged results for combination 4–B with the default design options as wellas (A), (C), and (E).

S-I-0-110 S-I-A-110 S-I-C-110 S-I-E-1103 Ring 3 Ring 3 Ring 3 Ring

Specific Energy Km 2.09E+02 2.09E+02 2.10E+02 2.09E+02(W·h/kg)
























4–D also produces a non-physical, invalid result. Investigation of the objective/constraint

combinations 4–B, 4–C, and 4–D have shown that the shape factor of 0.5 as given in Figure

2.2 extend to multiple ring case as well as the isotropic single ring case as given in Equation

5.2.

89

5.1.10 3–C: Material Volume Energy Density, Fixed Outer Radius

The analysis on 3–C is virtually identical to the analysis of 4–C, with the exception of

the factor of density. Table 5.4 shows that the multiple ring rotor program also displays

infinitesimal thin ring behavior. It shows a few of the converged solutions after 30,000

optimization runs were made. Design option (B) was not used due to manufacturing failure,

which is discussed in Section 5.2.2. Design option (D) is also not presented in Table 5.4

due to the fact that the solution did not converge, with a maximum Kmv for the 3-Ring

rotor being 268 kW· h/m3. These results show that the 0.5 mm limit set by the code was

reached for these cases.

With this information, another single ring analysis was conducted as shown in Figures

5.12 and 5.13. The single difference in this analysis compared to combination 4–C lies in a

factor of density. This factor is shown in Equations 5.4 and 5.5

limri→ro

(Kmv) =XT

2(5.4)

limri→ro

(Kmv) =1

4XT

[1 +

(riro

)2]

(5.5)

Save for the factor of density ρ, this section is identical in refuting all possible options

for which a multiple ring rotor could produce a superior Kmv than a single, infinitesimally

thin ring rotor. Therefore, it can be stated that a multi-ring rotor cannot be physically

constructed that is superior in XT to a single, infinitesimally thin ring using the highest XT

material. This analysis combined with the multiple ring rotor program analysis allows for

the following statement: option 3–C produces a non-physical solution, as shown in Figure

5.11.

90


D-O-0-110 D-O-A-110 D-O-C-110 D-O-E-1103 Ring 3 Ring 3 Ring 3 Ring

Mat. Vol. Energy3.46E+02 3.46E+02 3.46E+02 3.49E+02

Density Kmv (kW· h/m3)
























5.1.11 3–B: Material Volume Energy Density, Fixed Inner Radius

Initial multi-ring analysis had also revealed that many of the combination 3–B optimized

solutions were converging to the minimum radial thickness required by the multiple ring

rotor program. A few of the converged solutions after 30,000 optimization runs were made

are shown in Table 5.5. Again, design option (B) was not used due to manufacturing failure,

which is discussed in Section 5.2.2. Design option (D) is also not presented in Table 5.3 due

91

Figure 5.12: The maximum Kmv for a single-material ring of fixed outer radius as a functionof inner radius with a series of curves corresponding to P values of –0.8, 0, and +0.8 of YT .

to the fact that the solution did not converge, with a maximum Kmv for the 3-Ring rotor

being 298 kW· h/m3. This information poses the same interesting result as with 4–C: the

optimization is moving toward an infinitesimally thin ring, with the 0.5 mm limit set by

the code reached every time. Because this behavior is so similar to that of Section 5.1.10,

the Figures used there apply here as well.

Figure 5.12 shows the peak Kmv as ri is changed compared to the ro. However, this

graph can be used for any outer radius, regardless of scaling. The reason for this is that

optimizing on Kmv is scale-invariant. That is why Figure 5.13 is a column graph instead of

a plot of behavior as the radius changes. This scale-invariant attribute of the Kmv forces

the optimization to an infinitesimally thin ring given by Equation 5.2 regardless of whether

92

Figure 5.13: Optimized Kmv for a single-material ring with various P values ranging from–0.8 to 0 times the value YT with both ri and ro allowed to vary.

it is the ri or ro that is fixed. So long as one of the radii can move, a non-physical thin ring

solution results. Therefore, the optimized result of combination 3–C is non-physical and

thus not valid, as shown in Figure 5.11.

5.1.12 3–D: Material Volume Energy Density, Constraint-Free Radii

As discussed in Sections 5.1.11 and 5.1.10, Kmv is a scale-invariant objective. If one of the

radii can vary, the objective approaches the solution given in Equation 5.4. When both

radii are allowed to vary, the optimized solution does not change. Therefore, combination

3–D also produces and non-physical, invalid result. Investigation of the objective/constraint

combinations 3–B, 3–C, and 3–D have shown that the shape factor of 0.5 applies material

93


D-I-0-110 D-I-A-110 D-I-C-110 D-I-E-1103 Ring 3 Ring 3 Ring 3 Ring

Mat. Vol. Energy3.44E+02 3.44E+02 3.44E+02 3.46E+02

Density Kmv (kW· h/m3)
























volume energy density for a multiple ring case as given in Equation 5.4.

5.1.13 The Remaining Combinations

There are three remaining combinations: 1–A, 1–C, and 4–A. These combinations have

undergone preliminary testing with the multiple ring rotor program and all 5 design options

to ensure that no non-physical or unbounded optimization behavior results. It is noted that

two of these valid objective/constraint combinations have both rotor radii fixed. This could

94

be the reason that the many of the multiple ring rotor analyses seen in Chapter 2 required

both radii to be fixed as well. Previous researchers may have attempted to allow one or

two of the rotor radii to vary, only to find unwanted results. Thus, they restructured their

analysis in a more constrained way. These three combinations are used in the Phase II

Investigation.


In this section, the results and discussion on the design options are presented. First, all five

design options are discussed and results presented. Next, the valid objective/constraint com-

binations are compared to each other. Last, some of the best design options are reviewed.

In addition to the coding schemes given in Table 3.1 and Figure 5.1, it is also important to

refer to the material listings given in Tables B.2 and B.6 to identify the materials used.

5.2.1 Variable Fiber/Matrix Option

As can be seen in Table 5.6, varying the fiber matrix ratio with no other design options pro-

vides very little useful benefit. The fiber volume fractions Vf typically rise to the maximum

of 65% fairly consistently. Since this is the default fiber fraction when design option (A)

is not used, there is almost no improvement when adding (A) in alone verses the default

fiber fraction of 65% for the K objective, and only a slight improvement when using the

Km objective. This is shown in Table 5.6 with most fiber fractions being set to 65% by the

optimization search. The accompanying performance values are given in Tables D.1, D.3,

and D.4. As can be seen in Table 5.6, design option (A) was useful for the Km by reducing

the fiber volume fraction of the inner ring. This option is useful because the density of the

epoxy is lower than the density of the glass fibers and reducing the mass of the inner ring

raises Km by placing a greater proportion of the mass of the rotor into the outer radial

regions of the rotor.

Looking at the graphs in Appendix D, a few more insights can be obtained. Most

95

Table 5.6: Best designs with (A) for three rings on each of the three valid objec-tive/constraint combinations.

E-O-A-110 E-IO-A-110 S-IO-A-1103 Ring 3 Ring 3 Ring

Angular Speed ω (rad/s) 4.33E+03 8.63E+02 9.18E+02

Ring Radius 1 (m) 1.82E-01 1.00E-01 1.00E-01

Ring Radius 2 (m) 2.06E-01 1.51E-01 1.29E-01

Ring Radius 3 (m) 2.17E-01 1.83E-01 1.81E-01

Ring Radius 4 (m) 2.50E-01 2.50E-01 2.50E-01

Fiber 1 Material 1 # 8 9 9



Fiber Fraction 1 V f 6.50E-01 6.50E-01 6.00E-02



benefit seen from (A) comes mainly when it is combined with option (E). This can be

seen in Figures D.5, D.15, and D.20. Another issue shown in these figures is that design

options (A) and (C) seem to overlap in capabilities and are competitive with each other.

As shown in Figure D.15, design combination (ACE) overlaps directly with (CE). Also,

design combination (AC) overlaps with both (A) and (C) for the K objective. For Km,

there is some improvement when these two options are used in concert, but not much. For

objective/constraint combination 1–A, option combination (AE) is superior to (CE). How-

ever, for 1–C, (CE) is superior to (AE). For 4–A, (CE) and (AE) are comparable to each

other. Given that varying the fiber/matrix ratio significantly is technically difficult for fly-

wheel rotor manufacturing using filament winding as well as other composite manufacturing

techniques, it is difficult to justify the use of this design option.

5.2.2 Radial/Hoop Fiber Option

Varying the radial/hoop fiber ratio with no other design options provides no benefit and

a severe penalty to even the default optimization with no design options selected. All of

96

the figures given in Appendix D show that any design option combination that includes

(B) suffers severe performance deficits. Both the variation on the number of rings as well

as the objective/constraint combinations offer no assistance to the poor performance of

this design option. The reason for this is shown in Figure 5.14 and Table B.4. As can be

seen in the figure, the residual loading between the two plies results in a roughly 9 MPa

tensile radial stress in the hoop ply, and a 9 MPa compressive radial stress in the radial ply.

However, Table B.4 shows that the limit for transverse tensile strength is 5 MPa. The high

modulus, low thermal expansion coefficient of the radial ply fibers are preventing the hoop

plies from thermally contracting in the radial direction. This result indicates that for the 5

MPa transverse strength, the hoop plies will fail in radial tension.

This result is not the case for the normal filament-wound rotors, so the natural question

results: Why are the radial fiber not simply eliminated by the optimization program? The

answer to this question lies in Figure 5.15. As fibers decrease in one direction or the other,

fewer fibers must hold under same force increasing the stress on them until the last fiber.

Zero fibers in one direction or the other may be peaks, but the algorithm is always pushed

away from those solutions. To deal with same stress, fewer fibers must take on more force.

When last fiber is removed, the strength associated with the epoxy is used instead of the

strength of the single remaining fiber, allowing for new stress state.

While the transverse strength limit of 5 MPa is low for multi-ply composite laminates,

there are factors that can justify this limit. Under centrifugal loading at high angular speeds,

high frequency vibrations can exist that can encourage growth in any micro-cracks that may

have been created in the hoop plies. If manufacturers cannot find a way to eliminate these

micro-cracks, the fatigue life of the flywheel rotor may be compromised. This may be the

reason that experimental manufacturing of rotors with both hoop and radial plies has been

limited. Most research up to now has been theoretical with neglect of the residual stresses.

However, with these results, it is suggested that research should switch back to performing

experimental validation that these rotors can survive both the manufacturing process and

97

Figure 5.14: Hoop and radial ply stresses for the radial direction under processing temper-ature change loading of ∆T = −110 ◦C without any centrifugal loading corresponding tothe optimal case E-O-B-110 with one ring.

preliminary testing before more theoretical work is performed to optimize rotors based on

this design option.

5.2.3 Two-Fiber Co-mingling Option

As a standalone design option, fiber co-mingling produces some benefits over the default

optimization. In many of the cases, two different fibers are chosen. However, occasional fiber

repeats occur. This can be seen in Table 5.7. The accompanying performance values are

given in Tables D.3, D.1, and D.4. Looking at the graphs in Appendix D, a few more insights

can be obtained. Option (C) provides some minor benefit when used alone as can be seen in

98

Figure 5.15: Notational illustration of the optimization behavior of design option (B),showing that a step change in the optimization as the last fiber in a direction is removed.

Figures D.15, D.5, and D.20. As discussed before, there is overlap and competition between

options (A) and (C). Figure D.5 shows that combination (AC) provides little to no benefit

over either (A) or (C) alone. Combination (CE) provides minor benefit over (E) alone, but

in a rare case, combination (ACE) provides significant benefit. The combination of (ACE)

also provides significant benefit for objective/constraint combination 4–A as shown in Figure

D.20. However, Figure D.15 shows that combination (ACE) is nearly identical to (CE),

meaning that when one of the radii are allowed to vary, the additional design option (A) is

no longer needed to help optimize the rotor. Given that fiber co-mingling is a reasonably

simple addition to the manufacturing process of filament winding for composite rotors, the

additional benefit of this design option shows that it is worthy of further investigation for

optimization.

99

Table 5.7: Best designs with (C) for three rings on each of the three valid objec-tive/constraint combinations.

E-O-C-110 E-IO-C-110 S-IO-C-1103 Ring 3 Ring 3 Ring


Ring Radius 1 (m) 1.82E-01 1.00E-01 1.00E-01

Ring Radius 2 (m) 1.98E-01 1.50E-01 1.31E-01

Ring Radius 3 (m) 2.25E-01 1.80E-01 1.73E-01

Ring Radius 4 (m) 2.50E-01 2.50E-01 2.50E-01










Fiber 1 Fraction 1 V f1 2.33E-01 3.36E-01 5.50E-02



5.2.4 Press-fitting Option

Of all the design options, press-fitting provides the most benefit when the inner and outer

radii are constrained. Design option (D) produces some amazing results and is clearly a

front runner from a theoretical standpoint, which is shown in Figures D.1, D.2, D.3, D.16,

D.17, and D.18. However, this ability to compensate for a highly constrained geometry

does not extend to the objective/constraint combination of 1–C, which is shown in Figures

D.11, D.12, and D.13. Single option designs are shown in Table 5.8. It must also be

noted that the search space for press-fitting may contain very small, sharp peaks that are

difficult for the search algorithm to identify. Many times for the constrained radii cases,

adding other design options made the space too complex to identify these high-performance

regions. Also, there have been cases where the press-fitting interferences are very small, as

100

shown for E–O–D–110 in Table 5.8 with the first intereference set to 18.5 microns. Given

that the small search peaks are difficult to find combined with the very small interference

fits that may be required, these two facts indicate that even slight deviations away from

the assigned interference can have a substantial impact on performance. Manufacturing

tolerances may not allow the optimized result to be reliably produced. Nonetheless, the

benefit of press-fitting is too great to simply discard. Other alternatives to incorporate this

design option are given in the next chapter.

Table 5.8: Best designs with (D) for three rings on each of the three valid objec-tive/constraint combinations.

E-O-D-110 E-IO-D-110 S-IO-D-1103 Ring 3 Ring 3 Ring


Ring Radius 1 (m) 1.70E-01 1.00E-01 1.00E-01

Ring Radius 2 (m) 1.87E-01 1.53E-01 1.53E-01

Ring Radius 3 (m) 2.27E-01 2.09E-01 2.09E-01

Ring Radius 4 (m) 2.50E-01 2.50E-01 2.50E-01







Interference 1 (m) 1.85E-05 2.53E-04 2.53E-04


5.2.5 Matrix Ballasting Option

Of the five options investigated, matrix ballasting is the most reliable benefit provider. A

list of designs incorporating (E) alone is given in Table 5.9. All three objective/constraint

combinations show significant benefit when (E) is used. For the highly constrained objec-

tive/constraint combinations 1–A and 4–A, (E) alone is superior to (A) and (C) alone, and

is comparable if not superior to (AC). This is shown in Figures D.5 and D.20. Option (E) is

101

only inferior to option (D). However, the greatest performance seen is when options (D) and

(E) are combined. For the objective/constraint combination 1–C, (E) alone is only slightly

below (D), and they provide comparable performance. This is shown in Figures D.11, D.12,

D.13, and D.15. When combined with (C) for (CE), this design combination is the second

highest seen, with (CDE) being only slightly higher in performance.

Table 5.9: Best designs with (E) for three rings on each of the three valid objec-tive/constraint combinations.

E-O-E-110 E-IO-E-110 S-IO-E-1103 Ring 3 Ring 3 Ring


Ring Radius 1 (m) 1.68E-01 1.00E-01 1.00E-01

Ring Radius 2 (m) 1.81E-01 1.50E-01 1.35E-01

Ring Radius 3 (m) 2.22E-01 1.84E-01 1.86E-01

Ring Radius 4 (m) 2.50E-01 2.50E-01 2.50E-01







Ballast Material 1 # 10 10 10



Ballast Fraction 1 V b · V m 8.70E-02 8.75E-02 8.74E-02



Given that the performance of this design option is generally competitive, it is not known

why it has not been investigated further after identified by Genta in the mid-1980s [18, 13].

However, since it is capable of being included into existing manufacturing processes, design

option (E) is worthy of further investigation.

102

5.2.6 Comparing the Valid Objective/Constraint Combinations

By comparing the results shown in Appendix D concerning the objective/constraint com-

binations, valuable information can be extracted. Option (D) seems to be very valuable for

compensating for the added geometric constraint of 1–A and 4–A, but when this correc-

tion is not needed, as is the case with 1–C, option (D) provides reduced additional benefit.

Although the fixed radii situation provides a relatively thick rotor of 15 cm with λ = 0.4,

it was not expected that the difference between the best results of 1–A and 1–C would

be so large. Clearly, fixing both radii arbitrarily can impose significant detriment to the

optimization. However, it is anticipated that adding more material rings with option (D)

can allow a designer to partially compensate for this penalty.

Looking at the results of objective/constraint combination 1–C alone, it seems that

there is some plateauing behavior for a variety of design option combinations. The greatest

increase in performance lies in the transition from one material ring to two. Since the

transition from two rings to three rings provides a lower performance increase, this behavior

indicates a leveling off. Full plateauing effects would be anticipated in either a 4-ring or

5-ring scenario.

Along with objective/constraint combination 1–A, 4–C also is severely penalized by the

relatively thick rotor with λ = 0.4. This penalty is especially severe since a thick rotor

opposes the goal of an infinitesimally thin ring. Tables 5.6 and 5.8 provide good indication

of the fact that the highest XT /ρ materials were not used because of this thickness penalty.

5.2.7 Best Rotor Design Option Combinations

In Table 5.10 is a summary of the best designs seen in the multiple ring rotor analysis

for the three objective/constraint combinations. As can be seen, there are significant fluc-

tuations in material choices, interferences and various volume fractions. When combined

with the fact that none of these results have converged to satisfaction, the explicit designs

themselves are not the emphasis of this section. What is emphasized is the fact that for the

103

objective/constraint combinations of 1–A and 4–A, high performance materials may not be

what is needed to optimize a design with fixed radii. However, when one of the rotor radii

can vary, the highest strength materials can be selected.

Table 5.10: Best designs for three rings on each of the three valid objective/constraintcombinations.

E-O-CDE-110 E-IO-ADE-110 E-IO-CDE-1103 Ring 3 Ring 3 Ring


Ring Radius 1 (m) 1.68E-01 1.00E-01 1.00E-01

Ring Radius 2 (m) 1.81E-01 1.60E-01 1.48E-01

Ring Radius 3 (m) 2.22E-01 2.05E-01 2.15E-01

Ring Radius 4 (m) 2.50E-01 2.50E-01 2.50E-01










Fiber 1 Fraction 1 V f1 6.25E-01 5.17E-02











Chapter 6

Conclusions & Recommendations

With this investigation, many conclusions can be made that would be highly useful to fly-

wheel rotor designers in terms of optimization methodology and design techniques. This

investigation has provided much information that has not been seen. This chapters starts

with the relevant conclusions concerning the objective/constraint investigation, then moves

on to conclusions made from the design option investigation. In the design option investiga-

tion, insight is presented on the value of these design options, the most valuable materials

used for the optimized designs, how the chosen objective/constraint combinations affect the

results, and what can be learned from the convergence issues. Then, general guidance for

flywheel designers is given concerning initial design decisions. Last, future investigation

possibilities are presented.


A summary of the objective/constraint combinations is given in Table 6.1. Just as before

with Table 3.1, objective/constraint notation can be referenced here. This table provides

necessary information to a flywheel rotor designer who must optimize based on various

design considerations. This analysis revealed that under certain objective/constraint com-

binations, non-physical solutions would result, even for multi-ring rotors. This analysis also

104

105

revealed that certain combinations produce identical designs and are thus equivalent in their

design solutions. Of the sixteen possible combinations, five combinations have been shown

to produce equivalences in their optimized design, six produce non-physical optimization

results, one produces an unbounded optimization, one is conditionally valid, and three are

considered fully valid.

Table 6.1: Optimization objective validity and equivalency for different radius constraintcombinations.

(A) (B) (C) (D)

Fixed RadiiFixed Inner Fixed Outer Constraint-

Radius Radius Free Radii

1. Total StoredValid

ConditionalValid Unbounded

Energy K Validity

2. Encl. Vol. Energy= 1–A

Non-physical= 2–C = 2–C

Density Kev = 3–D

3. Mat. Vol. Energy= 1–A

Non-physical Non-physicalNon-physical

Density Kmv = 3–D = 3–D

4. SpecificValid

Non-physical Non-physicalNon-physical

Energy Km = 4–D = 4–D

Novel contributions concerning the Phase I investigation include the following:

• It has been shown that four objective/constraint combinations produce equivalent

optimization designs to other combinations without producing non-physical results.

• Objective/constraint combination 1–D has been revealed to produce an unbounded

optimization.

• It has been discovered objective/constraint combination 1–B is conditionally valid. If

the composite material is allowed to change into a material similar to an isotropic

material, such as through the design option of variable fiber/matrix ratios, the outer

rotor radius can expand out to a few orders of magnitude beyond the set inner radius.

This results in a non-realistic rotor design.

106

• While it has been known that the optimal specific energy for a single-material, thin-

ring rotor is Km = XT2ρ , it was not known that this factor also provides the upper limit

for multi-ring rotors with design options that include of variable fiber/matrix ratios,

two-fiber co-mingling, press-fitting, and matrix ballasting.

• It has been discovered that the material volume energy density optimization objective

for multi-ring hoop-direction fiber ring rotor flywheels exhibits highly similar behavior

to that of specific energy, resulting in an infinitesimally thin ring with an optimal value

of Kmv = XT2 that is invariant to scaling.


In this section, the utility of five design options is summarized. Also, conclusions and rec-

ommendations concerning the material selections are given. A note is made concerning how

the selection of valid objective/constraint combinations affects the design. Last, information

on restricting the optimization parameters to better ensure convergence is given.

6.2.1 Design Options

For information concerning the design option descriptions, please refer to Section 3.2 and

Figure 5.1. Also, refer to Table 6.1 for identification of the objective/constraint combina-

tions.

(A) When the variable fiber/matrix design option is used independently of other options,

little-to-no benefit is found. However, when used in conjunction with other design

options, occasionally some additional benefit is produced. It is most useful in spe-

cific energy because it allows the density to be reduced. This design option is best

applied when the rotor radii are both fixed. However, in the least restrictive valid

objective/constraint combination of 1–C, it provides no additional benefit when used

in conjunction with design option combination (CE). Considering the difficulty of

107

varying the fiber/matrix ratio for the filament winding process, it is not advised to

further pursue this design option.

(B) This investigation has shown the radial/hoop multi-ply design option to be a very poor

performer due to the fact that thermal stresses of the standalone rotor indicate radial

failure in the hoop ply. This due to the high stiffness and low thermal expansion in the

radial direction created by the radial plies. Although a very low YT was placed into

the material model database which may not be representative of multi-ply laminate,

micro-cracks can be produced for even higher YT if higher processing temperatures

are used to produce the flywheel. This may also shed light on the fact that, while

many theoretical analyses have been performed on multi-ply laminate flywheel rotors,

reliable testing data seems limited. Only if the manufacturing process for design

option (B) can prove to produce a reliable flywheel should this design option be

further investigated. Otherwise, it is advised that this design option not be pursued

further.

(C) The performance of hybrid fiber-comingling is comparable to design option (A). When

used in conjunction with other design options, some benefit is usually produced. How-

ever, it is much more effective given the objective/constraint combination of 1–C. Be-

cause incorporating this design option in the existing filament-winding manufacturing

process is feasible, further investigation of this design option is warranted.

(D) This investigation has shown that the press-fitting option is the best design option to

use. Given a highly constrained optimization of fixed radii, press-fitting can substan-

tially compensate for these constraints to the point where it can definitively outper-

form all other design options. However, when the lower constraint of only a fixed outer

radius is applied, other design option combinations become competitive. Also, many

of the results shown in the highly constrained cases indicate that the high performance

peaks in the search space can be very difficult to find while occasionally producing

108

interferences below 0.1 mm. Because these tolerances are a so small, some of the

high performance press-fitting results are not considered feasible. Therefore, it is not

advised to pursue this design option concerning press-fitting variables in its current

form. However, because the benefits can be significant, it is advised to incorporate

at least one assigned press-fit interference in the middle of the rotor and attempt to

optimize on that.

(E) Matrix ballasting is the second most beneficial design option investigated. When

optimizing on specific energy, its benefits are comparable to options (A) and (C).

However, when optimizing on total stored energy, its performance is solidly above

(A) and (C). When used in conjunction with (A) and/or (C), additional benefit is

seen. In the objective/constraint combination of 1–C, the performance of design

combinations using this design option are comparable, and in some cases superior,

to the performance of combinations using press-fitting. Although this design option

has been largely abandoned after its presentation by Genta in the 1980s [18, 13],

matrix ballasting has been shown to be highly useful. Because incorporating ballast

particles into a matrix is a well established manufacturing technique, this author

advises aggressive continued investigation of this design option.

Another design option to discuss would be the number of rings to use. While this

research was not able to investigate a higher number of material rings, valuable information

can be obtained from three rings alone. For the objective/constraint combination 1–C,

the start of strong plateauing behavior is shown in all cases. This behavior would suggest

drastic diminishing returns if one or two additional material rings are added. However, if

the optimization constraints include fixed rotor radii, this is not the case. The analysis

has shown that when using press-fitting in conjunction with fixed rotor radii, no plateauing

effects are noticed for 3 rings. This observation indicates that if the inner rotor radius is

allowed to vary, fewer rings are required.

109

6.2.2 Material Selections

For the material numbers given here, please refer to Tables B.2 and B.6. Concerning the

material selections, only a few materials ended up in the optimization. These were the high

strength carbon fiber materials #1 and #5 for the outer ring, and lower stiffness, yet high-

strength, fibers such as the glass fiber materials #8 and #9. For the highly constrained

cases with thick rotor requirements, lower performance materials such as #8 and #9 were

used exclusively. Occasionally, the high density steel fiber #12 was used in the middle and

inner rings. However, once matrix ballasting was included, the high density fibers were

not needed. These observations support all previous analyses suggesting that increasing

the stiffness/density ratio as the radius increases is a valid design approach. However, the

ultra-high stiffness carbon fibers #6 and #7 were not used. This result would indicate

that in order for the highest stiffness/density fibers to be useful, they must also possess the

highest fiber strength. This support previous investigative work of Arvin [64].

For the ballasting material, the highest density material #10 was selected for the inner

material rings. For the outer material rings, either #1, #2, or #3 was used. This selection

suggests that, in addition to density manipulation, stiffness and thermal expansion tuning

are also useful.

6.2.3 Objective Constraint Combinations

For the total stored energy objective, it has been shown that there is substantial benefit

to allowing one of the rotor radii to vary and be subjected to optimization. This benefit

alone is comparable to using the press-fitting design option. For specific energy of multi-

ring rotor designs, it is now known that the optimal design still calls for an infinitesimally

thin rotor. Adding any rotor thickness detracts from the optimal solution. Nonetheless,

the total stored energy objective investigation has shown that allowing one of the radii to

vary can produce substantial benefits. In order to balance the competing specific energy

problems of a trivial solution when allowing one rotor radii to vary, compared to the overly

110

constrained decision to fixed both radii, the following solution is proposed:

• Specific energy, fixed outer radius, with a minimum total stored energy constraint

This solution allows the inner radius to vary but requires some radial thickness to store

energy. Another solution to allow a designer to optimize on specific energy would be to

include a hub into the design of the rotor. The fact that a realistic thickness of composite

material is necessary to compensate for any displacements made by the hub under spin

loading could generate a realistic design.

6.2.4 Convergence Issues

Even for as few as three material rings, there were problems getting the solutions to con-

verge to satisfaction. There are three reasons for these problems. The first reason includes

issues with design option (B). Because a viable flywheel rotor could not be constructed

with (B) and the assigned strength values, many faulty designs were calculated. The sec-

ond reason was associated with design option (D). While including the press-fitting option

resulted in fantastic performance results compared to other design option combinations,

those fantastic results were very difficult for the search algorithm to find, indicating that

the solutions corresponding to very small, sharp peaks in the solution space. The last, and

most comprehensive reason that convergence proved difficult was that so many material

options were given, allowing for a large number of discrete search space hyper-cubes to be

investigated.

For fiber co-mingling and matrix ballasting on 3-ring rotors, (12 · 12 · 10)3 = 2.99 billion

discrete material search spaces must be investigated. If future investigations are conducted,

a much lower number of materials should be used. For composite material selections, one

high stiffness fiber, one medium stiffness fiber, and one low stiffness fiber could be selected.

With the varying degrees of stiffness, it is important to make sure all fibers have a high

strength. For ballasting materials, it is important to have one that is high density material,

such as #10, and one material that is of lower density, such as #1. It is possible to add in

111

high and low stiffness characteristics as well for a total of 4 ballasting materials. If theses

guidelines are followed, then the number of discrete material search cubes can be reduced

drastically: (3 · 3 · 4)3 = 46, 656. Other options to drastically reduce the number of discrete

search cubes would be to remove fiber the co-mingling and/or matrix ballasting design

options.

6.3 Avenues for Future Work

There are many avenues for further investigation. One investigation of merit would be

to determine how many more rings would be required to obtain an optimization limit. If

the number of materials in the material database are suitably reduced, it may be possible

to obtain converged results for a higher number of material rings within a rotor. Objec-

tive/constraint combinations of 1–A and 1–C can be compared for a higher number of rings

to determine the difference in the number of material rings required for reaching the upper

limit of optimization. Also, the limits of 1–A and 1–C could be compared to each other

for a large number of material rings. The objective/constraint combination of 4–A could

be compared to the proposed combination of specific energy, fixed outer radius, with a

minimum energy storage constraint in a fashion similar to the comparison of 1–A and 1–C.

With the number of material choices reduced, it would also be feasible to more critically

investigate some of the best design option combinations. Combination (CDE) can be com-

pared to the case of (CE). These two combinations can be compared to the case of (CE)

with an array of assigned press-fit values, which could account for manufacturing tolerances.

Any or all of the valid objective/constraint combinations could be applied in these analyses.

The goal of all this research would be to find a high-performance rotor design strategy that

is attractive for a rotor designer.

Bibliography

[1] Mavizen. AMP20M1HD-A: A123 20Ah LIFEP04 prismatic pouch cell. Online. 28 May2013. http://www.mavizen.com/products/amp20/, 2012.

[2] Maxwell Technologies. DATASHEET 160V MODULE: BMOD0006 E160 B02. On-line. 28 May 2013. http://www.maxwell.com/products/ultracapacitors/docs/

maxwell_160v_datasheet_v4.pdf, 2013.

[3] Compressed air energy storage. Online, 15 July 2011 http://en.wikipedia.org/

wiki/Compressed_air_energy_storage, 2011.

[4] A. P. Coppa and S. V. Kulkarni. Composite Flywheels: Status and Performance Assess-ment and Projections. Technical Report UCRL-89085, Lawrence Livermore NationalLaboratory, 1983.

[5] Flybrid Systems. CFT KERS. Online. 28 May 2013 http://www.flybridsystems.

com/CFTKERS.html, 2013.

[6] Volvo Car Group. Volvo Cars tests of flywheel technology confirm fuel savings of upto 25 percent. Online. 28 May 2013 https://www.media.volvocars.com/global/

enhanced/en-gb/Media/Preview.aspx?mediaid=48800, 25 April 2013.

[7] Williams Hybrid Power. WHP’s Flywheel Technology. Online. 10 May 2011. http://www.williamshybridpower.com/technology/whps-flywheel-technology, 2010.

[8] Torotrak PLC. Design and Specification. Online. 11 May 2011. http://www.torotrak.com/content/16/design-and-specification.aspx1, 2010.

[9] J. J. R. Hilton, D. Isaac, and L. Cross. High Speed Flywheel Seal. Patent, US2010/0059938 A1, 2010.

[10] H. Daneshi, N. Sadrmomtasi, and M. Khederzadeh. Wind power integrated with com-pressed air. In IEEE International Conference on Power and Energy, pages 634–9,2010.

[11] W. F. Pickard, N. J. Hansing, and A. Q. Shen. Can large-scale advanced-adiabaticcompressed air energy storage be justified economically in an age of sustainable energy?Journal of Renewable and Sustainable Energy, 1, 2009.

112

113

[12] Beacon Power Corporation. Smart Energy 25 Flywheel. Online. 11 May 2011. http://www.beaconpower.com/products/smart-energy-25.asp, 2011.

[13] G. Genta. Kinetic energy storage: Theory and practice of advanced flywheel systems.Butterworth & Co. ltd., 1985.

[14] R. M. Christensen and E. M. Wu. Optimal design of anisotropic (fiber-reinforced)flywheels. Journal of Composite Materials, 11:395–404, 1977.

[15] E. L. Danfelt, S. A. Hewes, and T.-W. Chou. Optimization of composite flywheeldesign. International Journal of Mechanical Science, 19(2):69–78, 1977.

[16] P. B. Mohr and C. E. Walter. Flywheel Rotor and Containment Technology Devel-opment - Final Report. Technical Report UCRL-53448, Lawrence Livermore NationalLaboratory, 1983.

[17] K. Ikegami, J.-I. Igarashi, and E. Shirator. Composite flywheels with rim and hub.International Journal of Mechanical Science, 25(1):59–69, 1983.

[18] G. Genta. On the design of thick rim composite material filament would flywheels.Composites, 15(1):49–55, 1984.

[19] K. Miyata. Optimal structure of fiber-reinforced flywheels with maximized energydensity. Bulletin of Japanese Society of Mechanical Engineers, 28(238):565–70, 1985.

[20] G.G. Portnov and I. A. Kustova. Energy capacity of composite flywheels with contin-uous chord winding. Mechanics of Composite Materials, 5:688–94, 1988.

[21] D. H. Curtiss, P.P. Nongeau, and R. L. Puterbaugh. Advanced composite flywheelstructural design for a pulsed disk alternator. IEEE Transactions on Magnetics, 31(1):26–31, 1995.

[22] C. W. Gabrys and C. E. Bakis. Design and testing of composite flywheel rotors.Composite Materials: Testing and Design, 13:1–22, 1997.

[23] S. G. Lekhnitskii. Anisotropic Plates. Gordon & Breach Science Publishers, 1968.

[24] S. K. Ha, H.-M. Jeong, and Y.-S. Cho. Optimum design of thick-walled composite ringsfor an energy storage system. Journal of Composite Materials, 32(9):851–73, 1998.

[25] I. M. Daniel and O. Ishai. Engineering Mechanics of Composite Materials. OxfordUniversity Press, 2006.

[26] S. K. Ha, H.-I. Yang, and D.-J. Kim. Optimal design of a hybrid composite flywheelwith a permanent magnet rotor. Journal of Composite Materials, 33(16):1544–75, 1999.

[27] D. Eby, R. C. Averill, W. F. Punch, and E. D. Goodman. The optimization of flywheelsusing an injection island genetic algorithm. Online. 2 Feb. 2011 http://garage.cse.

msu.edu/papers/GARAGe99-04-01.pdf, 1999.

114

[28] R. P. Emerson and C. E. Bakis. Viscoelastic behavior of composite flywheels. InProceedings of the 45th SAMPE Symposium and Exhibition, 21 May 2000. Corrected:4 April 2001.

[29] J. Tzeng. Mechanics of composite rotating machines for pulsed power applications.IEEE Transactions on Magnetics, 37(1):328–31, 2001.

[30] S. K. Ha, D.-J. Kim, and T.-H. Sung. Optimum design of multi-ring composite flywheelrotor using a modified generalized plane strain assumption. International Journal ofMechanical Sciences, 43:993–1007, 2001.

[31] Y. Gowayed, F. Abel-Hady, G. T. Flowers, and J. J. Trudell. Optimal design of multi-direction composite flywheel rotors. Polymer Composites, 23(3):433–41, 2002.

[32] R. P. Emerson and C. E. Bakis. Viscoelastic model of anisotropic flywheels. CompositeMaterials: Testing & Design, 14:75–92, 2003.

[33] C. K. Corbin, J. M. Ganley, and S. W. Tsai. Composite flywheel rotor technologydevelopment overview. Earth & Space, pages 890–97, 2004.

[34] G. G. Portnov, C. E. Bakis, and R. P. Emerson. Some aspects of designing multirimcomposite flywheels. Mechanics of Composite Materials, 40(5):397–408, 2004.

[35] G. G. Portnov, A.-N. Uthe, I. Cruz, and R.P. Fiffe F. Arias. Design of steel-compositemulitrim cylindrical flywheels manufactured by winding with high tensioning and insitu curing. Mechanics of Composite Materials, 41(2):139–152, 2005.

[36] A. C. Arvin and C. E. Bakis. Optimal design of press-fitted filament wound compositeflywheel rotors. Composite Structures, 72:47–57, 2006.

[37] B. C. Fabien. The influence of failure criteria on the design optimization of stacked-plycomposite flywheels. Structural Multidisciplinary Optimization, 33:507–17, 2007.

[38] M. Strasik, P. E. Johnson, A. C. Day, J. Mittleider, M. D. Higgins, J. Edwards, J. R.Schindler, K. E. McCrary, C. R. Melver, D. Carlson, J. F. Gonder, and J. R. Hull.Design, fabrication, and test of a 5-kWh/100-kW flywheel energy storage utilizing ahigh-temperature superconducting bearing. IEEE Transactions on Applied Supercon-ductivity, 17(2):2133–7, 2007.

[39] M. A. Arslan. Flywheel geometrydesign for improved energy storage using finite ele-ment analysis. Materials & Design, 29:514–8, 2008.

[40] S. K. Ha, J. H. Kim, and Y. H. Han. Design of a hybrid composite flywheel multi-rimrotor system using geometric scaling factors. Journal of Composite Materials, 42(8):771–85, 2008.

[41] S. K. Ha, H. H. Han, and Y. H. Han. Design and manufacture of a composite flywheelpress-fit multi-rim rotor. Journal of Reinforced Plastics and Composites, 27(9):953–65,2008.

115

[42] M. Krack, M. Secanell, and P. Mertiny. Cost optimization of hybrid composite flywheelrotors for energy storage. Structural and Multidisciplinary Optimization, 41:779–95,2010.

[43] K.-C. Lin, J. Gou, C. Ham, S. Helkin, and Y. H. Joo. Flywheel energy storage systemwith functionally gradient nanocomposite rotor. In 5th IEEE Conference on IndustrialElectronics and Applications, pages 611–3, 2010.

[44] J. L. Prez-Aparicio and L. Ripoll. Exact, integrated and complete solutions for com-posite flywheels. Composite Structures, 93(9):1404–15, 2011.

[45] M. Krack, M. Secanell, and P. Mertiny. Cost optimization of a hybrid compositeflywheel rotor with a split-type hub using combined analystical/numerical models.Structural and Multidisciplinary Optimization, 44:57–73, 2011.

[46] S. K. Ha, M. H. Kim, S. C. Han, and T.-H. Sung. Design and spin test of a hybridcomposite flywheel rotor with a split type hub. Journal of Composite Materials, 40(23):2113–18, 2006.

[47] S. K. Ha, S. J. Kim, S. U. Nasir, and S. C. Han. Design optimization and fabricationof a hybrid composite flywheel rotor. Composite Structures, 94:32909, 2012.

[48] P. J. J. van Rensburg, A. A. Groenwold, and D. W. Wood. Optimization of cylindricalcomposite flywheel rotors for energy storage. Structural Multidisciplinary Optimization,47:13547, 2013.

[49] M. H. Sadd. Elasticity: Theory, Applicaitons, and Numerics. Academic Press, 2009.

[50] R. M. Christensen. Mechanics of Composite Materials. John Wiley & Sons, Inc., 1979.

[51] S. W. Tsai and H. T. Hahn. Introduction to Composite Materials. Technomic PublisingCompany, 1980.

[52] R. M. Christensen. A critical evaluation for a class of micro-mechanics models. Journalof the Mechanics and Physics of Solids, 38(3):379–404, 1990.

[53] J. Kennedy and E. Russell. Particle swarm optimization. In IEEE International Con-ference on Neural Networks Proceedings - Conference Proceedings, volume 4, pages1942–8, 1995.

[54] R. Storn and K. Price. Differential evolution a simple and efficient heuristic for globaloptimization over continuous spaces. Journal of Global Optimization, 11:341–59, 1997.

[55] G. R. Harik and F. G. Lobo. A parameter-less genetic algorithm. In Proceedings of theGenetic and Evolutionary Computation Conference, volume 1, pages 258–65, 1999.

[56] G. R. Harik, F. G. Lobo, and D. E. Goldberg. The compact genetic algorithm. IEEETransactions on Evolutionary Computation, 3(4):287–97, 1999.

116

[57] K. Deb, A. Anand, and D. Joshi. A computationally efficient evolutionary algorithmfor real-parameter optimization. Evolutionary Computation, 10(4):371–395, 2002.

[58] N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolutionstrategies. Evolutionary Computation, 9(2):159–95, 2001.

[59] N. Hansen. and A. Auger. A restart CMA evolution strategy with increasing populationsize. In Proceedings of the IEEE Congress on Evolutionary Computation, pages 1769–76, 2005.

[60] N. Hansen. and A. Auger. Performance evaluation of an advanced local search evolu-tionary algorithm. In Proceedings of the IEEE Congress on Evolutionary Computation,pages 1777–84, 2005.

[61] N. Hansen., A. Auger, R. Ros, S. Finck, and P. Posik. Comparing results of 31 algo-rithms from the black-box optimization benchmarking. In Genetic and EvolutionaryComputation Conference. Association for Computing Machinery, July 2010.

[62] N. Hansen. The CMA Evolution Strategy: A Tutorial. Online. 20 Jul. 2011. http://www.lri.fr/~hansen/cmatutorial.pdf, 2011.

[63] C. E. Bakis. personal correspondence. 22 December 2012.

[64] A. Arvin. Design of rotating machinery using anisotropic elasticity solutions and sim-ulated annealing. Master’s thesis, The Pennsylvania State University, 2004.

[65] E. Kreyszig. Advanced Engineering Mathematics. John Wiley & Sons, Inc., 1999.

[66] Hexcel. HexTow IM10 Carbon Fiber. Online. 12 January 2013. http://www.hexcel.com/resources/datasheets/carbon-fiber-data-sheets/im10.pdf.

[67] Toray Carbon Fibers America. T1000G Data Sheet. Online. 12 January 2013. http://www.toraycfa.com/pdfs/T1000GDataSheet.pdf.

[68] C. E. Bakis. personal correspondence. 25 May 2012.

[69] Toray Carbon Fibers America. T700G Preliminary Data Sheet. Online. 12 January2013. http://www.toraycfa.com/pdfs/T700GDataSheet.pdf.

[70] Toray Carbon Fibers America. M46J Data Sheet. Online. 12 January 2013. http://www.toraycfa.com/pdfs/M46JDataSheet.pdf.

[71] Toray Carbon Fibers America. M60J Data Sheet. Online. 12 January 2013. http://www.toraycfa.com/pdfs/M60JDataSheet.pdf.

[72] MatWeb. Aluminum 7075-T6; 7075-T651. Online. 15 January 2013. http://www.

matweb.com, 2013.

[73] MatWeb. Titanium Ti-6Al-6V-2Sn (Ti-6-6-2). Online. 15 January 2013. http://www.matweb.com, 2013.

117

[74] MatWeb. AISI Grade 18Ni (350) Maraging Steel, Aged, at RT, tested longitudinal,round bar 16 mm. Online. 15 January 2013. http://www.matweb.com, 2013.

[75] MatWeb. Ceradyne Ceralloy 546 Boron Carbide. Online. 14 February 2013. http://www.matweb.com, 2013.

[76] MatWeb. Silicon Carbide, Alpha SiC. Online. 14 February 2013. http://www.matweb.com, 2013.

[77] MatWeb. Overview - Alumina, 98%. Online. 14 February 2013. http://www.matweb.com, 2013.

[78] MatWeb. Titanium Carbide, TiC. Online. 14 February 2013. http://www.matweb.com,2013.

[79] MatWeb. Zirconium Carbide. Online. 14 February 2013. http://www.matweb.com,2013.

[80] Iron. Online, 8 June 2011 http://en.wikipedia.org/wiki/Iron, 2012.

[81] Lead. Online, 17 June 2011 http://en.wikipedia.org/wiki/Lead, 2012.

[82] MatWeb. Tantalum Carbide, TaC. Online. 14 February 2013. http://www.matweb.com/search, 2013.

[83] MatWeb. Tungsten Carbide, WC. Online. 14 February 2013. http://www.matweb.com,2013.

Appendix A

Analytic Formulations

The stress and failure formulations presented for the single ring rotor program are based

on the Gabrys & Bakis article [22] as well as books by Genta [13] and Lekhnitskii [23]. The

stress and failure formulations presented for the multiple ring rotor program are based on

multiple articles by Ha et al. [24, 26, 30] and the book by Daniel and Ishai [25]. In these

formulations, each symbol is not denoted below the equations as with the main chapters.

This is done to save space and prevent redundancy in denoting the symbols. However, all

the symbols are given in the List of Symbols.

A.1 Single Ring Formulation

Equations A.1 through A.4 define parameters that are used to simplify the formulas for

stresses in Equations A.5 and A.6.

χ =r

ro(A.1)

λ =riro

(A.2)

118

119

µ =

√EhEr

(A.3)

L =λ−µ−1 − λ2

λ−µ−1 − λµ−1(A.4)

σh = ρω2r2o

(3 + νhr9− µ2

)[µLχµ−1 + µ (L− 1)χ−µ−1 − χ2

(µ2 + 3νhr3 + νhr

)]+

µ

1− λ2µ

[(W − Pλµ+1

)χµ−1 +

(Wλµ−1 − P

)λµ+1χ−µ−1

](A.5)

σr = ρω2r2o

(3 + νhr9− µ2

)[Lχµ−1 − (L− 1)χ−µ−1 − χ2

]+

1

1− λ2µ

[(W − Pλµ+1)χµ−1 − (Wλµ−1 − P )λµ+1χ−µ−1

](A.6)

For the inner and outer boundary radial stresses, P and W , a positive value represents

radial tension, while a negative value represents radial compression. Equations A.1 through

A.6 are used to determine the radial and hoop stress, σr and σh, at a radial location r.

To determine the maximum stresses for given radial dimensions, it is necessary to eval-

uate Equations A.5 and A.6 where their first derivatives are zero and also at the inner and

outer boundaries to determine the maximum hoop and radial stress locations. With the

maximum stress locations known, it is possible to calculate the maximum angular speeds

for the given maximum strength using the maximum stress failure criterion as shown in

Equations A.7 and A.8.

ωh =

√√√√ 1

ρr2o

(9− µ2

3 + νhr

)XT − µ

1−λ2µ

[(W − Pλµ+1

)χµ−1 + λµ+1

(Wλµ−1 − P

)χ−µ−1

]µLχµ−1 + µ (L− 1)χ−µ−1 − χ2

(µ2+3νhr3+νhr

) (A.7)

ωr =

√1

ρr2o

(9− µ2

3 + νhr

)YT − 1

1−λ2µ

[(W − Pλµ+1

)χµ−1 − λµ+1

(Wλµ−1 − P

)χ−µ−1

]Lχµ−1 − (L− 1)χ−µ−1 − χ2

(A.8)

For these equations, the maximum angular speed based on hoop stress ωh can be com-

120

pared to the radial stress maximum angular speed ωr to determine which is more limiting

(lower). Note that the hoop tensile strength XT and radial tensile strength YT are used

in these equations as well. Because W is zero for the outermost ring and the magnitude

of P is always lower than the magnitude of the radial tensile strength YT , the compressive

strengths XC and YC are not considered due to the more conservative boundary stress limits

provided by the magnitude of YT . The only cases where the compressive strengths are used

are when a compressive P is applied such that the boundary is not subject to a compres-

sive radial stress greater than the maximum stress failure limits allow. Other issues with

these equations include material-related non-physical behavior such as µ = 3 and νhr = −3.

These specific property values are not used in this analysis.

A.2 Multiple Ring Formulation

This formulation first starts with defining the Thermo-elastic Stress—Strain Rela-

tions. Using the plane stress assumption σz = 0, and the fact that shear stresses are not

included in this analysis, these relations can be defined according to Equation A.9.

(σθσr

)=

[Qθθ QθrQrθ Qrr

]{(εθεr

)+

(αθαr

)(∆T )

}(A.9)

The next set of equations to develop are the Force Equilibrium Equations. For a

thin disk, cylindrical coordinates are used. For centrifugal loading under equilibrium, there

are no shear stresses involved. Also, the body forces for centrifugal loading are in the radial

direction only. Therefore, only the radial force component equilibrium equation with the

shear stresses set to zero is relevant. These assumptions result in Equation A.10.

dσrdr

+σr − σθ

r+ ρrω2 = 0 (A.10)

The next step is to determine and simplify the relevant Strain—Displacement Re-

121

lations. For the plane stress assumption with centrifugal loading, only two of the strain

components are needed. These are the radial and circumferential strains. Since the only

displacements for centrifugal loading are in the radial direction, only radial displacements

exist, providing the relevant relations as given in Equations A.11 and A.12

εθ =urr

(A.11)

εr =∂ur∂r

(A.12)

The previous equations give us the ability to relate forces to stresses (Equilibrium Equa-

tion), stresses to strains (Stress—Strain Relations), and strains to displacements (Strain—

Displacement Relations). These equations, with boundary conditions, are all that is needed

to completely define the system. The goal is to determine the stresses, strains, and dis-

placements in terms of equations that are only dependent on the radial coordinate r. In

order to do this, a number of mathematical steps are performed:

1. Use the Stress—Strain Relations and Strain—Displacement Relations to develop equa-tions for the stresses in terms of displacements.

2. Take the newly-formed Stress—Displacement Relations and place them in the Equi-librium Equation to obtain an Ordinary Differential Equation (ODE) in terms of theradial displacement ur and radial coordinate r.

3. Solve the second-order ODE. Both the homogeneous and particular solutions are rel-evant.

4. Using the radial displacement solution in terms of the radius, solve for the stressesand strains in terms of the radius.

The Stress—Displacement Relations are given in Equations A.13 and A.14.

σθ = Qθθurr

+Qθr∂ur∂r− (Qθθαθ +Qθrαr) ∆T (A.13)

122

σr = Qθrurr

+Qrr∂ur∂r− (Qθrαθ +Qrrαr) ∆T (A.14)

These equations are placed into the Equilibrium Equation to produce what is given in

Equation A.15.

r2d2urdr2

+ rdurdr− QθθQrr

ur = − 1

Qrr

{ρr3ω2 + [(Qθθ −Qθr)αθ + (Qθr −Qrr)αr] r∆T

}(A.15)

For this ordinary differential equation, the homogeneous solution is of the form of an

Euler-Cauchy equation and can be evaluated as such. For the particular solution, the

Method of Variation of Parameters is applied, where the Wronskian is found and used [65].

The solution of the radial displacement in terms of the radial coordinate alone is given in

Equation A.19, with material property factors given in Equations A.16, A.17, and A.18.

κ =

√QθθQrr

(A.16)

φ0 =1

Qrr

(1

κ2 − 9

)(A.17)

φα1 =1

Qrr

(1

κ2 − 1

)[(Qθθ −Qθr)αθ + (Qθr −Qrr)αr] (A.18)

ur = Carκ + Cbr

−κ + φ0ρr3ω2 + φα1r (∆T ) (A.19)

Knowing the radial displacement in terms of the radial coordinate, all the radial and

circumferential stress and strains can be determined in terms of the radial coordinate as

well. These are given in Equations A.28 through A.31, with additional material property

factors given in Equations A.20 through A.27.

123

φ1 = Qθr + κQrr (A.20)

φ2 = Qθr − κQrr (A.21)

φ3 = φ0 (Qθr + 3Qrr) (A.22)

φ4 = Qθθ + κQθr (A.23)

φ5 = Qθθ − κQθr (A.24)

φ6 = φ0 (Qθθ + 3Qθr) (A.25)

φα2 = (Qθr +Qrr)φα1 − (Qθrαθ +Qrrαr) (A.26)

φα3 = (Qθθ +Qθr)φα1 − (Qθθαθ +Qθrαr) (A.27)

εr = κCarκ−1 − κCbr−κ−1 + 3φ0ρr

2ω2 + φα1 (∆T ) (A.28)

εθ = Carκ−1 + Cbr

−κ−1 + φ0ρr2ω2 + φα1 (∆T ) (A.29)

σr = Caφ1rκ−1 + Cbφ2r

−κ−1 + φ3ρr2ω2 + φα2 (∆T ) (A.30)

σθ = Caφ4rκ−1 + Cbφ5r

−κ−1 + φ6ρr2ω2 + φα3 (∆T ) (A.31)

Now that the equations for stress, strain, and displacement are known for a centrifugally

loaded ring, the next step is to construct a series of equations to solve for the unknown

solution constants with boundary conditions set as the adjacent elements. This is performed

in the following steps:

1. For a single ring, two displacement equations for the inner and outer radial boundariesare constructed as a matrix equation.

2. For a single ring, two normalized force equations for the inner and outer radial bound-

124

aries based on the stress equations are constructed as a matrix equation.

3. The displacement and normalized force matrix equations are rewritten and set equalto the unknown constant vector.

4. The displacement and normalized force matrix equations are combined to produceforce-displacement equations for a single ring.

5. The equations for multiple rings are combined for a global force-displacement equation.

For the inner and outer ring radius, the boundary displacement vector can be constructed

from the radial displacement equations to produce Equation A.37 with factor definitions

given in Equations A.32 through A.36.

u =

(uriuro

)(A.32)

uω = φ0ρω2

(r3i

r3o

)(A.33)

u∆T = φα1 (∆T )

(riro

)(A.34)

g =

(rκi r−κirκo r−κo

)(A.35)

c =

(CaCb

)(A.36)

u = uω + u∆T + g c (A.37)

The boundary force vector can also be constructed for the outer and inner edges of the

ring. The axial length is removed because it is constant throughout the equations, but

the radial lengths are not. This is shown in Equation A.43 with factor definitions given in

Equations A.38 through A.42.

125

f b =

(−riσrroσr

)(A.38)

fσ = φ3ρω2

(−r3

i

r3o

)(A.39)

f∆T = φα2 (∆T )

(−riro

)(A.40)

I∗ =

(−1 00 1

)(A.41)

Φ =

(φ1 00 φ2

)(A.42)

f b = fσ + f∆T + I∗ gΦ c (A.43)

These equations can be rearranged and set equal to each other to eliminate some factors

as shown in Equation A.47 with factor definitions given in Equations A.44 through A.46.

ζ =riro

(A.44)

ζ1 = ζ−κ − ζκ (A.45)

k = I∗ gΦ g −1 =1

ζ1

(φ1ζ

κ − φ2ζ−κ φ2 − φ1

φ2 − φ1 φ1ζ−κ − φ2ζ

κ

)(A.46)

f b − fσ − f∆T = k u− k uω − k u∆T = I∗ gΦ c (A.47)

Setting the previous force and displacement equations equal allows us to eliminate the

constant vector and rearrange the previous equations into a form given by Equation A.50

with factor definitions given in Equations A.48 and A.49.

126

fω = −fσ + k uω (A.48)

f ε∆T = −f∆T + k u∆T (A.49)

k u = f b + fω + f ε∆T (A.50)

From here, the next step is to define the boundary conditions between each material

ring. Equation A.51 is a statement that the radial stresses at the boundary condition of

each ring interface must match. Equation A.52 is the statement that ring displacement at

the boundary condition of each ring interface must match, unless there is a press-fit condi-

tion which produces a set small displacement that must be added into the ring interface.

Equation A.53 is the loading term associated with the interference displacement. Note that

the notation for n is such that it counts upward from the inner radius, meaning the inner

ring is denoted as Ring 1, and the outer ring is denoted as Ring N . Also, the assigned ring

radii are also defined as increasing in number from inner to outer radius.

σ(n)ro = σ(n+1)

ri (A.51)

u(n)ro + δ(n) = u(n+1)

ri (A.52)

f(n)δ = k

(n)(

0

δ(n)

)=

(f

(n)δ1

f(n)δ2

)(A.53)

With these conditions, the global system of equations can be constructed to solve for

all of the displacements. This is shown in Equation A.60 with factor definitions given in

Equations A.54 through A.59.

127

U =

u(1)r1...

u(n−1)r1

u(n)r1...

u(N)r1

u(N)r2

(A.54)

K =

N∑n=1

k (n) =

k(1)11 · · · 0 0 · · · 0...

. . ....

... . .. ...

0 · · · k(n−1)22 + k

(n)11 k

(n)12 · · · 0

0 · · · k(n)21 k

(n)22 + k

(n+1)11 · · · 0

... . .. ...

.... . .

...

0 · · · 0 0 · · · k(N)22

(A.55)

F b =N∑n=1

f(n)b =

−r1σ(1)r1

0

...

0

−rN+1σ(N)r2

(A.56)

Fω =N∑n=1

f(n)b =

f(1)ω1

...

f(n−1)ω2 + f

(n)ω1

f(n)ω2 + f

(n+1)ω1

...

f(N)ω2

(A.57)

128

F ε∆T =N∑n=1

f (n)ε∆T

=

f(1)ε∆T1

...

f(n−1)ε∆T2

+ f(n)ε∆T1

f(n)ε∆T2

+ f(n+1)ε∆T1

...

f(N)ε∆T2

(A.58)

F δ =N∑n=1

f(n)δ =

f(1)δ1...

f(n−1)δ2

+ f(n)δ1

f(n)δ2

+ f(n+1)δ1

...

f(N)δ2

(A.59)

U = K −1(F b + Fω + F ε∆T + F δ

)(A.60)

With this global displacement vector information, each displacement pair for one mate-

rial ring can be used to calculate c corresponding to each material ring. This is shown in

Equation A.62, which corresponds back to a single material ring. Equation A.61 corresponds

to the displacement vector resulting from the set interference.

uδ =

(0δn

)(A.61)

c = g−1

(u− uω − u∆T − uδ) (A.62)

Solving this series of equations provides all of the boundary condition constants for the

129

stress Equations A.30 and A.31. These stress equations can be used directly for a single

ply composite where all the fibers are oriented in one direction. However, if there are

both radial and hoop fiber orientations, a muli-ply stress calculation must be used. For

the multi-ply stress calculation, the overall strains given in Equations A.28 and A.29 must

first be calculated. With the strains and the individual ply stiffness matrices known, the

stresses can be calculated from the Stress—Strain Relations given in Equation A.9. Once

these stresses are known, failure criteria can be applied to all radial locations for which

the stresses were calculated. With the maximum stress criterion, it is a simple matter of

comparing the stress values to the material strengths. However, for the Tsai-Wu failure

criterion, the calculations given in Equations A.63 through A.67 must be performed. This

failure criterion is satisfied when R = 1.

σT Z σ + Z σR−R2 = 0 (A.63)

R =

Z σ +

√(Z σ)2

+ 4(σT Z σ

)2

(A.64)

σ =

(σθσr

)(A.65)

Z =

(1

XT− 1

XC

1

YT− 1

YC

)(A.66)

Z =

1

XTXC−1

2

√1

XTXC

1

YTYC

−1

2

√1

XTXC

1

YTYC

1

YTYC

(A.67)

A.2.1 Stress Calculation Procedure

All of the information required to calculate the stresses for all the small flywheel thin shells

is given. The computational procedure is as follows:

130

1. Determine flywheel speed ω, rim radii r(n), rim interface displacements δ(n), andcomposite material properties.

2. Calculate material property constants and obtain the spring constant coefficients for

the stiffness matrix K.

3. Calculate all the force terms F b, Fω, F ε∆T , and F δ; with the known information.

4. Determine all of the displacements by evaluating the Equation A.60.

5. Determine the integration constants C(n)a and C

(n)b for each material ring using Equa-

tion A.62 and the global displacement information.

6. With all the integration constant information obtained, determine the radial andcircumferential plane stresses for each ply using either Equations A.30 and A.31; orEquations A.9, A.28, and A.29.

7. Apply the maximum stress and 2-D Tsai-Wu failure criteria on all stresses calculated.

Appendix B

Composite Material Properties

This appendix contains the material properties used in all calculations. In the single ring

rotor program, the material properties are not allowed to vary according to any material

models. Therefore, the information required is much less than the multiple ring program.

The material properties were obtained from a variety of sources, which are referenced the

tables of this appendix. For the strength values given in Table B.4, a 30% fatigue knockdown

penalty (70% of original property value) was incorporated for all carbon fiber composites,

and a 50% fatigue knockdown penalty was incorporated for all other fiber materials.

131

132

Table B.1: Material properties of selected unidirectional fiber composite materials for thesingle material ring rotor optimization program [25, 66].

E-Glass/EpoxyHigh StrengthCarbon/Epoxy

Fiber Volume Fraction Vf 0.55 0.60

Mass Density ρ (g/cm3) 1.97 1.60

Hoop Modulus Eh (GPa) 41 190

Radial Modulus Er (GPa) 10.4 9.9

Hoop Tensile Strength Xt (MPa) 1140 3310

Radial Tensile Strength Yt (MPa) 39 62

Hoop Comp. Strength Xc (MPa) 620 1793

Radial Comp. Strength Yc (MPa) 128 200

Poissons Ratio νθr 0.28 0.35

Table B.2: Densities, composite volume fractions, elastic moduli, and Poisson’s ratios forthe multiple ring rotor program composite materials database.

Material LabelVolume Density Poisson’sFraction ρ (g/cm3) Ratio νθr

1 [67, 68] T1000G-9405,9470EPOXY 0.55 1.57 0.30

2 [69, 68] T700G-9405,9470EPOXY 0.55 1.57 0.34

3 [25] AS4,3501-6EPOXY 0.63 1.60 0.27

4 [25] IM6G,3501-6EPOXY 0.66 1.62 0.31

5 [25] IM7,977-3EPOXY 0.65 1.61 0.35

6 [70, 68] M46J-9405,9470EPOXY 0.55 1.59 0.30

7 [71, 68] M60J-9405,9470EPOXY 0.55 1.64 0.30

8 [25, 68] S2GLASS-9405,9470EPOXY 0.60 1.94 0.28

9 [25, 68] EGLASS-9405/9470EPOXY 0.60 2.02 0.28

10 [72, 68] Al7075-9405/9470EPOXY 0.60 2.15 0.358

11 [73, 68] Ti-6Al-6V-2Sn-9405/9470EPOXY 0.60 3.19 0.352

12 [74, 68] 350MaragSteel-9405/9470EPOXY 0.60 5.31 0.34

133

Table B.3: Longitudinal and transverse elastic moduli and coefficients of thermal expansionfor the multiple ring rotor program composite materials database.

Long. Mod. Trans. Mod. Long. CTE Trans. CTEE1 (GPa) E2 (GPa) α1 ( ◦C−1 · 10−6) α2 ( ◦C−1 · 10−6)

1 [67, 68] 1.65E+2 7.50 7.00E-1 3.50E+1

2 [69, 68] 1.25E+2 7.80 7.00E-1 3.50E+1

3 [25] 1.47E+2 1.03E+1 -9.00E-1 2.70E+1

4 [25] 1.69E+2 9.00 -9.00E-1 2.50E+1

5 [25] 1.90E+2 9.90 -9.00E-1 2.20E+1

6 [70, 68] 2.45E+2 6.90 -4.00E-1 3.50E+1

7 [71, 68] 3.30E+2 5.90 -4.00E-1 3.50E+1

8 [25, 68] 5.30E+1 1.80E+1 5.00 2.00E+1

9 [25, 68] 4.50E+1 1.80E+1 5.00 2.00E+1

10 [72, 68] 4.50E+1 1.79E+1 2.43E+1 3.253E+1

11 [73, 68] 7.21E+1 1.99E+1 9.44 2.59E+1

12 [74, 68] 1.22E+2 2.15E+1 1.175E+1 2.72E+1

Table B.4: Longitudinal and transverse tensile and compressive strengths for the multiplering rotor program composite materials database.

Long. Tensile Trans. Tensile Long. Comp. Trans. Comp.Strength Xt(MPa)Strength Yt(MPa)Strength Xc(MPa)Strength Yc(MPa)

1 [67, 68] 2.13E+3 5.00 7.85E+2 1.40E+2

2 [69, 68] 1.72E+3 5.00 7.85E+2 1.40E+2

3 [25] 1.60E+3 5.00 8.60E+2 1.40E+2

4 [25] 1.57E+3 5.00 8.40E+2 1.40E+2

5 [25] 2.28E+3 5.00 7.95E+2 1.40E+2

6 [70, 68] 1.51E+3 5.00 4.90E+2 1.40E+2

7 [71, 68] 1.23E+3 5.00 3.90E+2 1.40E+2

8 [25, 68] 8.00E+2 5.00 5.60E+2 1.40E+2

9 [25, 68] 6.05E+2 5.00 5.60E+2 1.40E+2

10 [72, 68] 1.58E+2 5.00 1.58E+2 1.40E+2

11 [73, 68] 3.73E+2 5.00 3.73E+2 1.40E+2

12 [74, 68] 7.20E+2 5.00 7.20E+2 1.40E+2

134

Table B.5: Epoxy matrix density, Poisson’s ratios, elastic moduli, and coefficients of thermalexpansion for the multiple ring rotor program composite materials database.

Material LabelDensity Poisson’s Modulus CTE α2

ρ (g/cm3) Ratio νθr E1 (GPa) ( ◦C−1 · 10−6)

1 [68] SHELL-EPON9405,9470EPOXY 1.16 0.40 4.83 4.00E+1


3 [25] 3501-6EPOXY 1.27 0.35 4.30 4.50E+1

4 [25] 3501-6EPOXY 1.27 0.35 4.30 4.50E+1

5 [25] 977-3EPOXY 1.28 0.35 3.70 4.00E+1








Table B.6: Ballast material density, Poisson’s Ratios, elastic moduli, and coefficients ofthermal expansion for the multiple ring rotor program composite materials database.

Material LabelDensity Poisson’s Modulus CTE α2

ρ (g/cm3) Ratio νθr E1 (GPa) ( ◦C−1 · 10−6)

1 [25] S-2 Glass 2.460 0.23 8.69E+1 1.60

2 [75] BoronCarbide 2.500 0.17 4.60E+2 5.60

3 [76] SiliconCarbide 3.100 0.14 4.10E+2 3.90

4 [77] AluminumOxide 3.900 0.22 3.40E+2 8.10

5 [78] TitaniumCarbide 4.940 0.185 4.49E+2 7.70

6 [79] ZirconiumCarbide 6.560 0.20 3.95E+2 7.00

7 [80] PureIron 7.874 0.29 2.11E+2 1.18E+1

8 [81] PureLead 1.134E+1 0.44 1.60E+1 2.89E+1

9 [82] TantalumCarbide 1.430E+1 0.24 4.82E+2 6.30

10 [83] TungstenCarbide 1.570E+1 0.24 6.83E+2 6.25

Appendix C

Composite Material Models

The composite material database given in Appendix B provides both composite and stand-

alone epoxy material information. With these models and material information concerning

the epoxy resin and unidirectional composite, the equivalent fiber properties can be back

calculated for determining composite information of an arbitrary fiber volume fraction. The

epoxy material information is also needed to calculate the new modified matrix material

that incorporates ballasting interstitial particles. If the two-fiber co-mingling design option

is used, the properties of both fibers as well as the epoxy material are used in modified

micromechanics models. Last, if there are stacked radial and hoop plies, then a stiffness

matrix for the multi-ply laminate is constructed. Material models are derived from many

references [25, 50, 52, 51].

C.1 Preliminary Calculation Models

This section calculates the equivalent fiber properties for the given composite and epoxy

database information. It includes the fiber density ρf , longitudinal modulus Ef1 , Poisson’s

ratio νf12, transverse modulus Ef2 , longitudinal tensile strength F fT , matrix shear modulus

Gm, fiber longitudinal coefficient of thermal expansion αf1 , and the transverse coefficient of

thermal expansion αf2 [25]. For the transverse tensile strengths YT , flywheel testing data

135

136

shows that they are significantly lower than predicted, usually resulting in premature failure.

Therefore, an an engineering estimate is used for all composites as shown in Table B.4. For

the transverse modulus E2, the Halpin-Tsai semi-empirical model is used with ξT = 1.68.

This factor was determined by performing a least-squares fit transverse modulus data for

glass-fiber composites of various fiber volume fractions given Figure 9.5 in the book by Tsai

in [51]. These properties are calculated according to Equations C.1 through C.8.

ρf =ρ− ρmV m

V f(C.1)

Ef1 =E1 − EmV m

V f(C.2)

νf12 =ν12 − νmV m

V f(C.3)

Ef2 =(Em)2 ξT

(V f − 1

)+ E2E

m(V f + ξT

)E2 (V f − 1) + Em (V fξT + 1)

(C.4)

F fT ≈XTE

f1

Ef1Vf + EmV m

(C.5)

Gm ≈ XCVm (C.6)

αf1 =α1

(Ef1V

f + EmV m)− EmαmV m

Ef1Vf

(C.7)

αf2 =1

V f

[α2 − νf12α

f1V

f − αmV m (1 + νm) +(νf12V

f + α1νmV m

)](C.8)

137

C.2 Ballast Calculations

If ballast material is used, the next step after the preliminary calculations would be to

calculate the density, elastic modulus, Poisson’s ratio, and thermal coefficient of expansion

for the new matrix material. For the density, the rule of mixtures is used. For the elastic

modulus and he Poisson’s ratio, the article by Christensen is used [52]. In this article,

the elastic modulus and Poisson’s ratio are calculated first by calculating the bulk and

shear modulus. The book by Christensen is used for the thermal coefficient of expansion

[50]. Note that both V b and V e are given as volume fractions of the matrix material V m.

Therefore, the ballast fraction of the entire composite material is V b · V m. Calculations for

these properties are given in Equations C.9 through C.24.

ρm = ρbV b + ρeV e (C.9)

Ke =Ee

3 (1− 2νe)(C.10)

Kb =Eb

3 (1− 2νb)(C.11)

Ge =Ee

2 (1 + νe)(C.12)

Gb =Eb

2 (1 + νb)(C.13)

Km = Ke +V b(Kb −Ke

)1 + (1− V b)

Kb −Ke

Ke + 43G

e

(C.14)

η1 =

(Gb

Ge− 1

)(7− 10νe)

(7 + 5νb

)+ 105

(νb − νe

)(C.15)

138

η2 =

(Gb

Ge− 1

)(7 + 5νb

)+ 35

(1− νb

)(C.16)

η3 =

(Gb

Ge− 1

)(8− 10νe) + 15 (1− νe) (C.17)

A = 8

(Gb

Ge− 1

)(4− 5νe) η1

(V b) 10

3 − 2

[63

(Gb

Ge− 1

)η2 + 2η1η3

](V b) 7

3

+ 252

(Gb

Ge− 1

)η2

(V b) 5

3 − 50

(Gb

Ge− 1

)[7− 12νe + 8 (νe)2

]η2V

b

+ 4 (7− 10νe) η2η3 (C.18)

B = − 4

(Gb

Ge− 1

)(1− 5νe) η1

(V b) 10

3+ 4

[63

(Gb

Ge− 1

)η2 + 2η1η3

](V b) 7

3

− 504

(Gb

Ge− 1

)η2

(V b) 5

3+ 150

(Gb

Ge− 1

)(3− νe) νeη2V

b

+ 3 (15νe − 7) η2η3 (C.19)

C = 4

(Gb

Ge− 1

)(5νe − 7) η1

(V b) 10

3 − 2

[63

(Gb

Ge− 1

)η2 + 2η1η3

](V b) 7

3

+ 252

(Gb

Ge− 1

)η2

(V b) 5

3+ 25

(Gb

Ge− 1

)[(νe)2 − 7

]η2V

b

− (7 + 5νe) η2η3 (C.20)

Gm =Ge

2A

(−B +

√B2 − 4AC

)(C.21)

Em =9KmGm

3Km +Gm(C.22)

νm =3Km − 2Gm

2 (3Km +Gm)(C.23)

αm = αe +αb − αe(1

Kb− 1

Ke

) ( 1

Km− 1

Ke

)(C.24)

139

C.3 Single Fiber Calculations

For a single fiber composite, calculating the material properties for a composite of arbitrary

fiber volume fraction is simply the reverse of the pre-calculations, with the fiber volume

fractions set by the algorithm, or assigned to the default value of 65% if the fiber fraction is

not allowed to vary [25]. To maintain consistency in the models used, the following factor

remains the same: ξT = 1.68.

ρ = ρfV f + ρmV m (C.25)

E1 = Ef1Vf + EmV m (C.26)

ν12 = νf12Vf + νmV m (C.27)

ηT =Ef2 − Em

Ef2 + ξTEm(C.28)

E2 = Em(

1 + ξT ηTVf

1− ξTV f

)(C.29)

XT ≈ F fT

(V f + V mE

m

Ef1

)(C.30)

XC ≈Gm

V m(C.31)

α1 =αf1E

f1V

f + αmEmV m

Ef1Vf + EmV m

(C.32)

α2 = αf2Vf

(1 + νf12

αf1

αf2

)+ αmV m (1 + νm)− α1

(νf12V

f + νmV m)

(C.33)

140

Q =

[Qθθ QθrQrθ Qrr

]=

1

E1−ν12

E1

−ν12

E1

1

E2

−1

(C.34)

α =

(αθαr

)=

(α1

α2

)(C.35)

C.4 Two-Fiber Calculations

For the fiber co-mingling model, two fibers are used. The preliminary calculation values

are used to compute hybrid composite values using modified micro-mechanical models. For

the density, longitudinal modulus, and Poisson’s ratio; the rule of mixtures is used. For

the transverse modulus, an expanded version of the Halpin-Tsai equations is used, except

that the transverse fiber modulus is replaced by a series spring model of the two fibers [25].

Because the two fiber composites used may include different matrix materials, the matrix

properties of the two composite materials m1 and m2 must be averaged in addition to the

fiber properties f1 and f2. Note that for Equation C.44, the first fiber is assumed to have

the lowest strain to failure.

ρm =ρm1V f1 + ρm2V f2

V f1 + V f2(C.36)

ρ = ρf1V f1 + ρf2V f2 + ρmV m (C.37)

Em =Em1V f1 + Em2V f2

V f1 + V f2(C.38)

E1 = Ef11 V f1 + Ef2

1 V f2 + EmV m (C.39)

νm =νm1V f1 + νm2V f2

V f1 + V f2(C.40)

141

ν12 = νf112V

f1 + νf212V

f2 + νmV m (C.41)

Ef2 =Ef1

2 Ef22

(V f1 + V f2

)Ef1

2 V f2 + Ef22 V f1

(C.42)

E2 = Em

[1 + ξT ηT

(V f1 + V f2

)1− ηT (V f1 + V f2)

](C.43)

XT ≈ F f1T

(V f1 + V f2E

f2

Ef1+ V m E

m

Ef1

)(C.44)

XC ≈Gm1V f1 +Gm2V f2

V m (V f1 + V f2)(C.45)

αm =αm1V f1 + αm2V f2

V f1 + V f2(C.46)

α1 =αf1

1 Ef11 V f1 + αf2

1 Ef21 V f2 + αmEmV m

Ef11 V f1 + Ef2

1 V f2 + EmV m(C.47)

α2 = V f1(αf1

2 + νf112α

f11

)+ V f2

(αf2

2 + νf212α

f21

)+αmV m (1 + νm)− α1

(νf1

12Vf1 + νf2

12Vf2 + νmV m

)(C.48)

C.5 Two-Ply Laminate Calculations

This multi-ply laminate analysis is derived from the book by Daniel & Ishai [25]. However,

it is highly simplified compared to the book calculation procedures. This is due to the fact

that the laminate model is designed for a symmetric, balanced laminate with an overall

laminate thickness set to unity, h = 1. For these calculations, the radial volume fraction Vr

of the total laminate thickness is needed.

142

Q =

1

E1−ν12

E1

−ν12

E1

1

E2

−1

(1− Vr) +

1

E2−ν12

E1

−ν12

E1

1

E1

−1

(Vr) (C.49)

α =

Qθθ Qθr

Qrθ Qrr

−1 α1

α2

1

E1−ν12

E1

−ν12

E1

1

E2

−1

(1− Vr)

+

α2

α1

1

E2−ν12

E1

−ν12

E1

1

E1

−1

(Vr)

(C.50)

Appendix D

Multiple Ring Rotor Optimization

Program Results

The results shown in this appendix represent the best results after thousands of optimiza-

tion runs for each data point. For the objective/constraint combinations that are considered

completely valid shown in Sections D.1, D.3, and D.4; 75,000 optimization runs were per-

formed. For the objective/constraint combination that is considered conditionally valid is

shown in Section D.2; 35,000 optimization runs were performed because of the generation

of non-reasonable results. For each run, anywhere from 1,000 functional evaluations for

low search dimension cases to the upper limit of 200,000 were made. For Tables D.1, D.2,

D.3, and D.4; the gray shaded listings indicate solutions that did not converge, indicating a

low probability that the global optimum was found. In all cases, a processing temperature

change of ∆T = −110 ◦C was used to include temperature-related residual stresses.

143

144

D.1

Tota

lS

tore

dE

nerg

y,

Fix

ed

Rad

ii

Tab

leD

.1:

En

ergy

obje

ctiv

ew

ith

both

rad

iifi

xed

at10

0m

man

d25

0m

m.

Bes

tre

sult

saf

ter

75,0

00

op

tim

izati

on

run

s.G

ray

shad

ing

ind

icate

sco

nve

rgen

cefa

ilu

re.

Un

its

are

inkW·h

.

# o

f R

ings

E-I

O-0

-11

0E

-IO

-A-1

10

E-I

O-B

-110

E-I

O-C

-110

E-I

O-D

-11

0E

-IO

-E-1

10

11.5

2E

-01

2.6

9E

-01

9.1

0E

-13

1.5

2E

-01

1.5

2E

-01

3.6

8E

-01

21.3

9E

+00

1.3

9E

+00

9.3

3E

-13

1.4

5E

+00

2.6

9E

+00

1.7

8E

+00

31.7

1E

+00

1.7

1E

+00

9.2

9E

-13

1.7

4E

+00

1.2

7E

+01

2.2

4E

+00

E-I

O-A

B-1

10

E-I

O-A

C-1

10

E-I

O-A

D-1

10

E-I

O-A

E-1

10

E-I

O-B

C-1

10

E-I

O-B

D-1

10

E-I

O-B

E-1

10

E-I

O-C

D-1

10

E-I

O-C

E-1

10

E-I

O-D

E-1

10

11.1

3E

-01

3.7

4E

-01

2.6

9E

-01

3.8

7E

-01

9.1

0E

-13

9.1

0E

-13

9.0

1E

-02

1.5

2E

-01

3.6

8E

-01

3.6

8E

-01

21.1

6E

-01

1.4

5E

+00

2.6

9E

+00

2.3

6E

+00

9.3

3E

-13

9.3

5E

-13

8.9

5E

-02

2.7

2E

+00

1.8

7E

+00

3.5

4E

+00

31.1

5E

-01

1.7

4E

+00

1.2

9E

+01

3.6

2E

+00

9.3

2E

-13

9.1

2E

-13

8.1

5E

-02

1.3

2E

+01

2.2

9E

+00

1.8

6E

+01

E-I

O-A

BC

-110

E-I

O-A

BD

-110

E-I

O-A

BE

-110

E-I

O-A

CD

-110

E-I

O-A

CE

-110

E-I

O-A

DE

-110

E-I

O-B

CD

-11

0E

-IO

-BC

E-1

10

E-I

O-B

DE

-110

E-I

O-C

DE

-110

11.1

3E

-01

1.1

3E

-01

1.2

6E

-01

3.7

4E

-01

4.2

5E

-01

3.8

7E

-01

9.1

0E

-13

9.0

2E

-02

9.0

1E

-02

3.6

8E

-01

21.1

6E

-01

1.1

6E

-01

1.3

1E

-01

2.6

9E

+00

2.9

7E

+00

2.9

7E

+00

9.2

8E

-13

9.0

3E

-02

8.8

9E

-02

3.7

3E

+00

31.1

6E

-01

1.1

6E

-01

1.3

0E

-01

1.8

0E

+00

4.5

5E

+00

2.3

0E

+01

8.3

8E

-13

8.7

6E

-02

3.1

9E

-02

1.3

5E

+01

E-I

O-A

BC

DE

-110

E-I

O-A

BC

D-1

10

E-I

O-A

BC

E-1

10

E-I

O-A

BD

E-1

10

E-I

O-A

CD

E-1

10

E-I

O-B

CD

E-1

10

11.2

6E

-01

1.1

3E

-01

1.2

6E

-01

1.2

6E

-01

4.2

5E

-01

9.0

2E

-02

21.3

6E

-01

1.1

6E

-01

1.3

4E

-01

1.3

5E

-01

2.9

8E

+00

8.8

8E

-02

31.2

8E

-01

1.1

7E

-01

1.3

0E

-01

1.2

9E

-01

4.4

9E

+00

3.4

1E

-04

145

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

O-0

-11

0

E-I

O-A

-11

0

E-I

O-B

-11

0

E-I

O-C

-11

0

E-I

O-D

-11

0

E-I

O-E

-11

0

Fig

ure

D.1

:E

ner

gy

ob

ject

ive

wit

hb

oth

rad

iifi

xed

at10

0m

man

d25

0m

m.

Incl

ud

esze

roan

don

ed

esig

nop

tion

sele

cted

.

146

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

O-A

B-1

10

E-I

O-A

C-1

10

E-I

O-A

D-1

10

E-I

O-A

E-1

10

E-I

O-B

C-1

10

E-I

O-B

D-1

10

E-I

O-B

E-1

10

E-I

O-C

D-1

10

E-I

O-C

E-1

10

E-I

O-D

E-1

10

Fig

ure

D.2

:E

ner

gyob

ject

ive

wit

hb

oth

rad

iifixed

at10

0m

man

d25

0m

m.

Incl

ud

estw

od

esig

nop

tion

sse

lect

ed.

147

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

O-A

BC

-11

0

E-I

O-A

BD

-11

0

E-I

O-A

BE

-11

0

E-I

O-A

CD

-11

0

E-I

O-A

CE

-11

0

E-I

O-A

DE

-11

0

E-I

O-B

CD

-11

0

E-I

O-B

CE

-11

0

E-I

O-B

DE

-11

0

E-I

O-C

DE

-11

0

Fig

ure

D.3

:E

ner

gyob

ject

ive

wit

hb

oth

rad

iifi

xed

at10

0m

man

d25

0m

m.

Incl

ud

esth

ree

des

ign

op

tion

sse

lect

ed.

148

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

O-A

BC

DE

-11

0

E-I

O-A

BC

D-1

10

E-I

O-A

BC

E-1

10

E-I

O-A

BD

E-1

10

E-I

O-A

CD

E-1

10

E-I

O-B

CD

E-1

10

Fig

ure

D.4

:E

ner

gyob

ject

ive

wit

hb

oth

rad

iifi

xed

at10

0m

man

d25

0m

m.

Incl

ud

esfo

ur

and

all

five

des

ign

op

tion

sse

lect

ed.

149

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

O-0

-11

0

E-I

O-A

-11

0

E-I

O-C

-11

0

E-I

O-E

-11

0

E-I

O-A

C-1

10

E-I

O-A

E-1

10

E-I

O-C

E-1

10

E-I

O-A

CE

-11

0

Fig

ure

D.5

:E

ner

gyob

ject

ive

wit

hb

oth

radii

fixed

at10

0m

man

d25

0m

m.

Incl

ud

esal

lth

ed

esig

nop

tion

sex

cep

tfo

rd

esig

nop

tion

s(B

)an

d(D

).

150

D.2

Tota

lS

tore

dE

nerg

y,

Fix

ed

Inn

er

Rad

ius

Tab

leD

.2:

En

ergy

obje

ctiv

ew

ith

the

inn

erra

diu

sfi

xed

at10

0m

m.

Bes

tre

sult

saf

ter

35,0

00

op

tim

izati

on

run

s.G

ray

shad

ing

ind

icat

esco

nve

rgen

cefa

ilu

re.

Un

its

are

inkW·h

.

# o

f R

ings

E-I

-0-1

10

E-I

-A-1

10

E-I

-B-1

10

E-I

-C-1

10

E-I

-D-1

10

E-I

-E-1

10

11.3

4E

+00

5.1

3E

+05

1.1

9E

-04

1.3

9E

+00

1.3

4E

+00

1.3

6E

+00

25.6

9E

+00

4.0

8E

+05

1.0

3E

-04

5.9

5E

+00

8.9

1E

+00

2.9

2E

+02

36.2

4E

+00

2.8

3E

+05

1.0

3E

-04

7.4

3E

+00

1.8

4E

+01

4.6

5E

+02

E-I

-AB

-110

E-I

-AC

-11

0E

-I-A

D-1

10

E-I

-AE

-110

E-I

-BC

-110

E-I

-BD

-110

E-I

-BE

-11

0E

-I-C

D-1

10

E-I

-CE

-11

0E

-I-D

E-1

10

11.0

1E

+07

8.2

0E

+05

5.3

5E

+05

5.2

8E

+05

1.4

9E

-04

1.1

6E

-04

9.8

3E

+05

1.3

9E

+00

3.4

6E

+01

6.6

9E

+00

23.8

8E

+06

4.3

7E

+05

3.4

8E

+05

2.5

7E

+05

6.5

9E

-05

4.7

0E

-05

1.5

3E

+06

9.5

3E

+00

2.7

8E

+02

2.4

8E

+01

38.7

9E

+05

3.3

5E

+03

2.0

4E

+05

3.6

2E

+04

1.8

2E

-05

1.8

8E

-05

9.4

4E

+04

1.8

0E

+01

3.1

2E

+02

1.8

6E

+02

E-I

-AB

C-1

10

E-I

-AB

D-1

10

E-I

-AB

E-1

10

E-I

-AC

D-1

10

E-I

-AC

E-1

10

E-I

-AD

E-1

10

E-I

-BC

D-1

10

E-I

-BC

E-1

10

E-I

-BD

E-1

10

E-I

-CD

E-1

10

14.2

7E

+06

5.2

6E

+06

2.9

0E

+06

7.2

3E

+05

3.8

3E

+05

5.2

8E

+05

1.4

9E

-04

7.1

7E

+06

9.8

3E

+05

3.4

6E

+01

23.8

9E

+06

4.0

4E

+06

8.3

0E

+05

3.2

5E

+05

2.6

1E

+04

2.5

3E

+05

4.7

3E

-05

1.6

5E

+06

5.8

5E

+05

1.9

8E

+01

34.7

9E

+04

9.9

3E

+05

1.5

0E

+04

7.6

7E

+04

8.5

6E

+02

3.5

6E

+04

1.5

6E

-06

3.7

4E

+05

3.6

4E

-05

1.1

0E

+02

E-I

-AB

CD

E-1

10

E-I

-AB

CD

-110

E-I

-AB

CE

-11

0E

-I-A

BD

E-1

10

E-I

-AC

DE

-11

0E

-I-B

CD

E-1

10

11.8

5E

+06

4.2

7E

+06

1.8

5E

+06

2.9

0E

+06

3.8

6E

+05

5.0

3E

+06

22.5

3E

+05

1.0

4E

+06

2.5

3E

+05

9.8

9E

+05

1.4

9E

+05

1.8

6E

+01

34.0

6E

+02

5.2

3E

+05

4.0

6E

+02

7.0

8E

+05

8.9

0E

+02

2.6

1E

-14

151

1.E

-04

1.E

-03

1.E

-02

1.E

-01

1.E

+00

1.E

+01

1.E

+02

1.E

+03

1.E

+04

1.E

+05

1.E

+06

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

-0-1

10

E-I

-A-1

10

E-I

-B-1

10

E-I

-C-1

10

E-I

-D-1

10

E-I

-E-1

10

Fig

ure

D.6

:E

ner

gyob

ject

ive

wit

hth

ein

ner

rad

ius

fixed

at10

0m

m.

Incl

ud

esze

roan

don

ed

esig

nop

tion

sele

cted

.

152

1.E

-05

1.E

-04

1.E

-03

1.E

-02

1.E

-01

1.E

+00

1.E

+01

1.E

+02

1.E

+03

1.E

+04

1.E

+05

1.E

+06

1.E

+07

1.E

+08

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

-AB

-11

0

E-I

-AC

-11

0

E-I

-AD

-11

0

E-I

-AE

-11

0

E-I

-BC

-11

0

E-I

-BD

-11

0

E-I

-BE

-11

0

E-I

-CD

-11

0

E-I

-CE

-11

0

E-I

-DE

-11

0

Fig

ure

D.7

:E

ner

gyob

ject

ive

wit

hth

ein

ner

rad

ius

fixed

at10

0m

m.

Incl

ud

estw

od

esig

nop

tion

sse

lect

ed.

153

1.E

-06

1.E

-05

1.E

-04

1.E

-03

1.E

-02

1.E

-01

1.E

+00

1.E

+01

1.E

+02

1.E

+03

1.E

+04

1.E

+05

1.E

+06

1.E

+07

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

-AB

C-1

10

E-I

-AB

D-1

10

E-I

-AB

E-1

10

E-I

-AC

D-1

10

E-I

-AC

E-1

10

E-I

-AD

E-1

10

E-I

-BC

D-1

10

E-I

-BC

E-1

10

E-I

-BD

E-1

10

E-I

-CD

E-1

10

Fig

ure

D.8

:E

ner

gyob

ject

ive

wit

hth

ein

ner

rad

ius

fixed

at10

0m

m.

Incl

ud

esth

ree

des

ign

opti

on

sse

lect

ed.

154

1.E

-14

1.E

-13

1.E

-12

1.E

-11

1.E

-10

1.E

-09

1.E

-08

1.E

-07

1.E

-06

1.E

-05

1.E

-04

1.E

-03

1.E

-02

1.E

-01

1.E

+00

1.E

+01

1.E

+02

1.E

+03

1.E

+04

1.E

+05

1.E

+06

1.E

+07

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

-AB

CD

E-1

10

E-I

-AB

CD

-11

0

E-I

-AB

CE

-11

0

E-I

-AB

DE

-11

0

E-I

-AC

DE

-11

0

E-I

-BC

DE

-11

0

Fig

ure

D.9

:E

ner

gy

ob

ject

ive

wit

hth

ein

ner

rad

ius

fixed

at10

0m

m.

Incl

ud

esfo

ur

and

all

five

des

ign

op

tion

sse

lect

ed.

155

1.E

+00

1.E

+01

1.E

+02

1.E

+03

1.E

+04

1.E

+05

1.E

+06

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-I

-0-1

10

E-I

-A-1

10

E-I

-C-1

10

E-I

-E-1

10

E-I

-AC

-11

0

E-I

-AE

-11

0

E-I

-CE

-11

0

E-I

-AC

E-1

10

Fig

ure

D.1

0:

En

ergy

obje

ctiv

ew

ith

the

inner

rad

ius

fixed

at10

0m

m.

Incl

ud

esal

lth

ed

esig

nop

tion

sex

cep

tfo

rd

esig

nop

tions

(B)

an

d(D

).

156

D.3

Tota

lS

tore

dE

nerg

y,

Fix

ed

Ou

ter

Rad

ius

Tab

leD

.3:

En

ergy

obje

ctiv

ew

ith

the

oute

rra

diu

sfi

xed

at25

0m

m.

Bes

tre

sult

saf

ter

75,0

00

op

tim

izati

on

run

s.G

ray

shad

ing

ind

icat

esco

nve

rgen

cefa

ilu

re.

Un

its

are

inkW·h

.

# o

f R

ings

E-O

-0-1

10

E-O

-A-1

10

E-O

-B-1

10

E-O

-C-1

10

E-O

-D-1

10

E-O

-E-1

10

17.1

4E

+00

7.1

4E

+00

9.5

9E

-13

7.4

3E

+00

7.1

4E

+00

7.2

4E

+00

21.9

7E

+01

1.9

7E

+01

1.0

2E

-12

1.9

7E

+01

2.2

3E

+01

2.1

6E

+01

31.9

8E

+01

1.9

8E

+01

1.0

5E

-12

2.1

5E

+01

2.5

0E

+01

2.4

5E

+01

E-O

-AB

-11

0E

-O-A

C-1

10

E-O

-AD

-110

E-O

-AE

-11

0E

-O-B

C-1

10

E-O

-BD

-110

E-O

-BE

-110

E-O

-CD

-11

0E

-O-C

E-1

10

E-O

-DE

-110

11.1

8E

-01

7.4

3E

+00

7.1

4E

+00

7.2

4E

+00

9.5

9E

-13

9.5

9E

-13

9.5

0E

-02

7.4

3E

+00

7.5

2E

+00

7.2

4E

+00

21.2

7E

-01

1.9

7E

+01

2.1

6E

+01

2.1

9E

+01

1.0

2E

-12

1.0

3E

-12

9.7

4E

-02

2.2

6E

+01

2.2

1E

+01

2.1

8E

+01

31.3

0E

-01

2.1

5E

+01

2.0

0E

+01

2.4

8E

+01

9.9

3E

-13

1.0

3E

-12

9.1

3E

-02

2.5

2E

+01

2.6

2E

+01

2.4

5E

+01

E-O

-AB

C-1

10

E-O

-AB

D-1

10

E-O

-AB

E-1

10

E-O

-AC

D-1

10

E-O

-AC

E-1

10

E-O

-AD

E-1

10

E-O

-BC

D-1

10

E-O

-BC

E-1

10

E-O

-BD

E-1

10

E-O

-CD

E-1

10

11.2

0E

-01

1.1

9E

-01

1.7

4E

-01

7.4

3E

+00

7.5

2E

+00

7.2

4E

+00

9.5

9E

-13

9.5

0E

-02

9.5

0E

-02

7.5

2E

+00

21.2

8E

-01

1.2

7E

-01

1.9

4E

-01

2.0

6E

+01

2.2

1E

+01

2.1

9E

+01

1.0

3E

-12

9.5

2E

-02

9.5

1E

-02

2.2

3E

+01

31.3

0E

-01

1.3

1E

-01

1.9

5E

-01

2.1

5E

+01

2.6

2E

+01

2.4

9E

+01

9.6

1E

-13

8.3

3E

-02

9.1

0E

-02

2.6

3E

+01

E-O

-AB

CD

E-1

10

E-O

-AB

CD

-110

E-O

-AB

CE

-110

E-O

-AB

DE

-110

E-O

-AC

DE

-11

0E

-O-B

CD

E-1

10

11.7

7E

-01

1.2

0E

-01

1.7

7E

-01

1.7

4E

-01

7.5

2E

+00

9.5

0E

-02

21.9

7E

-01

1.2

8E

-01

1.9

8E

-01

1.8

4E

-01

2.2

1E

+01

9.2

7E

-02

31.9

5E

-01

1.3

1E

-01

1.8

6E

-01

1.8

9E

-01

2.5

9E

+01

6.1

5E

-16

157

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-O

-0-1

10

E-O

-A-1

10

E-O

-B-1

10

E-O

-C-1

10

E-O

-D-1

10

E-O

-E-1

10

Fig

ure

D.1

1:E

ner

gyob

ject

ive

wit

hth

eou

ter

rad

ius

fixed

at25

0m

m.

Incl

ud

esze

roan

done

des

ign

op

tion

sele

cted

.

158

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-O

-AB

-11

0

E-O

-AC

-11

0

E-O

-AD

-11

0

E-O

-AE

-11

0

E-O

-BC

-11

0

E-O

-BD

-11

0

E-O

-BE

-11

0

E-O

-CD

-11

0

E-O

-CE

-11

0

E-O

-DE

-11

0

Fig

ure

D.1

2:

En

ergy

ob

ject

ive

wit

hth

eou

ter

rad

ius

fixed

at25

0m

m.

Incl

ud

estw

od

esig

nop

tion

sse

lect

ed.

159

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-O

-AB

C-1

10

E-O

-AB

D-1

10

E-O

-AB

E-1

10

E-O

-AC

D-1

10

E-O

-AC

E-1

10

E-O

-AD

E-1

10

E-O

-BC

D-1

10

E-O

-BC

E-1

10

E-O

-BD

E-1

10

E-O

-CD

E-1

10

Fig

ure

D.1

3:

En

ergy

ob

ject

ive

wit

hth

eou

ter

rad

ius

fixed

at25

0m

m.

Incl

udes

thre

ed

esig

nop

tion

sse

lect

ed.

160

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-O

-AB

CD

E-1

10

E-O

-AB

CD

-11

0

E-O

-AB

CE

-11

0

E-O

-AB

DE

-11

0

E-O

-AC

DE

-11

0

E-O

-BC

DE

-11

0

Fig

ure

D.1

4:E

ner

gyob

ject

ive

wit

hth

eou

ter

rad

ius

fixed

at25

0m

m.

Incl

udes

fou

ran

dal

lfi

ved

esig

nop

tion

sse

lect

ed.

161

0

5

10

15

20

25

30

0

1

2

3

4

Energy (kW-h)

Nu

mb

er o

f R

ing

s

E-O

-0-1

10

E-O

-A-1

10

E-O

-C-1

10

E-O

-E-1

10

E-O

-AC

-11

0

E-O

-AE

-11

0

E-O

-CE

-11

0

E-O

-AC

E-1

10

Fig

ure

D.1

5:

En

ergy

obje

ctiv

ew

ith

the

oute

rra

diu

sfi

xed

at25

0m

m.

Incl

ud

esal

lth

ed

esig

nop

tion

sex

cep

tfo

rd

esig

nop

tion

s(B

)an

d(D

).

162

D.4

Sp

eci

fic

En

erg

y,

Fix

ed

Rad

ii

Tab

leD

.4:

Sp

ecifi

cen

ergy

obje

ctiv

ew

ith

bot

hra

dii

fixed

at10

0m

man

d25

0m

m.

Bes

tre

sult

saf

ter

75,0

00

op

tim

izati

on

run

s.G

ray

shad

ing

ind

icat

esco

nver

gen

cefa

ilu

re.

# o

f R

ings

S-I

O-0

-110

S-I

O-A

-110

S-I

O-B

-11

0S

-IO

-C-1

10

S-I

O-D

-110

S-I

O-E

-110

14.1

2E

-01

1.2

8E

+00

2.4

7E

-12

4.1

2E

-01

4.1

2E

-01

9.5

2E

-01

23.8

7E

+00

3.8

7E

+00

2.5

3E

-12

4.0

3E

+00

5.5

4E

+00

4.2

6E

+00

33.9

4E

+00

4.4

4E

+00

2.5

5E

-12

4.1

0E

+00

4.3

4E

+01

4.3

9E

+00

S-I

O-A

B-1

10

S-I

O-A

C-1

10

S-I

O-A

D-1

10

S-I

O-A

E-1

10

S-I

O-B

C-1

10

S-I

O-B

D-1

10

S-I

O-B

E-1

10

S-I

O-C

D-1

10

S-I

O-C

E-1

10

S-I

O-D

E-1

10

15.4

2E

-01

1.8

0E

+00

1.2

8E

+00

1.3

8E

+00

2.4

7E

-12

2.4

7E

-12

2.3

3E

-01

4.1

2E

-01

9.5

2E

-01

9.5

2E

-01

25.5

6E

-01

4.0

3E

+00

5.5

8E

+00

4.4

9E

+00

2.5

3E

-12

2.5

4E

-12

2.3

5E

-01

6.5

0E

+00

4.4

5E

+00

8.6

0E

+00

35.5

9E

-01

4.7

4E

+00

4.8

1E

+01

5.5

6E

+00

2.4

6E

-12

2.5

5E

-12

1.7

8E

-01

4.4

3E

+01

4.6

6E

+00

5.3

4E

+01

S-I

O-A

BC

-11

0S

-IO

-AB

D-1

10

S-I

O-A

BE

-110

S-I

O-A

CD

-110

S-I

O-A

CE

-11

0S

-IO

-AD

E-1

10

S-I

O-B

CD

-11

0S

-IO

-BC

E-1

10

S-I

O-B

DE

-11

0S

-IO

-CD

E-1

10

15.4

7E

-01

5.4

2E

-01

5.7

7E

-01

1.8

0E

+00

1.8

0E

+00

1.3

8E

+00

2.4

7E

-12

2.3

3E

-01

2.3

3E

-01

9.5

2E

-01

25.6

0E

-01

5.5

7E

-01

5.7

7E

-01

6.4

8E

+00

5.7

9E

+00

4.4

9E

+00

2.5

4E

-12

2.3

4E

-01

2.3

4E

-01

1.0

1E

+01

35.6

0E

-01

5.6

1E

-01

5.7

5E

-01

4.9

1E

+00

8.5

3E

+00

1.6

2E

+01

2.4

0E

-12

2.1

9E

-01

8.6

0E

-02

5.0

6E

+01

S-I

O-A

BC

DE

-11

0S

-IO

-AB

CD

-11

0S

-IO

-AB

CE

-11

0S

-IO

-AB

DE

-110

S-I

O-A

CD

E-1

10

S-I

O-B

CD

E-1

10

15.8

5E

-01

5.4

7E

-01

5.8

5E

-01

5.7

7E

-01

1.8

0E

+00

2.3

3E

-01

25.8

5E

-01

5.6

1E

-01

5.8

5E

-01

5.8

0E

-01

5.8

1E

+00

2.3

0E

-01

35.7

7E

-01

5.5

9E

-01

5.8

5E

-01

5.6

3E

-01

8.3

6E

+00

1.7

6E

-01

163

0

10

20

30

40

50

0

1

2

3

4

Specific Energy (W-h/kg)

Nu

mb

er o

f R

ing

s

S-I

O-0

-11

0

S-I

O-A

-11

0

S-I

O-B

-11

0

S-I

O-C

-11

0

S-I

O-D

-11

0

S-I

O-E

-11

0

Fig

ure

D.1

6:

Sp

ecifi

cen

ergy

obje

ctiv

ew

ith

bot

hra

dii

fixed

at10

0m

man

d25

0m

m.

Incl

ud

esze

roan

don

ed

esig

nop

tion

sele

cted

.U

nit

sar

ein

W·h

/kg.

164

0

10

20

30

40

50

0

1

2

3

4


Nu

mb

er o

f R

ing

s

S-I

O-A

B-1

10

S-I

O-A

C-1

10

S-I

O-A

D-1

10

S-I

O-A

E-1

10

S-I

O-B

C-1

10

S-I

O-B

D-1

10

S-I

O-B

E-1

10

S-I

O-C

D-1

10

S-I

O-C

E-1

10

S-I

O-D

E-1

10

Fig

ure

D.1

7:

Sp

ecifi

cen

ergy

obje

ctiv

ew

ith

bot

hra

dii

fixed

at10

0m

man

d25

0m

m.

Incl

ud

estw

od

esig

nop

tion

sse

lect

ed.

165

0

10

20

30

40

50

0

1

2

3

4


Nu

mb

er o

f R

ing

s

S-I

O-A

BC

-11

0

S-I

O-A

BD

-11

0

S-I

O-A

BE

-11

0

S-I

O-A

CD

-11

0

S-I

O-A

CE

-11

0

S-I

O-A

DE

-11

0

S-I

O-B

CD

-11

0

S-I

O-B

CE

-11

0

S-I

O-B

DE

-11

0

S-I

O-C

DE

-11

0

Fig

ure

D.1

8:

Sp

ecifi

cen

ergy

obje

ctiv

ew

ith

bot

hra

dii

fixed

at10

0m

man

d25

0m

m.

Incl

ud

esth

ree

des

ign

op

tion

sse

lect

ed.

166

0

10

20

30

40

50

0

1

2

3

4


Nu

mb

er o

f R

ing

s

S-I

O-A

BC

DE

-11

0

S-I

O-A

BC

D-1

10

S-I

O-A

BC

E-1

10

S-I

O-A

BD

E-1

10

S-I

O-A

CD

E-1

10

S-I

O-B

CD

E-1

10

Fig

ure

D.1

9:S

pec

ific

ener

gyob

ject

ive

wit

hb

oth

rad

iifi

xed

at10

0m

man

d25

0m

m.

Incl

ud

esfo

ur

an

dall

five

des

ign

op

tion

sse

lect

ed.

167

0

1

2

3

4

5

6

7

8

9

0

1

2

3

4


Nu

mb

er o

f R

ing

s

S-I

O-0

-11

0

S-I

O-A

-11

0

S-I

O-C

-11

0

S-I

O-E

-11

0

S-I

O-A

C-1

10

S-I

O-A

E-1

10

S-I

O-C

E-1

10

S-I

O-A

CE

-11

0

Fig

ure

D.2

0:

Sp

ecifi

cen

ergy

ob

ject

ive

wit

hb

oth

rad

iifi

xed

at10

0m

man

d25

0m

m.

Incl

ud

esall

the

des

ign

op

tion

sex

cep

tfo

rd

esig

nop

tion

s(B

)an

d(D

).

Appendix E

Nontechnical Abstract

Within the last century, energy storing flywheel rotor technology has begun to approach the

power and energy densities required for industrial and transportation applications due to the

development of fiber-reinforced composite materials. However, this technology has yet to be

fully optimized given the design possibilities. There have been many design approaches that

have been published over the years, but no overall comparisons of the many manufacturable

design options have been put forth. This research investigates some of these design options

for increasing flywheel rotor performance. It also increases understanding of how to set up a

proper optimization problem which avoids falling into various optimization traps that may

produce unrealistic results.

Of the design options investigated, some have shown very promising results. For flywheel

rotors with many material rings, press-fitting them together has been shown to produce

some of the best results. The second best design option investigated is matrix ballasting,

where high-density particles can be placed into the epoxy matrix of a composite material to

vary the material properties. Another good design option investigated is fiber co-mingling,

where two fibers of different properties are placed together to alter a composite material.

This research can point flywheel designers in the direction of the optimization strategy most

useful to their application.

168

RESIDENCE: 3432 Littleleaf Place Laurel, MD 20724 Mobile: (405) 473-3233

Jacob Wayne Ross [email protected]

OFFICE: 7 Research West Building University Park, PA 16802 Phone: (814) 865-2289

EDUCATION

Ph.D., Engineering Science & Mechanics 2009 – 2013 The Pennsylvania State University University Park, PA (GPA: 3.96)

M.S., Nuclear Engineering 2007 – 2009 The Pennsylvania State University University Park, PA (GPA: 4.0)

B.S., Physics 2003 – 2006 The Pennsylvania State University University Park, PA (GPA: 3.88)

B.S., Nuclear Engineering 2003 – 2006 The Pennsylvania State University University Park, PA (GPA: 3.88) A.S., Mathematics 1999 – 2002 Rose State College, OK (GPA: 4.00)

A.A.S., Computer Networking 1999 – 2002 Rose State College, OK (GPA: 4.00)

PROFESSIONAL EXPERIENCE

Composite Flywheel Energy Storage Optimization Researcher Pennsylvania State University 09/2009 – Present University Park, PA

Nine Mile Point Nuclear Fuel Services Engineer Constellation Energy Nuclear Group, LLC 05/2009 – 08/2009 Lycoming, NY

Nuclear Weapons Counter-Proliferation Target Analyst National Security Agency [DoD] 05/2008 – 08/2008 Fort George G. Meade, MD

Researcher: Lead Boiling Superheat Reactor (BSR) Design Engineer Pennsylvania State University 01/2008 – 05/2009 University Park, PA

Teaching Assistant: Department of Mechanical and Nuclear Engineering The Pennsylvania State University 08/2007 – 05/2009 University Park, PA � Fall 2007: Design Principles of Reactor Systems � Spring 2008 & 2009: Radiation Detection and Measurement � Fall 2008: Experiments in Reactor Physics

Nuclear Engineer: Core Thermal Hydraulic Analyst Westinghouse Electric Company 04/2007 – 08/2007 Monroeville, PA

Researcher: Two-Phase Flow Dynamics for Hydrogen Fuel Cells The Pennsylvania State University 05/2004 – 09/2005 University Park, PA

on the optimization of composite flywheel rotors

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