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This is to certify that the
thesis entitled

ORDER-TUNED VIBRATION ABSORBERS FOR SYSTEMS
WITH CYCLIC SYMMETRY WITH APPLICATIONS TO
TURBOMACHINERY

presented by

BRIAN JOHN OLSON

has been accepted towards fulfillment
of the requirements for the

PHD. degree in MECHANICAL ENGINEERING

 

 

wily

Majo'r Professor’s Signature

 

9/27/06

 

Date

MSU is an Affirmative Action/Equal Opportunity Institution

 

‘ LIBRARY— Ff
Michigan State
University

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

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ORDER-TUNED VIBRATION ABSORBERS FOR
SYSTEMS WITH CYCLIC SYMMETRY WITH
APPLICATIONS TO TURBOMACHINERY

By

Brian John Olson

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2006

ABSTRACT

ORDER-TUNED VIBRATION ABSORBERS FOR
SYSTEMS WITH CYCLIC SYMMETRY WITH
APPLICATIONS TO TURBOMACHINERY

By
Brian John Olson

This work investigates the performance of centrifugally-driven, order-tuned ab-
sorbers for vibration reduction in a class of systems with cyclic symmetry. The
rotating flexible structures of interest are bladed disk assemblies, such as the fans,
compressors and turbines in a jet engine, which consist of a nominally cyclic array
of interconnected substructures. Under steady operation, these assemblies rotate at
a constant speed and are subjected to traveling wave dynamic loading (the so—called
engine order excitation), which is characterized by excitation frequencies that are
proportional to the mean rotational speed of the rotor. Such excitations result in
component vibrations and can lead to high cycle fatigue failure, noise, reduced per-
formance, and other undesirable effects. Since order-tuned absorbers feature natural
frequencies that scale directly with the rotor speed, they are ideally suited to address
these vibrations. However, at the time of writing, there has been no systematic an-
alytical treatment of absorber systems applied to cyclic rotating flexible structures
under engine order excitation. This thesis reports the first such study.

The aim of this investigation is threefold: to quantify and understand the underly-
ing linear resonance structure of a cyclically-coupled bladed disk assembly fitted with
order-tuned absorbers; based on these findings, to design the absorbers to eliminate
or otherwise reduce blade motions relative to the rotating hub; and to generalize
the linear theory, methodology, and design to include the basic, first-order effects of

nonlinearity.

The analysis is carried out assuming identical, identically-coupled substructures,
which gives rise to a linearized model with block circulant matrices. A standard
change of coordinates based on this cyclic structure essentially decouples the governing
equations, and it gives rise to closed form expressions from which analytical results can
be gleaned. The linear resonance structure is found to be surprisingly rich, a feature
that arises from the order-nature of the absorbers. One of the main findings of the
linear analysis, and indeed of this entire thesis, is the existence of a “no-resonance
zone,” that is, an entire spectrum of absorber designs for which there are no system
resonances over the full range of possible rotor speeds. By designing the absorbers
within this small, but finite spectrum, system resonances are avoided altogether and
there is at least some level of robustness to parameter uncertainties.

In the presence of weak nonlinearity, which is introduced via the absorber path
geometry, the underlying linear resonance structure is shown to qualitatively persist—
including the no-resonance zone—provided that the excitation strength is sufficiently
small. There does exist a nonlinear design strategy in which relative blade motions can
be eliminated completely, but it depends on both the rotor speed and force amplitude.
The design is thus effective for only a single set of operating conditions, which suggests
that nonlinearity cannot be exploited to further improve absorber performance in the
systems of interest. When nonlinearity cannot be avoided it is shown that softening
characteristics are more desirable than hardening; the former simply sets an upper
limit on the range of speeds over which the absorbers are effective while the latter
may give rise to problematic resonances. Finally, for the weakly coupled and lightly
damped systems under consideration, there may be a host of symmetry-breaking
instabilities involving the desired traveling wave response. However, none could be
identified. This is a very promising finding since bifurcations of this kind are highly

undesirable from an applications viewpoint.

To my wife, Julie, my parents, and to the countless

people who have supported me during this effort.

iv

The theory of matrices exhibits much that is visually attractive. Thus,
diagonal matrices, symmetric matrices, (0,1) matrices, and the like are
attractive independently of their applications. In the same category are
the circulants.

- Philip J. Davis

ACKNOWLEDGMENTS

This thesis symbolizes the entirety of my journey in higher education, which has
spanned over thirteen years of efforts and sacrifice, successes and failures, and un-
countable life lessons. In bringing these pages to closure I can identify with a letter

to Robert Hooke in February of 1676, in which Sir Isaac Newton wrote:

“IfI have been able to see further, it was only because I stood on the

shoulders of giants.”

This effort would not have been possible without the encouragement and support of
many individuals, to whom the following acknowledgments are dedicated.

First and foremost I wish to acknowledge my wife, Julie, for her continual patience,
understanding and encouragement, my parents, whose support has been seemingly
without bound, and my brother, Gregg, with whom I have enjoyed countless hours of
engaging conversation.

I am particularly indebted to my advisor, colleague and friend, Steve Shaw, who
has offered much guidance, insight and wisdom. It has been an absolute pleasure
to know and work with him. Thanks are also due to my doctoral committee mem-
bers Christophe Pierre (and also Matt Castanier), Alan Haddow, Brian Feeny, Cevat
Gokcek, and Hassan Khalil.

To my esteemed colleagues—Jeff Rhoads, Pat Staron, Umar Farooq, Nyesunthi
(Polarit) Apiwattanalunggarn, Gabor Csernak, and many more—your friendship and
camaraderie have truly enriched my years as a student. I wish also to thank Richard
Cohn and Jim Schramski.

In addition to my doctoral committee members, there have been many faculty
and staff at Michigan State University to whom I owe many thanks: Craig Somerton,
Manoochehr Koochesfahani, Ronald Rosenberg, Carol Bishop, Gaile Griffore, Craig

Gunn, Aida Montalvo, Elaine Bailey, and Lindsay Niesen. I am grateful to the

vi

many dedicated instructors at Northwestern Michigan College, including Jim “Chief
Tormentor” Coughlin, Ken “Papa Bear” Stepnitz, Tony Jenkins, Steve Drake, and
Ernie East, whose support and encouragement was pivotal in my early years as a
student. Thanks also to Gabor Stépan from the Budapest University of Technology
andflEcononncs

Finally, this work was supported by grant CMS-O408866 from the National Science
Foundation. I would also like to thank Donald E. Knuth for creating the TEX system
for typesetting [1], Leslie Lamport for extending this system to LATEX [2], and the
many contributors to the MTEX 25 implementation, which was used to typeset this
dissertation. I would not have had the patience to typeset this work using a WYSIWYG

package.

vii

TABLE OF CONTENTS

LIST OF TABLES .............................. x11
LIST OF FIGURES ............................. x111
SELECTED LIST OF OPERATORS ................... x1x
1 Introduction ................................ 1
1.1 Motivation ................................. 1
1.2 Order-Tuned Vibration Absorbers .................... 2
1.3 Systems with Cyclic Symmetry ..................... 5

1.4 Vibration Absorbers for Rotating Flexible Structures under Engine

Order Excitation ............................. 7
1.5 Dissertation Overview .......................... 8
Background ................................ 11
2.1 Introduction ................................ 11
2.2 Mathematical Preliminaries ....................... 11
2.2.1 The Kronecker Product ..................... 12
2.2.2 The Fourier Matrix ........................ 12
2.2.3 Circulant Matrices ........................ 13
2.2.4 Diagonalization of Circulants .................. 14
2.3 Engine Order Excitation ......................... 15
2.3.1 Mathematical Model ....................... 15
2.3.2 Traveling Wave Characteristics ................. 17
2.4 Vibration Characteristics of Cyclic Systems .............. 21
2.4.1 A Prototypical Model ....................... 23
2.4.2 Forced Response Under Engine Order Excitation ....... 25
2.4.3 Eigenfrequency Characteristics .................. 30
2.4.4 Normal Modes of Vibration ................... 32
2.4.5 Resonance Structure ....................... 34
2.5 Vibration Absorbers ........................... 37
2.5.1 Introduction ............................ 37
2.5.2 The Frequency-Tuned Dynamic Vibration Absorber ...... 39
2.5.3 The Order—Tuned Centrifugal Pendulum Vibration Absorber . 44
2.6 Concluding Remarks ........................... 45
Mathematical Models .......................... 47
3.1 Introduction ................................ 47
3.2 Modeling Approach ............................ 48

viii

3.2.1 Absorber and Blade Models ................... 48

3.2.2 Motivation for Lumped-Parameter Blade Models ........ 51
3.3 The Geometry of an Arbitrary Absorber Path ............. 51
3.3.1 Fundamental Relationship Between Path Variables ...... 52
3.3.2 Angle Between Radius Vector and Tangent ........... 54
3.4 Bladed Disk Assembly Fitted with General-Path Absorbers ...... 56
3.4.1 Lumped-Parameter Model .................... 57
3.4.2 Kinetic and Potential Energy .................. 60
3.4.3 Equations of Motion ....................... 61
3.4.4 A Generalized Two-Parameter Family of Paths ........ 65
3.5 Bladed Disk Assembly Fitted with Circular-Path Absorbers ..... 68
3.5.1 Equations of Motion ....................... 68
3.5.2 Linearized Model with Restrictions on the Absorber Amplitudes 70
3.6 Estimates of the Dimensionless Damping Constants .......... 72
3.7 Concluding Remarks ........................... 75
Forced Response of the Coupled Linear System .......... 76
4.1 Introduction ................................ 76
4.2 Mathematical Model ........................... 78
4.2.1 Sector Model ........................... 79
4.2.2 System Model ........................... 81
4.2.3 Special Case: The Blades Locked ................ 82
4.2.4 Special Case: The Absorbers Locked .............. 82
4.3 Forced Response ............................. 83
4.3.1 Response with the Absorbers Locked .............. 85
4.3.2 Response with the Absorbers Free ................ 90
4.4 Absorber Tuning ............................. 99
4.4.1 Exact Tuning ........................... 100
4.4.2 Absorber Detuning and the N o-Resonance Zone ........ 102
4.5 The Effects of Damping ......................... 108
4.6 Concluding Remarks ........................... 113
Forced Response of the Nonlinear System .............. 115
5.1 Introduction ................................ 115
5.2 Formulation ................................ 118
5.2.1 Equations of Motion ....................... 118
5.2.2 Scaling ............................... 120
5.2.3 Linear Resonance Structure of the Scaled System ....... 123
5.2.4 Averaging ............................. 124
5.3 Traveling Wave Response ......................... 130
5.3.1 Existence ............................. 131

ix

5.3.2 Stationary Points ......................... 132

5.3.3 Local Stability .......................... 137

5.4 Forced Response of the Isolated Nonlinear System ........... 140
5.4.1 Frequency Response ....................... 141

5.4.2 Criteria for Zero Blade Amplitudes ............... 151

5.5 Forced Response of the Coupled Nonlinear System ........... 155
5.5.1 Local Stability of the Traveling Wave Response ........ 156

5.5.2 Case Studies ............................ 159

5.6 Concluding Remarks ........................... 165

6 Conclusions ................................ 166
6.1 Summary of Contributions ........................ 167
6.2 Recommendations for Absorber Design ................. 168
6.2.1 Linear and Nonlinear Tuning .................. 169

6.2.2 Damping .............................. 171

6.2.3 Absorber Sizing and Placement ................. 171

6.2.4 Path Type ............................. 171

6.3 Directions for Future Work ........................ 172
APPENDICES ................................ 175
A Selected Topics from Linear Algebra ................. 175
A1 Introduction ................................ 175
A2 The Direct Sum and Direct Product ................... 175
A3 Special Matrices .............................. 177
A31 Hermitian and Unitary Matrices ................. 178

A32 Permutation Matrices ....................... 178

A4 Similarity Transformations ........................ 179

B The Theory of Circulants ........................ 183
B.1 Introduction ................................ 183
B2 Circulant Matrices ............................ 184
B3 Block Circulant Matrices ......................... 187
B4 Diagonalization of Circulants ...................... 188
8.41 N th Roots of Unity ........................ 189

B.4.2 The Fourier Matrix ........................ 190

B.4.3 Diagonalization of the Cyclic Forward Shift Matrix ...... 196

B.4.4 Diagonalization of a Circulant .................. 197

845 Block Diagonalization of a Block Circulant ........... 199

846 Some Generalizations ....................... 201

C Noninteger Engine Order Excitation ................. 204

D The Critical Linear Detuning and Tuning Order .......... 207

E Averaging: The Usual Approach ................... 209
E. 1 Introduction ................................ 209
E2 Detuning Scheme ............................. 210
E3 The Averaged Equations ......................... 210
E4 Forced Response of the Undamped System ............... 212

BIBLIOGRAPHY .............................. 216

xi

LIST OF TABLES

2.1 Sets of engine orders n (mod N) E N: NgwE UNOTW U FTW corre-
sponding to backward traveling wave (BTW), forward traveling wave

(F TW), and standing wave (SW) dynamics loads applied to the blades

for (a) odd N and (b) even N. These Ecan be visualized 1n Figure 2. 3i-ii.

2.2 Sets of mode numbers p E N '2 O’EUPBTW U PFTW corresponding
to standing wave (SW), backward traveling wave (BTW), and forward
traveling wave (FTW) normal modes of free vibration for (a) odd N
and (b) even N. .............................

2.3 Condition on the engine order n E Z+ to excite mode p E N for
N = 10. ..................................

3.1 Selected list of dimensionless variables and parameters. ........
3.2 Selected terms in the general-path equations of motion given by Equa-
tion (3.12) and their counterparts for the case of circular-path ab-
sorbers. ..................................

4.1 Elements of the sector mass, damping, and stiffness matrices M, C,
and K ....................................
4.2 Data to accompany Fig. 4.3 and Figs. 4.5-4.7 for a model with N = 10,
n = 3, a = 0.84, 6 = 0.67, [1 = 0.015, and Q, = Q, = 5c = 0. Then
ficr = —-0.00701 (from Equation (D2) of Appendix D) and fiapp =
—0.00706 (from Equation (4.46) of Section 4.4.2). ...........
4.3 Selected list of dimensionless undamped natural frequencies. .....

5.1 Data to accompany Figure 5.2. .....................
5.2 Example integer pairs (k, l) that yield distinct stationary points a and
i") of the averaged system. Also indicated are the corresponding values
of g and f and whether the resulting blade/ absorber motions are in
phase (IP) or out of phase (OP). ....................

A.1 Some types of special matrices. .....................
A.2 Some types of linear transformations. .................

E.1 Shorthand notation. ...........................

xii

21

32

37

64

71

81

90
96

124

2.1
2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

LIST OF FIGURES

The the axial gas pressure 19(6): ideal and (example) actual conditions.

An example illustration of the discrete temporal and continuous spatial
variations of the traveling wave excitation defined by Equation (2.14)
(in real form): (a) the discrete dynamic loads with amplitude F and
period T = 27r/nf2 applied to each sector; and (b) the continuous BTW
excitation with wavelength N / n and speed C = N D / 27r relative to the
rotating hub. ...............................
Engine orders n mod N corresponding to BTW, FTW and SW applied
dynamic loading for (1) odd N and (ii) even N (see also Table 2.1);
example plots of applied dynamic loading (represented by the dots) for
a model with N = 10 sectors and with (a) n = 1 (BTW), (b) n = 5
(SW), (c) n = 9 (FTW), and (d) n = 10 (SW). The BTW engine order
excitation is represented by the solid lines. ..............
A prototypical linear cyclic system with cyclic boundary conditions
$0 = ZEN and 11w“ = 2:1. ........................
Dimensionless natural frequencies dip in terms of the number of nodal
diameters (n.d.) versus mode number p for weak coupling (WC) and
strong coupling (SC): (a) N = 11 (odd) and (b) N = 10 (even). Also
indicated below each figure is, for general N, the number of n.d. at
each value of p and also the mode numbers corresponding to standing
waves (SW), backward traveling waves (BTW), and forward traveling
waves (FT W) ..............................
A backward traveling wave ap<I>p(i —— 1+ Cpr) 2 ap COS(L,0P(Z— 1)+<ijr)
with amplitude ap, wavelength 27r/cpp = N / (p - 1), and speed C'p =
(Zip/app. ..................................
Normal modes of free vibration for a model with N = 100 sectors.
Mode 1 consists of a SW, in which each sector oscillates with the same
amplitude and phase. Mode 51 also corresponds to a SW, but neigh-
boring oscillators oscillate exactly 180 degrees out of phase. Modes
2—50 (resp. 52—100) consist of BTWS (resp. FTWs). .........
(a) Campbell diagram and (b) corresponding frequency response curves
for N=10,11:05,f=0.01,andn=1,...,N. ...........
(a) Campbell diagram for N = 10, V = 0.5, f = 0.01, and n =
1, . . . , 20N and the corresponding frequency response curves for (b) n =
N,...,2N and (c) n=2N,...,3N. ..................
Tuned vibration absorbers: (a) DVA; (b) Circular Path CPVA; (c) Gen-
eral Path CPVA. .............................

xiii

16

19

22

23

31

33

2.11 Dimensionless amplitude of the primary system versus dimensionless
excitation frequency: effect of an undamped vibration absorber (c = 0)
on the response of the undamped primary system (C’ = 0). .....

2.12 Dimensionless amplitude of the primary system versus dimensionless
excitation frequency: effect of a damped vibration absorber (c aé 0) on
the response of the undamped primary system (C = 0). .......

3.1 (a) Finite element model of a bladed disk assembly (reproduced with
permission from [3]); (b) General cyclic system with N identical cells
and nearest-neighbor coupling. .....................

3.2 Geometry of a general absorber path. .................

3.3 Lumped parameter model of a rotating bladed disk assembly.

3.4 Sector model of a bladed disk assembly fitted with a general-path ab-
sorber. The mathematical details of the absorber path are given in
Section 3.3. ................................

3.5 A generalized family of absorber paths defined by Equation (3.19) for
a = 0.84, 6 = 0.67, a = 3 (’7 = 0.168 and r0 = 1.008), —0.47 g s, S
0.47, and for a range of nonlinear tuning from softening to hardening:
17 = —20, —10,0, 10,20. .........................

3.6 Sector model for a bladed disk assembly fitted with circular-path ab-
sorbers with limited rattling space. ...................

3.7 Approximate (a) torsional and (b) translational absorber damping
constant versus mass ratio [1 based on Equation (3.31) and Equa-
tion (3.33) for pa = 0.005, a = 0.84, 6 = 0.67, and engine orders
(6.0.)71 = 1,. . .,10. ...........................

4.1 (a) Model of bladed disk assembly and (b) sector model. .......
4.2 The topology of a bladed disk assembly in (a) physical space and
(b) modal space. The modal transformation x(r) = Ex(r) reduces
the cyclic array of N single—DOF coupled models 8, which together
form a N —DOF coupled system, to a set of N single—DOF decoupled
models 8p. ................................
4.3 (a) Campbell diagram for N = 10, a = 0.84, (5 = 0.67, 7 = 0.169,
u = 0.015, V = 0.5, and engine orders (e.o.) n = 1,2, . . .,N — 1 and
(b) the corresponding frequency response curves with f = 0.01, and
£1, = EC = 0. ................................
4.4 The topology of a bladed disk assembly fitted with absorbers in
(a) physical space and (b) modal space. The modal transformation
q(r) = (E <8) I)u(r) reduces the cyclic array of N, 2—DOF sector mod-
els (B, A), which together form a 2N-DOF coupled system, to a set of
N, 2—DOF block decoupled models (819, AP). .............

xiv

42

43

49
53
57

58

67

69

74

79

87

91

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

Campbell diagrams showing the engine order (e.o.) line no and the
dimensionless natural frequencies a]? , (1222 = no, and 13%.], versus the
dimensionless rotor speed a for N = 10, n = 3, a = 0.84, 'y = 0.127,
6 = 0.67, and u = 0.015 with: (a) V = 0.01; (b) V = 0.25; (c) V = 0.5.

Absorber and blade frequency response curves and Campbell diagrams
for N = 10, n = 3, a = 0.84, (5 = 0.67, it = 0.015, V = 0.5, f =
0.01, and zero damping: (a) B = —0.07 (undertuned); (b) 6 = +0.07
(overtuned); (— —) frequency response with the absorbers locked. See
Table 4.2 on page 91 for the corresponding tuning order data and
resonant rotor speeds ores. .......................
Absorber and blade frequency response curves and Campbell diagrams
for N = 10, n = 3, a = 0.84, (5 = 0.67, a = 0.015, V = 0.5, f = 0.01,
and zero damping: (a) 6 = ficr/2 = —0.00351 (slightly undertuned);
(b) B = 0 (zero, or perfect tuning); (— —) frequency response with the
absorbers locked. The critical absorber detuning is Ber = —0.00701.
See Table 4.2 on page 91 for the corresponding tuning order data and
resonant rotor speeds ores. .......................
Rotor speeds ores corresponding to resonance (—-—) and possible reso-
nant speeds (— -—) in terms of the absorber detuning 6 for N = 10,
n = 3, a = 0.84, 6 = 0.67, p = 0.015, and V = 0.50. The no—resonance
gap is defined by 5a < 5 S 0, where Her = —0.00701. ........
Critical undertuning —flcr x 100 of the absorbers versus (a) the di—
mensionless absorber mass it (with a = 0.84 and 6 = 0.67), (b) the
dimensionless distance from blade base to absorber base point a (with
6 = 0.67 and u = 0.015), (c) the dimensionless radius of rotor disk
(5 (with a = 0.84 and u = 0.015), and (d) the engine order n (with
or = 0.84, 6 = 0.67, and u = 0.015). The dashed line in (d) corre-
sponds to the large-n approximation of ficr given by Equation (D3) in
Appendix D. ...............................
(a) Campbell diagram showing the resonant rotor speeds ores and o L
and the corresponding (b) blade and (c) absorber frequency response
curves for a model with N = 10, n = 3, a = 0.84, 6 = 0.01, 6 = 0.67,
a = 0.05, V = 0.5, f = 0.01, and 5b = {c = 0. .............
Blade and absorber (free and locked) frequency response curves for
overtuned absorbers (6 = 0.01), for the same parameter values used in
Figure 4.10, and for various levels of the absorber damping 5a.

Blade and absorber (free and locked) frequency response curves for
absorbers tuned within the no—resonance zone (6 = flcr / 2 = —0.00351),
for the same parameter values used in Figure 4.10, and for various levels
of the absorber damping €a- .......................

XV

97

104

105

106

107

110

111

112

5.1
5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

(a) Model of bladed disk assembly and (b) sector model. .......
(a) The rotor speed ores corresponding to resonance for the linearized
system of Chapter 4 and the scaled system formed by Equation (5.4)
(with r} = 0) versus the linear absorber detuning parameter 6 for a
model with N = 1, n = 3, a = 0.84, 6 = 0.67, V = 0, and for mass
ratios 0.005 S u 3 0.025; (b) the corresponding percent error. The
no-resonance gap (predicted by the linearized system of Chapter 4) is
defined by ficr < 3 S 0, where the values of the critical undertuning
ficr are given in Table 5.1 for the various mass ratios. .........
Example Campbell diagram showing the speed and order detuning
parameters A and A. ...........................
Blade / absorber frequency response curves for the essentially undamped
isolated nonlinear system with a hardening absorber path (77 = 1),
for various detuning values 6, and for n = 3, a = 0.84, 5 = 0.67,
[,l = 0.0035 (5 = 0.0592), and F = 0.0001 (f = 0.117). ........
Blade / absorber frequency response curves for the essentially undamped
isolated nonlinear system with a softening absorber path (17 = —1),
for various detuning values 6, and for n = 3, a = 0.84, 6 = 0.67,
it = 0.0035 (5 = 0.0592), and F = 0.0001 (f = 0.117). ........
Blade and absorber frequency response curves for the same conditions
in Figure 5.5 except with nonzero damping: Eb = 2 x 10’3 and E- =
2 x 10‘6. .................................
Blade and absorber frequency response curves to accompany those
shown in Figure 5.4 for a hardening absorber path showing various
levels of the order detuning 6 relative to Her 2 —0.00164 inside and
near the no—resonance zone. .......................
Blade and absorber frequency response curves to accompany those
shown in Figure 5.5 for a softening absorber path showing various
levels of the order detuning 6 relative to ficr = -—0.00164 inside and
near the no-resonance zone. .......................
Blade and absorber frequency response curves for the essentially un-
damped isolated nonlinear system with a softening absorber path, for
linear tuning within the no-resonance zone (B = ficr/ 2 = —0.822 x
10‘3), for various levels of the force amplitude F, and for n = 3,
a = 0.84, 6 = 0.67, and u = 0.0035 (8 = 0.0592). ...........
Example blade and absorber frequency response curves for linear tun—
ing within the no-resonance zone (6 = 5m / 2), for several nonlinear
tuning values relative to ncr, and for a model with n = 3, a = 0.84,

A

5 = 0.67, it = 0.0035 (8 = 0.0592), f = 0.0001 (f = 0.117), and
Sb = {a = 0. ...............................

xvi

119

125

128

143

144

145

146

147

150

5.11

5.12

5.13

5.14

6.1

8.1

82

Cl

Blade / absorber frequency response curves for the essentially undamped
coupled nonlinear system with N = 5 (odd) sectors, softening absorber
paths (7) = —1), linear undertuning fl = —0.004, various coupling levels
V, and for n = 3, oz = 0.84, 6 = 0.67, it = 0.0035 (5 = 0.0592), and
F = 0.0001(f: 0.117). .........................
(a) Blade and (b) absorber backward traveling wave (BTW) time re-
sponses for the essentially undamped coupled nonlinear system with
N = 5 (odd) sectors, softening absorber paths (77 = -—1), linear under-
tuning fi = —0.004, coupling strength V = 0.1, and for n = 3, a = 0.84,
6 = 0.67, p. = 0.0035 (5 = 0.0592), F = 0.0001 (f = 0.117), and
o = 0.55. .................................
Blade / absorber frequency response curves for the essentially undamped
coupled nonlinear system with N = 6 (even) sectors, softening absorber
paths (7] = —1), linear undertuning fl = —0.004, various coupling levels
V, and for n = 3, a = 0.84, 6 = 0.67, a = 0.0035 (5 = 0.0592), and
F = 0.0001(f: 0.117). .........................
(a) Blade and (b) absorber standing wave (SW) time responses for
the essentially undamped coupled nonlinear system with N = 6 (odd)
sectors, softening absorber paths (7; = —1), linear undertuning fl =
—0.004, coupling strength V = 0.1, and for n = 3, or = 0.84, 6 = 0.67,

A

u = 0.0035 (5 = 0.0592), F = 0.0001 (f = 0.117), and o = 0.55.

Absorber design chart showing ideal tuning and qualitative regions of
desired, acceptable, possibly poor, and poor absorber performance due
to primary resonance (prim. res.) and/or auxiliary resonance (aux.
res.) in terms of the linear and nonlinear tuning parameters [3 and
n, the values of which span the range from under- to overtuning and
softening to hardening paths, respectively. ...............

The distinct Nth roots of unity 10% (k = 0, 1, 2, . . . , N — 1) arranged
on the unit circle in the complex plane (centered at the origin) for
N =1,2,...,9. Note that 10%, = 1 is real, as is tux/2 = —1 ifN is
even. The remaining roots appear in complex conjugate pairs.

Arrays showing the (i,k) elements of the (a) identity, (b) flip, and
(c) cyclic forward shift matrices of dimension N = 3 for i, k = 1, 2, 3.

Frequency response curves for the system shown in Figure 2.4 on
page 23 for N = 10, V = 0.5, and f = 0.01: (a) n = 3; (b) n = 3.001;
(c) n = 3.01; and (d) n = 3.1; and (e) corresponding Campbell dia-
gram for n = 3.1. In (d) and (e) the principle resonance corresponding
to n + 1 = 4.1 is indicated by the black circle. .............

xvii

161

162

163

164

170

191

193

El

E2

Blade and absorber frequency response curves for the same conditions
in Figure 5.4 on page 142 (hardening absorber path) based on the less
accurate model defined by Equation (E3). ..............
Blade and absorber frequency reSponse curves for the same conditions
in Figure 5.5 on page 143 (softening absorber path) based on the less
accurate model defined by Equation (EH3) ..............

xviii

SELECTED LIST OF OPERATORS

Relevant page numbers are given in the parentheses.

Inner / dot product

EB Direct sum (175)

(8 Direct / Kronecker product (176)
ll ' ll Norm

( - )! Factorial

(T) Complex conjugate

(‘lT Transpose

(°)H = (-)T Conjugate/Hermitian transpose

( - )_1 Inverse

(:) = ad?) Differentiation with respect to physical time t

(-)’ 2 d8 Differentiation with respect to dimensionless time T
A ( - ) Matrix spectrum

circ ( ) Circulant operator (13, 184)

diag ( ) Diagonal operator (176)

det-( ~ ) Determinant

Im ( - ) Imaginary part

log ( - ) Logarithmic function to the base 10

mod ( - ) Modulo

Re ( - ) Real part

tr ( « ) Trace

xix

CHAPTER 1

Introduction

1. 1 Motivation

The goal of this work is to investigate the performance of order-tuned absorbers for vi-
bration reduction in (nearly) cyclic systems. The applications of interest are rotating
flexible structures, and in particular turbine blades, bladed disks assemblies, blisks
(integral disk/ blade systems), and helicopter rotor blades. During steady operation
these systems rotate at a constant speed and are subjected to traveling wave dynamic
loading, which is characterized by excitation frequencies that are proportional to the
mean rotational speed of the rotor. Such excitations result in component vibrations
and can lead to high cycle fatigue (HCF) failure, noise, reduced performance, and
other undesirable effects. This is an ideal setting for the use of centrifugally-driven,
order-tuned vibration absorbers, yet their implementation to such systems has re-
ceived little attention to date. Much is already known about the dynamic behavior
of systems of vibration absorbers, and the same is true for systems with symmetries
in general and for rotating flexible structures in particular. This work aims to apply
the theory, methodology, and design of order-tuned absorbers to such systems.

We begin with a brief survey of order-tuned vibration absorbers in Section 1.2
and of systems with cyclic symmetry in Section 1.3. The application of vibration

absorbers to rotating flexible structures is discussed in Section 1.4 and the chapter

closes in Section 1.5 with the main objectives and contributions of this work and an

outline of the dissertation.

1.2 Order-Tuned Vibration Absorbers

Vibration absorbers were originally proposed by Frahm [4] in a United States patent
in 1911, but it was Den Hartog [5, 6] who first carried out systematic studies on
their characteristics and design, including an optimal choice of design parameters.
They come in two basic varieties: frequency-tuned and order-tuned. In the former,
the absorber is tuned to a given problematic frequency (typically near a resonance)
and damping is added so that it performs effectively in the neighborhood of that
frequency. Such absorbers rely on elastic elements to provide their restoring forces,
and they are designed so that these forces counter the translational or rotational
motion of the primary systems to which they are attached. In contrast, order-tuned
vibration absorbers exploit the centrifugal field from rotation of the primary system
[7,8]. Since rotating flexible structures are dominated by forces that occur at orders
of rotation (the so-called traveling wave or engine order excitations [9]), order-tuned
absorbers are ideally suited for such applications.

A class of order-tuned absorbers that has enjoyed considerable attention in recent
decades are centrifugal pendulum vibration absorbers, or CPVAs. They essentially
consist of mass particles that ride along designer-specified paths relative to the pri-
mary system and their parameters are chosen such that they counteract the fluctu-
ating loads applied to that system. Each employs the centrifugal field from rotation
(rather than an elastic element) for its restoring force and this results in absorber
natural frequencies that are proportional to the rotation rate, where the constant of
proportionality is dictated by geometric parameters that are chosen by design. The
selection of the absorber path shape and location relative to the center of rotation of

the primary system prescribes its linear tuning order, as well as its nonlinear response

characteristics.

The dynamic performance, characteristics, and features of CPVAS are well-
understood in typical situations, and there are numerous examples of their imple-
mentation. For example, they are widely-used for reducing torsional vibrations in
rotating machinery, where they are attached directly to the rotor. CPVAs have been
used in light aircraft engines [10] and helicopter rotors [11], and they are also finding
new applications including diesel camshafts [12] and variable displacement automo-
bile engines [13]. In nearly all applications, CPVAS employ circular paths due to the
simplicity of their manufacture and also due to a lack of knowledge about noncircular
paths.

Den Hartog described the basic features of CPVAs, including how one selects pa-
rameters for linear tuning, as well as a discussion of the frequency shifting that arises
from nonlinear softening effects in circular—path absorbers (that is, the decrease in fre-
quency as the amplitude of oscillation increases) [7]. He also suggested intentionally
detuning (in fact, overtuning) the absorbers to avoid the jump instabilities associated
with these nonlinear effects. This approach works well, but it comes at the expense of
reduced absorber performance [14]. Newland carried out a systematic analysis of the
nonlinear response of a CPVA and offered a strategy for selecting an appropriate level
of detuning for circular path absorbers [15]. Extensive linear-based design guidelines
exist for CPVAs applied to crankshafts of internal combustion engines [10].

Subsequent research on CPVAs focused on several issues, including the design
of the absorber path for optimal performance and the response of CPVA systems
composed of several absorbers fitted to a rotor. Research on CPVA paths begins
with the analysis of Madden who suggested that a cycloidal path would avoid the
nonlinear frequency shift that leads to jumps [16]. Denman carried out a systematic
study of various paths and showed that a cycloidal path is slightly hardening whereas

a particular epicycloidal path is neutral, that is, neither softening nor hardening

[17]. Further work showed that while the epicycloidal path leads to essentially linear
absorber motions over all amplitude ranges, the corresponding torque applied to the
primary system is not purely harmonic at the desired order, but contains higher
order harmonics that arise from nonlinear kinematic effects [18]. An examination of
general paths and their attendant torques led to the development of subharmonic
pairs of epicycloidal absorbers that generate a purely harmonic torque [19—21].
When a rotor is fitted with multiple, identically-tuned CPVAs the absorbers are
coupled through its inertia, and the coupling is inherently weak since the absorber
inertia is much smaller than that of the rotor. Moreover, the absorbers are designed
to be lightly damped and they are tuned (close to) the order of the excitation. These
features give rise to a system of internally resonant, weakly coupled, and weakly
damped oscillators—“a situation ripe with instabilities and rich dynamics. Such fea-
tures also lead to mathematical models with small nondimensional parameters that
are amenable to analytical treatment using perturbation methods. Systematic inves-
tigations of nonlinear responses have been carried out using averaging and symmetric
bifurcation theory for a range of path types [14, 22,23]. It was shown that the desired
synchronous response of the absorbers can undergo two basic types of instabilities.
The first type maintains the symmetry of the response but results in jumps, just as
in the case of a single absorber. For multiple absorbers, an additional instability
can occur wherein the symmetry of the response is broken. This results in a rich
bifurcation where multiple response branches arise, including some spatially localized
responses [22,24]. Similar analyses were carried out for CPVA systems composed of
multiple subharmonic pairs [25]. The effects of small imperfections among the ab-
sorbers, which are inevitable due to manufacturing, in-service wear, and so on, were
considered in the context of linear system models, where it was shown that these
systems experience localized free modes of vibration, as well as localized responses to

order excitation [26,27].

Early experimental work on CPVAs focused on particular applications, for exam-
ple, specific internal combustion engines. However, systematic experiments using a
dedicated test rig have recently been carried out for both circular and epicycloidal
path absorber systems [28—30]. These results confirm the linear and nonlinear be—
havior of CPVAs for these path types, and also demonstrate the rich behavior that
occurs if absorbers are tuned too closely to the excitation order [29,31]. In terms
of applications, CPVAs show great promise for use in advanced internal combustion
engines that offer increased fuel efficiency and reduced emissions [13].

Finally, impact dampers have recently attracted considerable attention in the
jet engine community as an effective means of reducing blade vibration amplitudes.
Such dampers typically consist of a single mass traveling back and forth in a cavity
machined in a turbine blade, where energy dissipation occurs when the mass impacts
the cavity walls. In this case, the impact damper is also designed to act as a tuned
absorber, with both effects contributing to vibration reduction. Much of this work
has been carried out experimentally for specific applications [32,33], although there
has been some systematic theoretical work as well [34]. Other impact damper designs
are also being explored by industry, such as dry particle damping systems [35] that
include a large number of tiny masses in a blade cavity, where energy dissipation
is achieved from multiple impacts between the masses. Such designs have proven
effective experimentally, although they suffer durability problems in the harsh jet

engine environment (e.g., rotation, extreme temperatures, and so on).

1.3 Systems with Cyclic Symmetry

Many rotating flexible structures consist of an array of interconnected constituent
parts (substructures) whose geometry and structural properties are rotationally peri-
odic, and they are said to have cyclic symmetry. In a bladed disk, for example, the

fundamental substructure is one blade plus the corresponding segment of the disk,

which is collectively referred to as a sector. The entire dynamics of these systems
can be captured by analyzing a single sector, so long as one applies the appropriate
phase conditions at the interfaces with adjacent sectors. This is a feature shared by
all perfectly cyclic systems, and it offers a tremendous computational savings in their
analysis.

The linear free response of a cyclic structure is characterized by identical motions
of each sector, except for a fixed sector—to-sector phase difference, and hence the mode
shapes are harmonic in the circumferential direction. For bladed disk assemblies, this
leads to nodal lines across the disk called nodal diameters, and the system mode
shapes are referred to as nodal diameter modes. An engine order excitation of order
n will generally excite only the modes with n nodal diameters. Due to the cyclic
symmetry of the system, most of the natural frequencies occur in repeated pairs and
these correspond to traveling wave modes, where (in the absence of damping) the
system kinetic energy remains fixed and is simply passed from one blade to another.
The distinct frequencies correspond to standing wave modes, where the system ki—
netic energy varies sinusoidally [36]. If the sectors are only weakly coupled to one
another, all of the system natural frequencies lie closely together, leading to a very
rich structure with high modal density and high sensitivity to imperfections.

Generally, the blades on a turbomachinery rotor are meant to be identical. In
practice, however, there are always small random uncertainties among the blades
due to manufacturing tolerances, in-service wear, material imperfections, and so on.
These small variations, referred to as mistuning, can lead to a confinement of vibration
energy to a few blades or even a single blade, a phenomenon known as localization
[37—40]. Due to this spacial confinement of energy, some of the blades may experience
higher amplitudes than what is predicted from the ideal, perfectly periodic system
[41—45]. Forced response amplifications of 200% or more can occur, resulting in high

cycle fatigue and eventual failure. This is a major cost, safety, and readiness concern

for commercial and military jet engines alike [46].

The features of a typical cyclic system also lead to a rich variety of nonlinear
dynamic behaviors. In particular, the repeated natural frequencies give rise to internal
resonances when the systems are weakly damped and the excitation of interest is
resonant. Additionally, there is the possibility of multiple interacting modes when
the substructures are weakly coupled. These characteristics lead to situations ripe
for instabilities, bifurcations, and a multitude of possible responses, just as in the
case of torsional CPVAs. Previous research in this area has dealt with the modal
interactions that arise from internal resonances [47—50] and nonlinear localization
[51—54]. However, it is important to note that localization of responses can occur in
perfectly tuned nonlinear system models, where the mistuning between subsystems
arises from the frequency dependence on amplitudes, rather than from imperfections.

A recent line of particularly interesting research in cyclic systems has focused on
the problem of how to design systems to reduce the harmful effects of localization. A
promising idea is to introduce into the system intentional patterns of mistuning that
can make the system robust with respect to the unavoidable imperfections. This has

been quite successful, at least for linear system models [55—58].

1.4 Vibration Absorbers for Rotating Flexible Structures

under Engine Order Excitation

Since order-tuned absorbers are designed to address system vibrations at a trouble—
some order (rather than a fixed frequency), it is natural to consider their use on
rotating flexible structures subjected to engine order excitation. In this context, the
desired system response is a traveling wave, where each sector behaves identically
except for a fixed phase difference among nearest neighbors; any other response type

implies decreased absorber performance. The ideal response is one in which the blades

remain stationary relative to the rotating hub and the absorbers respond in a trav-
eling wave to identically counter the engine order (traveling wave) excitation of the
blades.

Order-tuned absorbers have already been considered for helicopter rotor blades
[59], for an ideal flexible beam [60], and for potential implementation in hollow turbine
blades [61]. In addition, absorbers that employ order tuning at small amplitudes
and that transition into impact absorbers at larger amplitudes have recently been
experimentally [32, 33] and analytically [34] investigated. While these studies have
been promising, they have been limited in several key ways: (1) each considered very
specific applications; (2) the studies do not systematically address how to size and
tune the absorbers for optimal performance over a wide range of operating conditions;
and (3) all previous studies focused on the implementation of an absorber to an
individual structural element (e. g., a single blade). In fact, these studies were limited
to either theoretical results based on linear analyses or observations gleaned from
experiments or simulations.

This thesis reports the first systematic analytical treatment of order-tuned vibra-
tion absorbers applied to a fully-coupled cyclic structure under engine order excitation,
including detailed recommendations for both linear and nonlinear absorber design. In
this way, it serves not only as the first study of its kind to unite the individually ma-
ture bodies of research on absorber systems and cyclic systems, but it also provides
context and direction for what is sure to be a plentiful and rich course of ongoing

theoretical and experimental work.

1.5 Dissertation Overview

The goal the investigation is threefold: (1) to quantify and understand the under-
lying linear resonance structure of a cyclically-coupled bladed disk assembly fitted

with order-tuned absorbers; (2) based on these findings, to design the absorbers to

eliminate or otherwise reduce blade motions relative to the rotating hub; and (3) to
generalize the linear theory, methodology, and design to include the basic, first-order
effects of nonlinearity.

As we shall see, the underlying linear resonance structure of the cyclically cou-
pled system fitted with absorbers is surprisingly rich, a feature that arises from the
order-nature of the absorbers. This is manifested in the classical eigenvalue veering
phenomenon and gives rise to an ideal absorber design (perfect absorber tuning) in
which the absorber tuning order is set to identically match the engine order. In the
absence of damping, the result is a total elimination of the blade motions relative to
the rotating hub—independent of the rotor speed. One of the main findings of the
linear analysis, and indeed of this entire thesis, is the existence of an entire spectrum
of absorber designs for which there are no system resonances over the full range of
possible rotor speeds. This corresponds to a continuous set of absorber under-tuning
values, the so—called “no-resonance zone,” where resonance is avoided altogether. By
selecting a design within this small, but finite gap, there is at least some level of
robustness to parameter uncertainties, but at the expense of residual (zero damping)
or slightly increased (nonzero damping) blade motions.

Nonlinearity is introduced via the absorber paths and, for the desired traveling
wave response, it gives rise to additional frequency response branches and resonances,
but the fundamental linear resonance structure mentioned above is shown to qualita-
tively persist. There does exist a nonlinear tuning strategy that guarantees a branch
of solutions corresponding to zero (or otherwise reduced) blade motions. However,
unlike its linear counterpart, it is highly sensitive to parameter uncertainties. Even
more importantly, the nonlinear tuning is shown to depend on both the rotor speed
and force amplitude, and is thus effective only for a single set of operating conditions.
These findings suggest that it is not practical to exploit nonlinearity to further improve

the absorber performance in the cyclic systems under consideration. If nonlinearity

is inevitable, it is clearly shown that softening characteristics are more desirable than
hardening; the former simply sets an upper limit on the range of rotor speeds over
which the absorbers are effective, while the latter may give rise to problematic reso-
nances (especially if the damping is light). Finally, for the weakly coupled and lightly
damped systems under consideration, there may be a myriad of additional responses
other than the desired traveling wave variety. However, based on a number of case
studies and extensive numerical investigations, none could be identified. (Analytical
local stability results for the fully-coupled nonlinear system are essentially intractable,
even after a simplifying reduction of the Jacobian matrix is carried out.) This is, in
fact, very good news from a practical viewpoint since a traveling wave response of
the absorbers is desired.

The main body of the dissertation is organized as follows. Chapter 2 highlights
relevant background topics and material, including some mathematical preliminaries,
engine order excitation, vibration characteristics of cyclic systems, and frequency- and
order-tuned vibration absorbers. A suitable mathematical model for a bladed disk
assembly fitted with order-tuned vibration absorbers is developed in Chapter 3, from
which a number of specific models to be systematically analyzed in the two subsequent
chapters are gleaned. A linearized model is investigated in Chapter 4, which gives rise
to a linear absorber tuning strategy, and these results are generalized in Chapter 5 to
include the basic effects of nonlinearity. Finally, the dissertation closes with detailed
recommendations for absorber design, a summary of contributions, and directions for

future work in Chapter 6.

10

CHAPTER 2

Background

2. 1 Introduction

This chapter highlights pertinent background material that will be useful in the anal-
yses of subsequent chapters. Some mathematical preliminaries are considered first
in Section 2.2, including the Kronecker product, the Fourier matrix, and the basic
theory of Circulants. These sections are quite brief and are meant only to highlight
well-known, but fundamental results and properties. A more exhaustive treatment
of the theory of Circulants (including many proofs) is given in Appendix B, and Ap-
pendix A reviews some selected topics from linear algebra. A model for engine order
excitation is subsequently developed in Section 2.3 and its traveling wave nature
discussed. In order to characterize the basic free and forced response of cyclically-
coupled systems under engine order excitation, a cyclic chain of linear oscillators is
investigated in Section 2.4. Finally, the theory of frequency- and order-tuned vibra-
tion absorbers is given in Section 2.5, and the chapter closes in Section 2.6 with some

concluding remarks.

2.2 Mathematical Preliminaries

We make frequent use of the theory of circulant matrices throughout this work. Their

basic properties are outlined in this section, along with some other relevant mathemat-

11

ical preliminaries. The Kronecker product is defined first in Section 2.2.1, followed by
the Fourier matrix in Section 2.2.2. Circulant and block circulant matrices are defined
in Section 2.2.3, and it is shown how to diagonalize such matrices in Section 2.2.4.
These sections are meant as a quick reference and correspondingly the treatment is
brief and proofs are omitted. A more detailed account of the theory of Circulants is
given in Appendix B, which was distilled from the classical text by Davis [62] and
the work by Ottarsson [36], and some selected topics from linear algebra are given in

Appendix A.

2.2.1 The Kronecker Product

Let A E men and B E Cpxq. Then the Kronecker (direct) product of A and B is

the mp x no matrix

fallB a12B ~- alnB-
a B a B a B

A53: 3 3 . 3 an
LamlB am2B ' ' ' amnB_

 

 

Some selected useful properties of the Kronecker product are as follows.
1. If A, B, C, and D are square matrices such that AC and BD exist, then
(A (X) B)(C 09 D) = (AC) <8) (BD).

2. If A and B are invertible matrices, then (A <8) B)‘1 = A’1 <8) B"1.

3. If A and B are square matrices, then (A (X) B)H = AH <8) BH.

Here, ()H = (j)T denotes the Hermitian operator, or conjugate transpose.

2.2.2 The Fourier Matrix

Let j = \/—I and N = {1,2, . . .,N}. Then the N X N complex Fourier matrix is
defined as

wti—lkk—l) = Leak—1190., 23k 6 N (22)

E = EN = le'ikl ; eik = m

sl~

12

2 7r
where w = wN = eff is the primitive Nth root of unity (see Section B.4.1 of

Appendix B) and

__ 27r(i — 1) ,
992 — “N , 2 E N (2.3)

is the angle subtended from the positive real axis in the complex plane to the it”
power of wN. (See Figure 31 on page 191.) When the dimension of E N is clear, the
subscript N will be omitted. It is shown in Section 2.2.4 that all circulant matrices

share the same linearly independent eigenvectors

e, ___ -1_(1, w(i-1),w2(i—1), , , _ ,w(N-1)(i—1))T
W , iEN (2.4)
1 - . .' . - ,, T
: __(1,e]901,e.l2‘pz, . _ . [BAN-1M)
x/N
which compose the N columns (or rows) of E, that is, E = [e1,e2,...,eN]. An

important property of the Fourier matrix is that it is unitary and, therefore,

E'HE = EEH = I, (2.5)
where I is the identity matrix. Finally, the matrices EH, E001, and (E<E§>I)nH = EH®I
are also unitary.

2.2.3 Circulant Matrices

An N x N circulant matrix (or circulant for short) is formed from an N-vector by

cyclically permuting its entries and is of the form

P61 C2 .o. CN '-
CN 61 CN—i

C= . . . . (2.5)
t02 63 Cl _

 

 

Thus a circulant matrix is defined completely by an ordered set of generating elements
c1, c2, . . . ,cN. It is convenient to define the circulant operator circ ( ~ ) that takes as its
argument these generating elements and results in the array given by Equation (2.6),
that is,

C =circ(cl,c2,...,cN). (2.7)

13

The set of all such matrices will be designated by (EN. An N111 x NM block circulant
matrix is defined similarly to Equation (2.6) and has the representation given by
Equation (2.7), where each entry ck is replaced by the M x 111 matrix Ck for each
k E N. The ordered set of matrices C1, C2, . . . , C N are called its generating matrices.
The set of all NM x NM block circulant matrices with M x M blocks, which is
sometimes called a block circulant of type (M, N), will be denoted by Q‘KMW. Finally,

if a matrix is both circulant and symmetric it can be written as

circ (c1,c2,...,cN,cN+2,cN,...,C3,c2), N even
T 2 T
C: (2.8)
CirC(C1,C2,...,CN-1,CN+1,CN+1,CN_1,...,C3,C2), NOdd
T T T T

and necessarily has repeated generating elements; only (N + 2) / 2 are distinct if N
is even and (N + 1) / 2 are distinct if N is odd. The set of all N X N symmetric
Circulants is denoted by V‘KN. An Nil! x Nil! block circulant, block symmetric
matrix is obtained by replacing each ck in Equation (2.8) with Ck for each k E N.
The set of all NM x NM block circulant, block symmetric matrices with M x M
blocks will be denoted by fi‘é‘fif’MW.

2.2.4 Diagonalization of Circulants

A matrix C E (KN with generating elements c1, c2, . . . ,c N can be diagonalized via the

unitary (similarity) transformation

 

 

A1 0
H A2 .
- 0 /\N_
where
N
A; = chui(k_1)(i_1), i E N. (2.10)
k=1

As a consequence, all circulant matrices share the same eigenvectors, which are given

by Equation (2.4). Their eigenvalues are given by Equation (2.10) and depend only 011

14

the generating elements. Similarly, a matrix C 6 $511.51 can be block diagonalized

via the unitary transformation1

(Eff ‘8 1M)C(EN 69 1M) = .. , (2-11)

 

 

where 0 and I M are the M x M zero and identity matrices, respectively, and
N
A,- = chw(k—1)(i‘1>, i e N (2.12)
1:21

which depends only on the generating matrices C1, C2, . . . , C N- Since Equa-
tion (2.11) is a unitary transformation it preserves the eigenvalues of C. Hence its
eigenvalues are the eigenvalues of the N matrices A,- E (CM XM . If v,- is an eigenvector

of A,- then the corresponding eigenvector of C is u,- = e,- ® v,.

2.3 Engine Order Excitation

2.3.1 Mathematical Model

Ideally, the steady axial gas pressure in a jet engine might vary with radius but is
otherwise uniform in the circumferential direction, thus resulting in an identical force
field on each blade in a particular fan, compressor, or turbine within the engine. In
practice, however, flow entering an engine inlet invariably meets static obstructions,
such as struts, stator vanes, etc., in addition to rotating bladed disk assemblies in
its path to the exhaust. Even in steady operation, therefore, the flow slightly up-
stream of these bladed assemblies is non-uniform in pressure, temperature, and so
on. This results in a static pressure (effective force) field on the blades that varies
circumferentially, an example of which is shown in Figure 2.1.

Consider, for example, an engine in steady operation with n evenly-spaced struts

slightly upstream (or downstream) of a bladed assembly. As explained in [9] these

 

1Note that (EN (8) 1M)“ = (Ell (8) IM), which follows from Property 3 of Section 2.2.1.

15

 

 

 

ll
A Ideal Conditions
3
D. /\\ A
<1)
H
:1
a p, /
:0
x...
m V
8
U
E" Actual Conditions
<
0 7r 27r

Circumferential Postion 6

Figure 2.1. The the axial gas pressure 11(6): ideal and (example) actual conditions.

obstructions produce a circumferential variation upon the mean axial gas pressure
that is essentially proportional to cos n9, where 9 is an angular position. Thus a
blade rotating through this static pressure field experiences a force proportional to
cos th, where Q is the constant angular speed of the bladed disk assembly and t is
time. An adjacent blade experiences the same force, but at a constant fraction of
time later. This type of excitation is defined as engine order excitation and n is said
to be the order of the excitation.

To be more precise, the axial gas pressure of a steady flow through a jet engine
may be described by a function p(0) = [)(6 + 27r), where 0 is an angular coordinate
measured relative to a fixed origin on the machine. That is, the pressure field is
rotationally periodic and can therefore be expanded in a Fourier series with terms of

the form p0 cos n6. Then if the angular position of the it” blade relative to the same

origin is defined by

2
9,0) = 0t+§(i— 1), iEN

where N is the total number of blades and N = {1, 2, . . . , N} is the set of blade, or

sector numbers, it follows that the total effective force exerted on blade i due to the

16

nth harmonic of the pressure field p(6) can be captured by
Fcos(nf2t + 27rj'6(i — 1)), i E N. (2.13)
Upon complexifying this gives rise to
[7,-(t) = Fej‘f’iejnm, 2' e N (2.14)

which is a model for the nth predominant component of the excitation. It has period
T = 27r/nf2, strength F, and is said to have angular speed ft. The so-called inter-blade

phase angle is defined by

a, = 5]") = 27%“ — 1) = 7m, ie N (2.15)

where n E Z4. is the order of the excitation and go,- is the angle subtended from

blade 1 to blade i and is defined by Equation (2.3). Equation (2.14) is defined as
nth engine order (e.o.) excitation (72 cc. excitation) or traveling wave excitation and
is used to model the dynamic loading on models of bladed disk assemblies throughout

this work. The traveling wave characteristics of this type of excitation are considered

next.

2.3.2 'h'aveling Wave Characteristics

Equation (2.14) is a function of continuous time t and it is discretized in space via the
index i. This gives rise to two interpretations of engine order excitation relative to the
rotating hub (one discrete and the other continuous) and these can be visualized in
Figure 2.2, which shows a dissection of the excitation amplitudes along time and sector
axes. In the first and usual sense, Equation (2.14) is a discrete temporal variation
of the dynamic loading applied to individual blades. That is, under an engine order
n excitation each sector is harmonically forced with strength F and frequency n9,
but with a fixed phase difference relative to its nearest neighbors. Physically, one can

think of this as placing N different observers at the discrete sectors and having the it”

17

observer record the excitation strength applied to sector i as a function of time. Their
recorded time traces would resemble those shown in Figure 2.2a. In the second and
more general sense, Equation (2.14) can be viewed as a continuous spatial variation of
the excitation strength relative to the rotating hub (along the sector axis) that evolves
with increasing time, i.e., it is a propagating waveform or traveling wave. If a single
observer was placed on the rotating hub and recorded the strength of this traveling
wave as a function of i (taken here to be continuous), it would resemble the curve
shown in Figure 2.2b. In this context, the instantaneous loading applied to individual
blades is obtained by essentially “sampling” the continuous traveling wave at each
sector i E N and, as time evolves, these sampled points define N time-profiles of the
force amplitudes, which is equivalent to the discrete temporal interpretation described
above. However, the latter interpretation illuminates some important traveling wave
characteristics of the engine order excitation that are otherwise difficult to explain,
and in what follows these are systematically described.

To explain the traveling wave mathematically, it is convenient to define

(Pt-(X) = 008(21rgérflx) = COSWkX), (2-16)

which is a cosinusoidal waveform with wavelength 27r/cpk. Then for i E N Equa-

tion (2.14) can be written (in real form) as

F,(t) = Fcos(<pn+1(i — 1) + nflt), (2.17a)

= F<I>V,,+1(i — 1+ Ct), (2.17b)

which is a harmonic function with a wavelength of 27r/tpn+1 = N / n sectors (99,,“
is the wave number) and angular frequency n52. Equation (2.17b) shows that it is
a traveling wave (TW) in the negative i-direction (descending blade number) with
speed C = nQ/aan = N 9/ 27r, measured in sectors per second. An example plot
of this continuous backward traveling wave (BTW) is shown in Figure 2.2b and, as

described above, the applied loads can be obtained from this figure by continuously

18

F10) F20) FN(t

F“) A /. /\ )/\ A,
_..\/ v v v .v \/

Sector 1 Sector 2 .. . Sector N

 

 

 

 

 

— Discrete Temporal Variation
Continuous Spacial Variation (BTW)
- o 0 Applied Loads (SW, BTW, or FTW)

 

 

 

 

 

 

 

s T A; t;
AWN UN
3 zfiflflfifffftflifflb’)“ i’
1 2 3 4sect5‘1’8d6 [A49
rflrl
(b) 0

Sector Number i

Figure 2.2. An example illustration of the discrete temporal and continuous spatial vari-
ations of the traveling wave excitation defined by Equation (2.14) (in real form): (a) the
discrete dynamic loads with amplitude F and period T = 27r/nf2 applied to each sector; and
(b) the continuous BTW excitation with wavelength N /n and speed C = N Q/27r relative

to the rotating hub.

19

“sampling” the waveform at the discrete sector numbers as time evolves. Then the
engine order excitation applied to the individual blades consists of a wave composed of
these N discrete points, examples of which are shown in Figure 2.3a-d. Interestingly,
this gives rise to discrete SW or even forward traveling wave (FTW) applied dynamic
loads (depending on the value of n relative to N) even though Equation (2.17) is
strictly a backward traveling waveform relative to the rotating hub. These additional
possibilities arise due to aliasing of the “sampled points” just as it occurs in elementary
signal processing theory [63,64]. Before characterizing the traveling and standing
waveforms it is shown that one need only consider engine orders n 6 N.

The traveling wave nature of the discrete applied loads (i.e., SW, BTW, or FTW)

depends only on the value of n relative to N. To see this, let
nznmodNEN, nEZ+ (2.18)

and assume n = a + mN for some integer m. Then one can write <I>fi+mN+1(X) =
(anlX) and it follows that if n = it corresponds to a SW, BTW, or FTW then so
does a + mN for any m E Z+. In this sense, the traveling wave nature of the applied
dynamic loads is seen to alias relative to N. These features are characterized next for
engine orders n E N, where it is understood that the results can be applied to any
n > N simply by taking 72. modulo N (where appropriate).

For the special case when n = N the rotating blades become entrained with the
excitation since quN) =‘ 27rn(i -— 1) with i,n E Z+ and hence each is forced with
the same strength and phase. As illustrated in Figure 2.3d, this is effectively a SW
excitation where each blade is harmonically forced according to F,(t) = F cos th.

N/Z) = 7rn(i -—

Entrainment also occurs when n = N / 2 if N is even, in which case (DE
1), where (i -— 1) is odd (resp. even) for even (resp. odd) sector numbers i E N.
Accordingly, all blades with odd sector numbers are driven by F;(t) = F cos nflt, as

are the blades with even sector numbers, but with a 180-degree phase shift. As shown

in Figure 2.3b, this amounts to the same standing wave excitation as the n = N case,

20

Table 2.1. Sets of engine orders n (mod N) E N = N32,“: UN§,€, UNST'E, corresponding to
backward traveling wave (BTW), forward traveling wave (FTW), and standing wave (SW)
dynamics loads applied to the blades for (a) odd N and (b) even N. These can be visualized

in Figure 2.3 i-ii.

 

 

(a)0ddN (b) EvenN
NBOTW={n€Z+ilSnSflf—l} NgTw={n€Z+zlgngflf—2}
N1‘9'1‘w={n€Z+IflgflSnSN—l} NgTw={n€Z+zy—2fl$nSN—l}
Nefw ={N} NSEW ={-§,N}

 

except for a phase reversal in the excitation among adjacent blades. The engine
orders corresponding to SW excitations for odd and even N are denoted by the sets
NSOWE C N, which are defined in Table 2.1 and all other values of n E N correspond
to traveling waves. Engine orders n E NgTfiv (resp. n E NPQ'fEIlv) correspond to
BTW (resp. FTW) excitation, an example of which is shown in Figure 2.3a (resp.
Figure 2.3c), where Nngv and NSTEJN are also defined in Table 2.1. These sets can

be visualized in Figure 2.3 i—ii for odd and even N.

2.4 Vibration Characteristics of Cyclic Systems

The basic free vibration characteristics of cyclic systems are discussed next, in addition
to forced vibration under the engine order excitation described in Section 2.3. A
prototypical linear model is introduced in Section 2.4.1, which consists of a cyclic array
of N identical, identically coupled oscillators, each with a single degree of freedom
(DOF). Its forced response is considered in Section 2.4.2, including a decoupling
strategy based on the cyclic symmetry of the system. The details of its free response
are given in Section 2.4.3 and in Section 2.4.4, which discuss the eigenfrequency
characteristics and normal modes of vibration, respectively. Finally, conditions for
resonance are given in Section 2.4.5, along with a description of the underlying linear

resonance structure in terms of the engine order and angular speed of the excitation.

21

l———-BTW-—-l I-———FTW————-l SW

 

 

 

 

 

 

 

(i) Odd N . - . . , , . T . , , > nmodN
1 00- Ngl N2i1 000 N—l N
I'-—'—BTW———I SW I——FTW——"I SW

(ii) Even N . . v , . , . . , . : n modN
1 co. N‘—2 fl N+2 no. N—l N

2 2 2
F

(a)n=1

-—BTW 0

n-BTW

_F 4 L ‘ z
I 2 3 4 5 0 7 8 9 10
F

(b)n=5

—BTW 0

000 SW

_F 1 1 T—‘k z
I 2 3 4 5 6 7 8 9 10
F
(c)n=9
BTW 0
co. FTW
.__F 1 ‘ z
1 2 3 4 5 0 7 8 9 10
F

(d)n=10

—BTW 0

000 SW

——F

1 2 3 4 5 6 7 8 9 10

Figure 2.3. Engine orders n mod N corresponding to BTW, FTW and SW applied dynamic
loading for (i) odd N and (ii) even N (see also Table 2.1); example plots of applied dynamic
loading (represented by the dots) for a model with N = 10 sectors and with (a) n = 1
(BTW), (b) n = 5 (SW), (c) n = 9 (FTW), and (d) n = 10 (SW). The BTW engine order
excitation is represented by the solid lines.

22

 

 

 

Figure 2.4. A prototypical linear cyclic system with cyclic boundary conditions 2:0 = :1: N
and $N+1 2 1'1.

2.4.1 A Prototypical Model

The undamped cyclic system to be considered is shown schematically in Figure 2.4.
It consists of a cyclic chain of N single-DOF oscillators each of mass M, the dynamics
of which are captured by the transverse displacements 33,-, and these are uniformly
attached around the circumference of a (nonrotating) rigid hub via linear elastic
elements of stiffness kb and effective length L. Adjacent masses are elastically coupled
via linear springs, each with stiffness kc. It is assumed that the elastic elements are
unstressed when the oscillators are in a purely radial configuration, that is, when
cc,- = O for each i E N. An individual oscillator, together with the nearest—forward-
neighbor elastic coupling, forms one fundamental sector and there are N such sectors
in the overall system. Finally, the system is subjected to engine order excitation of
order n E Z... and angular speed 9, which can be modeled by Equation (2.14).

The linear equation of motion for sector i is obtained in the usual manner. It is
divided through by the inertia term ML and time is rescaled according to T 2: wot,
where rug 2 W is the undamped natural frequency of a single isolated sector.

Then if q,- = ac,- / L the dynamics of the ith sector are governed by
(1;, + (11+ V2(-qi—1 + 2‘12" (n+1) = fejgbiejnm, ie N (2-19)

where V = (Nee/kg, is a nondimensional coupling strength and ()’ = d( - )/d7'. The
dimensionless angular speed and strength of the engine order excitation are denoted

by a = Q/wo and f = F / Lkb, respectively, n is its order, and g5,- is the interblade

23

phase angle defined by Equation (2.15). In Equation (2.19) it is understood that

(10 = (IN and qN+1 = (11. (2-20)

which implies that the N th oscillator is coupled to the first.

By stacking the N coordinates q, into the configuration vector q =
(q1,q2, . . . ,qN)T, the governing equation of motion for the overall N—DOF system
takes the form

q” + Kq = rein“, i6 N (2.21)

where f = (fej‘f’l, fejbe, . . . , fej‘f’N)T is the system forcing vector, which accounts
for the constant phase difference in the dynamic loading from one sector to the next.

The N x N matrix

"1+2u2 —u2 0 0 —u2 '
-—1/2 1+21/2 —u2 0 0
0 —1/2 1+21/2 0 0
K = . . . . . . (2.22)
0 0 0 1+2);2 —u2
_ —z/2 0 0 —u2 1+21/2J

 

 

reflects the nondimensional stiffness of each sector relative to the hub (additive unity
along its diagonal) and also the inter-sector coupling (V2 along the super- and sub-
diagonal). The elements -I/2 appearing in the (1, N) and (N, 1) positions of K are
due to the cyclic boundary conditions given by Equation (2.20), and in particular the
(123:1 terms in Equation (2.19). (In their absence, the system represents a finite chain
of N oscillators.) Thus, in addition to being symmetric, Equation (2.22) is also a

circulant and can be written as2

K = circ (1+ 2V2, —1/2, 0, . . . ,0, —1/2) E yng, (2.23)

 

2This is a property shared by all linear(ized), perfectly cyclic systems with N sectors, a single—
DOF per sector, and nearest-neighbor coupling. In the more general case of multi—DOF per sector,
the system matrices are block circulant (and also block symmetric) and hence belong to fiféifiy’MN,
where M is the number of DOF per sector.

24

where the circ ( ) operation is defined in Section 2.2.3. In the absence of coupling
(that is, if 1/ = 0) K is diagonal and Equation (2.21) represents a decoupled set of N
harmonically forced, single—DOF oscillators.

The forced response of Equation (2.21) under engine order excitation is considered
next with emphasis on a modal analysis whereby the fully coupled system (that is,
one in which 1/ 5f 0) is reduced to a set of N single—DOF oscillators, only one of
which is (harmonically) excited. Such an analysis illuminates the basic vibration
characteristics of linear cyclic systems, including their eigenfrequency and resonance
structures. The approach taken here, and a generalization in which each sector has
many DOF, is applied to the linear system in Chapter 4 and also in Chapter 5 to

handle (block) reduction of the Jacobian matrices.

2.4.2 Forced Response Under Engine Order Excitation

The steady-state response of Equation (2.21) can be obtained using standard tech-

niques [65] and, for non—resonant forcing, it is given by
ss _ 2 2 —1 37107
q (7') — (K — n a I) fe , (2.24)

where I is the N x N identity matrix. However, this requires inversion of the
impedance matrix K — n2021, which can be computationally expensive for a large
number of sectors, and it offers little insight into the basic vibration characteristics.
In what follows, a transformation based on the cyclic symmetry of the system is
exploited to fully decouple the single, N -DOF system to a set of N, single—DOF os-
cillators from which the steady-state response can easily be obtained. The procedure
is similar to the usual modal analysis from elementary vibration theory. However, a
key difference is that the transformation matrix (and hence the system mode shapes)
is known a priori and, since the transformation is unitary (thus preserving the sys-
tem eigenvalues), the natural frequencies can be obtained after the transformation

is carried out. Moreover, due to orthogonality conditions between the normal modes

25

and forcing vector, the steady—state response of the overall system reduces to finding
the forced response of a single harmonically-forced, single—DOF oscillator in modal
space, which offers a clear advantage over the solution to Equation (2.24).

It was shown in Section 2.3.2 that an engine order excitation can be regarded
as traveling wave dynamic loading, and it is therefore reasonable to expect steady—
state solutions of the same type. We begin with a simple way to show the existence
of such a response, and then systematically describe it based on the results of the

aforementioned modal analysis.

EXISTENCE or A TRAVELING WAVE RESPONSE

Since the excitation is a traveling wave it is natural to search for traveling wave

steady-state solutions of the same form, that is,
(133(7) = Aej‘f’oeflm (2.25)

for each i E N. Equation (2.25) assumes that each sector responds with the same
amplitude A, but with a constant phase difference relative to its nearest neighbors,
and together all N such solutions form a traveling wave response among the sectors.
(In real form, Equation (2.25) can be written as qu(7’) = A<I>n+1(i — 1 + Ct), where
<I>( - ) is defined by Equation (2.16) and C = no / 99n+1 is the wave speed of the engine
order excitation.) By mapping this trial solution into Equation (2.19) and dividing

through by the common term ejdsiejn"T one obtains
—(na)2A + A + V2 (—Ae"j‘pn+1 + 2.4 — Aej‘pn‘fl) = f, (2.26)

where the identity
$241 - 952' = i30n+1 (227)

has been employed. Upon simplification, the amplitude A is found to be

 

f
A = , 2.28
1+ 2u2<1 — cocoon“) — (no)2 ( )

26

 

from which it follows that a)” +1 = \/1 + 21/Q(1 —— cos 90n+1l is one of the N natural
frequencies of the coupled system, and it corresponds to mode p = n + 1.3

Equation (2.28) highlights a fundamental result when a linear cyclic system with
nearest-neighbor elastic coupling is subjected to engine order excitation of order n:
mode 77. + 1 is excited. The reason for this is not clear from this approach, but it
can be described systematically via a modal analysis that considers the fully coupled

system.

MODAL ANALYSIS

It is well—known that circulant matrices, such as the stiffness matrix defined by Equa—
tion (2.23), can be diagonalized via a similarity transformation involving the Fourier
matrix, and in what follows this property is exploited to fully decouple the governing
equations of motion given by Equation (2.21). The theory is due to P..I. Davis (1979)
and is exhaustively developed in his seminal work, Circulant M atrices [62]. A detailed
development of the pertinent theory is given in Appendix B (and summarized without
proofs in Section 2.2) in a way that should be familiar to the vibrations engineer.
Diagonalization can be achieved by employing Equation (2.9), and in particular

Theorem B7 on page 197. To this end, the change of coordinates4
N
q(’r) = Eu('r) = Z ekuk(7) or qp(T) = eguh), p E N (2.29)
k=1

is introduced, where E is the N x N complex Fourier matrix and ep (given by Equa-
tion (2.4)) is its pth column, (,)T denotes transposition, and u = (u1,u2, . . . ,uN)T
is a vector of modal, or cyclic coordinates. Substituting Equation (2.29) into Equa-

tion (2.21) and multiplying from the left by EH yields

EHEu” + E'HKEu = Eerjm’T. (2.30)

 

3Strictly speaking, the excited mode is p = 71 mod N +1, which will be shown in the next section.
4The index p corresponds to the pm mode of vibration and shall be referred to as the mode

number.

27

Since EHE = I from Equation (2.5) (that is, E is unitary, which is proved in Sec-

tion B42) and in light of Equation (2.9), it follows that

11:1: (I)? 2 0 U1 egf

u (.72 U2 e f .

.2 + 2 , , = 2, aim", (231)
UK] 0 (DIQV UN efif

‘ scalar element of the N x 1 modal forcing vector EHf is eyf. Equa-

where the p”
tion (2.29) is a unitary (similarity) transformation and hence the system natural
frequencies are preserved, which is guaranteed by Theorem A.1 on page 180. For

each p E N they follow from Equation (2.10) and are given implicitly by

2
“Di; 5 (_) 21+ 2V2 — V2w(P—1)+ O + . . . + 0 — u2w(N‘1)(P‘1)
= 1+ 2,,2 _ V2 (we-1) + u,(N-1)(p-1))

= 1+ 2u2(1 — cos app), (2.32)

where w = 82%;: is the primitive N M root of unity (see Section B41) and the identity
"Mp-1) + w<N*l)<p-1) = 2 cos top has been employed. The overbar indicates that the
frequencies are in dimensionless form.
Equation (2.31) is a decoupled set of N, single—DOF harmonically forced modal
oscillators and, in the steady-state, the pth modal response is
2 1.2.3113— eJ'WT, p e N. (2.33)
p — (no)2

The steady state response of sector i (in physical coordinates) can be obtained from

the transformation given by Equation (2.29) and is given by qu(7) = ezTuSSU), or

N 1 )

ss _ j(i—1 99k, ss
q2 (T) —— 2 —_e ”k (7‘)
k=1 N
1 iv: elf-(f j(i—1)(,9k jnor IE N (2 34)
: -—-— e ‘6 , ll -
VN 16:16): — (no)

28

which reflects that the total system response is simply a superposition of individual
modal responses. Depending on the details of the modal forcing terms eyf Equa-
tion (2.34) shows that there are N possible resonances, and these arise if the excitation
frequency matches a system natural frequency. However, only a single mode survives
under an engine order excitation of order n, which is clear by applying Theorem 33

on page 192. Then the pth modal forcing term reduces to

83f = ,Z,(k—1><o—1)fwn<k—1>

N
Z w(k-—1)(n+1-—p)

{Wfl n+1—p=mN
= (2.35)

0, otherwise

(m is an arbitrary integer), which shows that the force vector f is mutually orthogonal
to all but one of the modal vectors ep, that is, only a single mode is excited. Therefore,

given an engine order n E Z+ and since )0 E N, the excited mode is
p = nmod N + 1. (2.36)

Finally, since (i —- 1)<p,-,+1 = (1),, Equation (2.34) can be written as

 

qfs(r) = _2 f( )2 ej‘piejnm, i E N (2.37)
wn+1- n0

which is recognized to be in agreement with the results of the previous section.
Indeed, the process described above is significantly more laborious than the direct
approach of the previous section, but many general features can be gleaned from
the analysis. The eigenfrequency characteristics are described next, followed by a
description of the normal modes in Section 2.4.4, and the system resonance structure

is detailed in Section 2.4.5.

29

2.4.3 Eigenfrequency Characteristics

The dimensionless natural frequencies follow from Equation (2.32) and are given by

 

top = 3% = (/1+ 2u2(1— cosep), p e N (2.38)

which clearly exhibit the effect of the coupling. For I/ = 0 we recover (DP = 1, or
(up = wo in dimensional form, which was used to nondimensionalize the model in
Section 2.4.1. In this case the sectors are dynamically isolated and each has the same
natural frequency. For nonzero coupling (1/ ¢ 0) it is clear that there will be repeated
natural frequencies, a. degeneracy that is due to the circulant structure of K. This is

manifested in the cyclic term
cos (op = cos(%—(%fl) = Re (tug/.1) , (2.39)

which can be obtained by projecting the powers of the N M roots of unity onto the real
axis. (See Figure B1 on page 191.) Multiplicity of the eigenfrequencies can also be
visualized in Figure 2.5, which shows the dimensionless natural frequencies in terms of
the diametral components (that is, the number of nodal diameters) in their attendant
mode shapes versus the mode number p for weak and strong coupling and for odd
and even N. These cyclic features are now described in terms of mode numbers in
the sets Pgi‘fiilv’ 793%,, and "Pg“? which are defined in Table 2.2. A description of
the BTW, FTW, and SW designations of these sets is deferred to the next section.
The natural frequency corresponding to mode p = 1 E 773,5: (zero harmonic of
Equation (2.39)) is distinct, but the remaining natural frequencies appear in repeated
pairs, except for the case of even N, in which case the p = (N +2) / 2 E ’Pgw frequency
(N / 2 harmonic) is also distinct. There are (N — 1) / 2 such pairs if N is odd, and these
correspond to mode numbers in "PST“, and PEQTWv respectively. For even N there are
(N — 2) / 2 repeated natural frequencies corresponding to mode numbers in ”PETW and
PETW. Finally, if k E 731%“le then the mode number of the corresponding repeated

eigenfrequency is N + 2 — k E Praia.

30

 

 

 

 

 

 

 

 

 

 

‘l
C
a P
(a) N=11 E 3-
“5
(Odd) E 2.
cs 1.
4 0’
A A A A A 4A A A A A A ; p
1 2 3 4 5 6 7 8 9 10 11
sw l————BTW————i l—-———FTW—————1
I p—l n.d.—«I l——N—p+1 n.d.——-—|
l I ' ' ' I l T ' ' I
1 2 000 N+1N+3 000 N
2 2
5’19
ll
’1‘ 5’
2 4»
(b) N=10 2 3»
“‘5
Even o.
( ) 3 2r
,5
a4 1»
0r
4 ‘ ‘ 19

SW l-——-— BTW——-i SW l-— FTW ——-l

 

 

 

 

+————p—l n.d. ,4 N—p+1n.d.—1

I I ' ' I l l T ' l

1 2 0.0 fl N+2 N+4 000 N
2 2 2

Figure 2.5. Dimensionless natural frequencies (2,, in terms of the number of nodal diameters
(n.d.) versus mode number p for weak coupling (WC) and strong coupling (SC): (a) N = 11
(odd) and (b) N = 10 (even). Also indicated below each figure is, for general N, the number
of n.d. at each value of p and also the mode numbers corresponding to standing waves (SW),
backward traveling waves (BTW), and forward traveling waves (FTW).

31

Table 2.2. Sets of mode numbers 19 E N = 733,33 U P03 U 195,13, corresponding to standing

BTW

wave (SW), backward traveling wave (BTW), and forward traveling wave (FTW) normal
modes of free vibration for (a) odd N and (b) even N.

 

 

(a) OddN (b) EvenN
pgTWL—{pez+z2$n£flgfl} P§T\V={pEZ+12SnSZ¥'}
og,wz{pez+;fl,+—3gc_<.zv} P§1~ocv={pEZ+=-2-N+4S"SN}

 

The normal modes of vibration are described next, and it will be shown that each

can be categorized as a SW, BTW, or FTW.

2.4.4 Normal Modes of Vibration

In Section 2.4.2 it was shown that Equation (2.21) can be decoupled via a unitary
(similarity) transformation involving the Fourier matrix E 2 [e1, e2, . . . ,eN] and as
a consequence ep is the pth normal mode of vibration corresponding to the natural
frequency 52,). In what follows these mode shapes are characterized by investigating
the free response of the sysem, and it is shown that they are of the SW, BTW, or
FTW variety.

The free response of the system in its pt" mode of vibration can be described by
q(p)(r) = apepej‘I’PT, where ap is a modal amplitude and the natural frequency Op
is defined by Equation (2.38). (There will generally be a phase angle as well, which
is omitted since its presence does not affect the arguments to follow.) Noting that
element i of ep can be written as won—IMF” = ej‘ppu’ll and for i,p E N the free

response of sector i can be written (in real form) as

(1,") l (T) = ap COS (9912(1' - 1) + W). (2.408.)

where Cp 2 (DP/(op and the function (Dk(x) is defined by Equation (2.16). Equa-

32

 

 

 

 

 

 

‘ 7.07; [0’1 >1
I I
up"
0..
_a.
P £0127-
; L .l
1 000 N1 0.. N1

Sector Number i

Figure 2.6. A backward traveling wave ap<I>p(i — 1 + Cpr) 2 ap cos((pp(i — 1) + (DPT) with
amplitude ap, wavelength 277/99,, = N/(p — 1), and speed C1,, = ‘p/(op.

tion (2.40) is a function of continuous time T and it is discretized according to the
sector number i. In this way, it is endowed with the same discrete temporal and con—
tinuous spatial duality that was described in Section 2.3.2 in the context of traveling
wave dynamic loading (engine order excitation). That is, it can be regarded as (l)
the time-response of individual (discrete) sectors, or (2) a continuous spacial variation
of displacements among the sectors that evolves with increasing time (i.e., a travel-
ing wave). The propagating waveform is strictly a BTW in the negative i-direction
(descending sector number) with wavelength 27r/cpp = N / (p — 1) and speed Cp, an il-
lustration of which is shown in Figure 2.6. However, depending on the value of p, this
gives rise to SW, BTW, or FTW mode shapes, a property that follows analogously
from the features described in Figure 2.3.

For the special case of p = 1 it is clear from Equation (2.40a) that each sector
behaves identically with the same amplitude and the same phase since (p1 = 0. An
additional special case occurs when p = (N + 2) / 2 if N is even. Then (0( N +2) /2 = 7r
and each sector has the same amplitude but adjacent sectors oscillate with a 180-
degree phase difference. Hence the vibration modes p E 738“? correspond to SW

mode shapes whose characteristics can be visualized in Figure 2.3b and Figure 2.3d

33

by replacing the amplitude F with op. The remaining mode shapes correspond to
repeated natural frequencies and are either BTWs or FTWs. In particular, the normal
modes p E nga (resp. p E qu‘gw are backward (resp. forward) traveling waves
and can be visualized in Figure 2.3a (resp. Figure 2.3c). If mode k E 773%, is a
BTW corresponding to a natural frequency 0k, then the attendant FTW mode is
N + 2 — k E PFQ'fEVJV corresponding to QN+2—k = ok.

Figure 2.7 illustrates the normal modes of free vibration for a model with N = 100
sectors. In this figure, the extent of the radial lines represents sector displacements;
those appearing outside (resp. inside) the hub are to be interpreted as being positively
(resp. negatively) displaced relative to their zero positions. Modes 1 and 51 are SWs
and modes 2—50 (resp. 52—100) consist of backward (resp. forward) traveling waves.
Finally, the number of nodal diameters can be clearly identified in Figure 2.7. For

example, modes 4 and 98 feature 3 n.d.

2.4.5 Resonance Structure

In general, there may be a system resonance whenever the excitation frequency
matches a natural frequency, that is, if no 2 (DP or n52 = wp in dimensional form.
These possible resonances can be conveniently identified in a Campbell diagram, an
example of which is shown in Figure 2.8a for a model with N = 10, V = 0.5, and
for engine orders n E N. (The general case of n 6 2+ is considered below.) In this
figure, the natural frequencies are plotted in terms of the dimensionless rotor speed
and several engine order lines no are superimposed. Possible resonances correspond
to intersections of the order lines and eigenfrequency loci, and there are (N + 2) / 2
(resp. (N + 1) / 2) such possibilities for each engine order when N is odd (resp. even).
In light of Equation (2.35), however, there is only a single resonance associated with
each n under the traveling wave dynamic loading of Section 2.3 and it corresponds

to the mode number given by Equation (2.36). These are indicated by black dots in

34

       

\\\“)‘)‘llllllllli(o,

\\

 

 

Figure 2.7. Normal modes of free vibration for a model with N = 100 sectors. Mode 1
consists of a SW, in which each sector oscillates with the same amplitude and phase. Mode

51 also corresponds to a SW, but neighboring oscillators oscillate exactly 180 degrees out
of phase. Modes 2-50 (resp. 52—100) consist of BTWs (resp. FTWs).

35

(1) (8)(7)(6 (5 (p=3)

 

 

 

 

 

 

 

Lap 7 6 5) 2 e.o.
2 , /

j/: / n.d. p co.

5 6 5
1.5 . Ii/J/r ”I 4 5’7 4’6
f/ - , 3 4,8 3,7
rL/[F // 1 2 3,9 2,8
U/ I 1 2,10 1,9

0 1 10

 

 

 

 

 

 

 

A A A A A A 0’
0.2 0.4 0.6 0.8 1 1.2
('5)
longl
2r
1 .
0
_. jlllllb
"
—2
A A L l 0'
0.2 0.4 0.6 0.8 1 1.2
(b)

Figure 2.8. (a) Campbell diagram and (b) corresponding frequency response curves for
N: 10, u=0.5, f =0.01, and n=1,...,N.

Figure 2.8a and the corresponding frequency response curves are shown in Figure 2.8b
for a model with f = 0.01. For example, a 3 e.o. (resp. 7 e.o.) excitation gives rise to
a resonance of the 4th (resp. 8th) mode, which is a BTW (resp. FTW) with 3 nodal
diameters. (The TW and n.d. designations can be verified in Figure 2.5.)

The basic resonance structure shown in Figure 2.8a for n = 1, . . . , N essentially
aliases relative to the total number of sectors, in the sense that the excited modes for

n =mN+1,...,(m+1)N withm E Z+ arethe same asthoseforn=1,...,N. This

36

Table 2.3. Condition on the engine order n E Z+ to excite mode p E N for N = 10.

 

 

 

Excited Mode Conditions on Engine Order n
1 mN = 10, 20, 30,...
2 1+mN=1,11,21,...
3 2+mN=2,12,22,...
N—l N—2+mN=8,18,28,...
N N—1+m.N=9,19,29,...

 

 

 

follows from the orthogonality condition given by Equation (2.35) and is manifested in
Equation (2.36), which gives a relationship for the excited mode in terms of the engine
order n and total number of sectors N. Since n > 0 by assumption (see Section 2.3)
the first mode (p = 1) is excited when n = mN = 10, 20, 30, . .. , the second mode
(p = 2) is excited when n = 1+mN = 1, 11,21, . . . , and so on. Table 2.3 summarizes
these conditions for a model with N = 10 sectors and the corresponding resonance
structure for n = N, . . . , 20N is shown in Figure 2.9a. Each collection of resonance
points n = mN +1, . . . , (m + 1)N is qualitatively the same in structure. However, for
m > 1 the resonances become increasingly clustered, which is shown in Figure 2.9b for
n = N, . . . , 2N and in Figure 2.9c for n = 2N, . . . ,3N. In terms of the sets defined
in Table 2.1 and Table 2.2, an engine order nmodN E NSWE excites a SW mode
p 6 773(5). Similarly, an engine order nmodN E Nngv (resp. nmod N .E NSTEV)

excites a FTW (resp. BTW) mode p E nga (resp. p E nga).

2.5 Vibration Absorbers

2.5.1 Introduction

When an engineering structure experiences unwanted levels of vibration due to peri-

odic excitations acting on its constituent parts it may be impractical (or even impos-

37

(I) 20N10N 5N 4N 3N e.o. 2N e.o.
p /

 

2 - ,.

1.5 P . A Z:
.........~w5i- ‘ —— 3
”(unlafi’m
muuf‘fl'w- x 2

 

  

0.5 *

 

 

0.02 0.04 0.06 0.08 0.1 0.12

 

 

_2. I
0.02 0.04 0.06 0.08 0.1 0.12
(5)
longl
2.

 

 

 

(C)

Figure 2.9. (a) Campbell diagram for N = 10, u = 0.5, f = 0.01, and n = 1,. . .,20N and
the corresponding frequency response curves for (b) n = N, . . . ,2N and (c) n = 2N , . . . , 3N.

38

sible) to change the makeup of the system to improve its vibratory characteristics, or
to change or eliminate the source of the excitation. In these cases tuned vibration
absorbers offer a possible solution.

The notion of a vibration absorber was introduced by Frahm [4] in a United States
patent in 1911, but it was Den Hartog [5, 6] who first carried out systematic studies
on tuned absorbers, including an optimal choice of parameters. Tuned vibration
absorbers are auxiliary components that are attached to a primary system to eliminate,
or otherwise reduce its steady-state motions. This is done through a particular choice
of absorber parameters, typically by setting the natural frequency of the absorber
close to the most problematic harmonic of the excitation. The absorber is said to be
exactly tuned if these frequencies match identically; otherwise, the absorber is said to
be detuned.

We shall discuss two varieties of vibration absorbers: the classical frequency-tuned
dynamic vibration absorber (DVA) in Section 2.5.2 and the order-tuned centrifugal
pendulum vibration absorber (CPVA) in Section 2.5.3. The DVA, which is shown in
Figure 2.10a, relies on an elastic element for its restoring force, whereas the CPVA,
examples of which are shown in Figure 2.10b (circular path) and Figure 2.10c (general
path), employ the centrifugal field due to rotation of the primary system. Order-tuned

absorbers play a key role in this dissertation.

2.5.2 The Frequency-Tuned Dynamic Vibration Absorber

Here we highlight the classical theory of the frequency-tuned DVA due to Den Hartog
[6]. Consider the 2——DOF system shown in Figure 2.10a. It consists of a primary
system (M, C, K) that is harmonically excited by f (t) = foeth, where f0 is the
strength of the excitation and w is its (constant) frequency, t is time, and j = \/——_1.
When this primary system is isolated (i.e., when the absorber is not attached) it has

a resonance at em; = MK/M (its undamped natural frequency), which is indicated

39

{4:4 74%;; or; «

0130

t
a}.
Primary .
M Tit) 1202;;
SYstem ' 1 / ’3 .m R(S)
Wt, c0) - o
DVA m L—] ‘ J
W) J ‘/9. 9 fl

(a) DVA (b) Circular Path CPVA (c) General Path CPVA

Figure 2.10. Tuned vibration absorbers: (a) DVA; (b) Circular Path CPVA; (c) General
Path CPVA.

 

 

 

 

 

 

 

 

 

 

by the dashed lines in Figure 2.11. A DVA subsystem (m, 0,10) is attached to M and
its parameters are chosen to attenuate the vibratory response of the primary system
near to = 021V. The undamped natural frequency of the isolated DVA is denoted by
can 2 k / m.
The governing equations of motion for the composite system in Figure 2.10a are
given by
M5: + 05: + Kx = fejw‘, (2.41)

where x = (51:,y)T and f 2 (f0, 0)T are displacement and forcing vectors and

K+k —k
a K_[_k k],

MO

0 m

C+c —c
—c c

M: , C:

 

 

 

 

are the mass, damping, and stiffness matrices. Assuming harmonic motion, the steady-

state solution to Equation (2.41) follows in the usual way and, for non-resonant forcing,

 

 

 

 

is given by
[:3 = if
where
X 2 f0 (k — ma)2 +jwc)
P (2.43)

 

are the steady-state amplitudes of the primary and absorber systems, respectively,

and
F = (K + k — ma)2 + j(C + c)w) (k — mw2 + jcw) — (k + jcw)2. (2.44)

We first consider a tuning strategy for the undamped DVA, that is, for c = 0. It will
be shown that the undamped absorber can be designed to completely eliminate the
steady—state motions of M ——independent of C—but that the resulting design is not
robust to frequency drift. It is then shown how this situation can be improved by

including damping in the absorber model.

ABSORBER TUNING: UNDAMPED DVA

In the absence of absorber damping (c = 0), the condition 022 = k/ m is sufficient to
eliminate steady-state motions of M, which is clear from the first element of Equa-
tion (2.43). Since we are interested in improving the response of the primary system

near its resonance to ”E to N = «K / M , the absorber should be designed such that

K k
022 = “12V = 423, or 012 = M = a, (2.45)

which is the absorber tuning. That is, the absorber is designed such that its natural
frequency matches that of the isolated primary system. This is accomplished by
choosing the absorber parameters m and k such that Equation (2.45) is satisfied.
If the undamped absorber is tuned according to Equation (2.45), then
X = 0 .
(2.46)
Y = “fa/k }
(independent of C) and the steady—state motions of the primary system are completely
eliminated. Under this tuning strategy the absorber oscillates out of phase with
respect to the excitation, which is clear from the sign of Y, and it exerts at all times
a force equal in magnitude to the applied force f (t). The resulting frequency response

of the primary system is given by the solid lines in Figure 2.11.

41

 

 

6' ' ' H
m / A4 = 0.25 I |
W12V = 00%; l l -
‘ — — Without Absorber
4 \ —— With Absorber
IQ:
f0 ‘

 

 

 

   

 

 

0.5 1 1.5 2
w/wN

Figure 2.11. Dimensionless amplitude of the primary system versus dimensionless exci-
tation frequency: effect of an undamped vibration absorber (c = 0) on the response of the
undamped primary system (C = 0).

From a design standpoint, there are two different ways to acheive the absorber
tuning:

1. If the allowable amplitude Y is prescribed, set 10 = |—fo/Y| = fo/ Y (from
Eq. (2.46)). Then the required absorber mass follows from Eq. (2.45) and is

given by m = k/w?V = (%)M.

2. If the absorber mass m is prescribed, set k = ma)?V = (77%)K (from Eq. (2.45)).
Then the resulting absorber amplitude is Y and is given by Eq. (2.46).

In the first strategy the absorber is assumed to have a limited space in which to
operate and the designer is not at liberty to arbitrarily choose its mass. Conversely,
if the absorber mass is specified (typically it is made as small as possible), then the

corresponding absorber amplitude is automatically prescribed.

ABSORBER TUNING: DAMPED DVA

The undamped DVA just described is successful in that it removes the original res-
onance peak in the response of the primary system, but it does so at the expense

of an additional resonance, which is shown in Figure 2.11. This may be undesirable

42

  

I
m = l
/M 025 H], l l —— wNzk/m,c=0
4 l l l l l l — - — Optimum Tuning, c = 0
1% I I l . — Optimum Tuning & damping

 

 

 

 

0.5 1 1.5 2

Figure 2.12. Dimensionless amplitude of the primary system versus dimensionless exci-
tation frequency: effect of a damped vibration absorber (c # 0) on the response of the
undamped primary system (C = 0).

for a machine that must pass through the first resonance in order to reach its steady-
state operating speed, and also if it exhibits significant frequency drift under normal
operation. The composite system can be made robust to these situations by includ-
ing damping in the absorber model. The basic idea is to first optimally detune the
absorber away from the perfect tuning given by Equation (2.45), which assures that
the two points shared by all frequency response curves of the composite system have
the same ordinate value. Then the absorber damping is adjusted to optimize the
two resonance peaks. Den Hartog [6] was the first to propose such an optimization
scheme, and the reader is referred to his work for details.5 See [67—69] for additional
work related to the optimum design of damped vibration absorbers. An example plot

of an optimally tuned system with a damped DVA is shown in Figure 2.12. ‘

 

5The reader should note that Equation (3.24) on page 96 of [6] has a typographical error. The
damping terms should be (2(c/cc)g)2, not (2(c/cc)g f )2. This equation is correctly reported in [66].

43

2.5.3 The Order-Tuned Centrifugal Pendulum Vibration Absorber

The classical frequency-tuned DVA of Section 2.5.2 is effective only at a particular
frequency and it works well for systems with steady operating speeds. However,
absorbers of this type are not suitable for many systems with rotating assemblies,
such as an automobile or jet engine, which are characterized by varying speeds and
forces that occur at orders of rotation [7]. Here we briefly highlight the essential
features of a centrifugal pendulum vibration absorber (CPVA) and indicate how it
can be tuned to a given order of rotation, rather than to a fixed frequency, and is
hence effective at all speeds.

The system shown in Figure 2.10b on page 40 captures the essential features of
a typical CPVA. It consists of a rigid rotor (primary system) with polar moment of
inertia J and radius R, which rotates about a fixed axis at O. The primary system
is harmonically excited by a torque of the form T(t) = To + Tejnm, where T is the
strength of the fluctuating excitation, n is its order, t is time, Q is the speed of the
rotor, and To is the mean torque. A pendulum absorber of length r and mass m is
attached to the periphery of the rotor and its parameters are chosen to reduce the
torsional oscillations of the primary system.

An analysis similar to the one carried out in Section 2.5.2 for the frequency-tuned
DVA shows that the steady-state torsional oscillations of the rotor can be eliminated
completely by setting the undamped natural frequency of the absorber to that of
the excitation. A key difference, however, is that the absorber’s natural frequency is

proportional to the rotor speed [70]. That is,

Mn: fig
7'

where a = VR/r is defined as the linear tuning order of the absorber. This gives

an, (2.47)

rise to a tuning condition n = n (the so-called order-tuning) which is independent of

the rotor speed. In this way, the absorber tuning is effective over the full range of

44

possible rotor speeds.

Similar statements can be made for the more general, arbitrary-path absorber
system shown in Figure 2.10c, which has been exhaustively studied by Shaw and
coworkers. The reader is referred to a wealth of existing literature for the theoretical

details [14, 18, 20—23, 25, 27, 71, 72] and experimental validation [28—30, 73—75].

2.6 Concluding Remarks

An overview of relevant theoretical background has been given, including some mathe-
matical preliminaries, a detailed account of engine order excitation and its application
to a prototypical cyclic system, and on the basic operation and features of frequency-
and order-tuned vibration absorbers.

Key mathematical concepts were briefly stated in Section 2.2, including the Kro-
necker product, the Fourier matrix, circulant matrices, and the diagonalization of
Circulants. This was done in a way that elicits ease of reference; a much more de-
tailed account of the required mathematical machinery is given in Appendix A and
Appendix B, including many of the proofs.

A mathematical model for engine order excitation was developed in Section 2.3.
It was described in terms of a discrete temporal variation of dynamic loading applied
to individual blades as well as a continuous spatial variation of the excitation strength
relative to the rotating hub, and the former was subsequently categorized as a BTW,
SW, or F TW. The essence of these two interpretations of the excitation is captured
in Figure 2.2.

A detailed account of the vibration characteristics of a generic linear cyclic system,
with nearest-neighbor elastic coupling, under engine order excitation was given in
Section 2.4, including a description of its rich eigenfrequency and resonance structures
for large engine orders. The essence of these features are captured in Figure 2.8 and

Figure 2.9.

45

Finally, the basic theory of frequency-tuned dynamic vibration absorbers was
highlighted in Section 2.5 and this was compared to that of order-tuned absorbers. A
key feature in the latter is that the absorbers are tuned to a given order of rotation,
rather than to a fixed frequency.

Mathematical models are developed next for arbitrary-path, order-tuned absorbers

fitted to a bladed disk assembly under engine order excitation.

46

CHAPTER 3

Mathematical Models

3. 1 Introduction

A general mathematical model is developed for the bladed disk assemblies of interest
fitted with arbitrary—path, order-tuned vibration absorbers, from which a number of
specific models to be considered in subsequent chapters are distilled. The chapter
begins with a brief overview of typical modeling approaches in Section 3.2, includ-
ing motivation for the simplified lumped-parameter blade models to be employed.
The geometry of an arbitrary absorber path is then described in Section 3.3, which
forms a kinematic model for a general-path absorber, and the desired model for a
nominally-cyclic bladed disk assembly fitted with such absorbers is formulated in Sec-
tion 3.4. This general model forms the basis for all of the analysis to follow, and a
number of simplifications and reductions are carried out to put it in a more tractable
form. Specifically, the equations of motion for the case of circular-path absorbers
with motion-limiting stops are derived in Section 3.5, and these are subsequently lin-
earized for small blade and absorber motions and cast into a form that is employed
in Chapter 4. Reduction of the full nonlinear equations of motion via scaling and av-
eraging is deferred to Chapter 5, where the nonlinear system dynamics are estimated
for the case of a single isolated blade/ absorber and for the cyclically-coupled system

with identical sectors. In these nonlinear models a specific two—parameter family of

47

paths is employed, which is derived in Section 3.4.4. Finally, blade and absorber
damping levels are estimated in Section 3.6 and a summary of the models is given in

Section 3.7.

3.2 Modeling Approach

3.2.1 Absorber and Blade Models

For the applications of interest an absorber realization may consist of a spherical mass
that rolls on a machined path (or more generally a surface) relative to the primary
system, in this case a blade [76]. By ignoring the effects of rotational inertia, it can
be modeled by a mass particle that translates along a prescribed curvilinear path
relative to the blade. (Realistic absorber masses are very small relative to those of
the blades due to strict limitations on rattling space.) If in addition the absorber path
is circular, then it can be modeled by a simple pendulum attached to the primary
structure.

Accurate modeling of the bladed disk assemblies of interest, a typical represen-
tation of which is shown in Figure 3.1a, can be significantly more complicated due
to their complex geometries. The blades are characterized by significant transverse
curvature (camber), in addition to variations in thickness, width, and curvature along
their chordwise lengths. They are generally attached at their roots to the periphery of
a circular disk by means of, for example, a dovetail joint and the composite assembly
forms one stage in a turbomachine [77]. As our understanding of the dynamics of
such bladed disk assemblies has improved, so too has the level of sophistication of the
attendant modeling and analysis techniques. Typical models generally fall into three
basic categories and they are briefly presented below in order of increasing complexity,
both in terms of the dynamic phenomena that they are able to capture and in the

corresponding analysis.

48

Blade

.4
. //'I
III)??? \
v I 'to

.0 ~ §
1'17"“
llllllfi
I, ..

Q‘
~
IԤ

   

Figure 3.1. (a) Finite element model of a bladed disk assembly (reproduced with per-
mission from [3]); (b) General cyclic system with N identical cells and nearest-neighbor
coupling.

The first and undisputedly simplest models assume lumped parameters and essen-
tially consist of a cyclic chain of nominally identical oscillators [36, 47, 58, 78—80]. (See
Figure 3.1b.) The analysis involved with such models is relatively simple, especially
if one assumes that each sector is identical, i.e., that the structure is perfectly cyclic.1
Then the fully coupled system can be reduced to a set of reduced-order models via a
transformation based on its cyclic symmetry, similarly to the way it was done in Sec-
tion 2.4.2. This often offers significant insight into the overall system behavior, even
if a very small number of DOF is employed. While these models are very attractive
due to their simplicity and are able capture some very rich dynamics of cyclically-
symmetric structures, they do have serious limitations. Clearly, one cannot expect
to capture all of the complicated mode shapes of the actual system, such as plate-
or shell-type modes of the hub. Moreover, parameter identification can be extremely

difficult if it is desired to use such models to predict specific behavior in an actual

 

1Such a structure is said to be perfectly tuned. When there are small differences among the
sectors, due to material tolerances, in-service wear, and so on, the structure is said to be mistuned.
This designation is not to be confused with intentional under- or over-tuning of the absorbers, which
is referred to as detuning. (See Section 4.4.)

49

structural system [36].

The second type of models employ distributed—parameter elements and are there-
fore able to satisfactorily capture more complicated normal modes, including those
that involve flexure of the rotating hub, but at the expense of a significantly more
involved analysis [9,81,82]. For example, the transverse blade vibrations may be
captured by cantilevered beams, provided that the disk is sufficiently stiff and the
blades are sufficiently long relative to the hub dimensions (large aspect ratio). Clas-
sical beam theory breaks down for smaller aspect ratios but can be replaced by a
more general shell theory. Again, while such an approach offers higher-fidelity mod—
eling, it is accompanied by more difficult and expensive analyses by means of, for
example, Rayleigh/Ritz/Galerkin methods, variational techniques, transfer matrices,
finite elements, and numerical methods [77].

Finally, the third modeling approach involves a full finite element representation
of a real bladed disk assembly [3,83—86], such as the compressor stage shown in
Figure 3.1a. A finite element model is typically generated for only one sector and, as-
suming that all sectors are identical, cyclic symmetry can be used to calculate the free
and forced response much more efficiently than by modeling the entire system. While
this approach offers the highest-fidelity modeling, the computation involved can be
prohibitively high. This is especially true for large industrial stages with complicated
geometry or if mistuning is included, which causes possibly drastic changes in the
dynamics due to a disruption of the cyclic symmetry. Many realizations of randomly
mistuned rotors must be run in order to accurately assess the full statistics of the
blade response. This is generally not feasible due to the size of an industrial finite
element model of a bladed disk, which can run into the millions of DOF. However,
advanced reduced-order modeling techniques have been developed by Pierre et al. and

others since the 19903 [85-89], which offer a more tractable finite element analysis.

50

3.2.2 Motivation for Lumped-Parameter Blade Models

The aim of this dissertation is to investigate the performance of centrifugally-driven,
order-tuned absorbers to attenuate vibrations in cyclic rotating flexible structures sub-
jected to traveling-wave dynamic loading. These two classes of systems are special
cases of systems of vibration absorbers and nominally cyclic systems, respectively,
both of which have been extensively studied in many contexts and, taken individu-
ally, each forms a very mature body of research. However, at the time of writing
there have been no systematic analytical treatments of vibration absorbers applied
to nominally cyclic systems, and it is thus appropriate to begin such an effort with
simplified, lumped-parameter models. In what follows, a number of such models are
developed, the analysis of which forms the remainder of this thesis. The applications
of interest are rotating flexible structures—bladed disk assemblies in particular—and
the work is carried out in this context, though the methodology and results should
also be applicable to address vibration issues in other systems with nominal cyclicity.

A kinematic model for the absorbers is developed first by quantifying the geometry

of an arbitrary path.

3.3 The Geometry of an Arbitrary Absorber Path

This section describes in general terms the geometry of the absorber paths, which
prescribe their positions relative to the primary systems, or blades. Key results are
relationships among the path variables, which are described in Section 3.3.1 and an
expression for the tangent angle of each path, which is given in Section 3.3.2. These
are employed in Section 3.4, where the equations of motion for a bladed disk assembly
fitted with general-path absorbers are derived.

Consider the ith general path shown in Figure 3.2. It can be described relative

to a basepoint O by the radius vector R,(S,-) = Ri(S,-)é{f, where S,- is the arc length

51

along the path relative to an origin at its vertex V. The angle subtended by S,- is
denoted by 19,(S,-) and the distance from the basepoint to the path vertex is given
by R,(0) = R0, = aiL, + d,, where a, and L,- are defined in Section 3.4 and d,- is
the local radius of curvature at V. Physically, the absorbers may have rattling space
limits, which are denoted by 3:50,, and in such cases their displacements are restricted
such that [5,] S 30,-. The path could also be described in terms of the length p, and
angle 2b,. A circular path is obtained by restricting p,- = d,- = constant, which is
indicated by the dashed lines in Figure 3.2. Finally, the paths are generally assumed
to be symmetric about their vertices at S,- = 0 implying that R,(S,-) = R,(—S,-), i.e.,
that each R,(S,~) is an even function in 3,. However, this assumption is formally
introduced in later developments when specific paths are chosen for the analysis. In
what follows, there are no assumptions built into the absorber paths other than their
gross placement relative to the blades.

Relationships between the path variables 12,-, Si, and 29,; are derived next and

R

the angle q,- between the radius and tangent unit vectors éz- and éiS is subsequently

quantified. We draw liberally from the development in [90].

3.3.1 Fundamental Relationship Between Path Variables

Let P be anarbitrary position along the general path corresponding to Bi, 5,, 19, > 0
and let Q correspond to R,- + 6R,-, S,- + 68,-, and 6',- + 619,-, where 6B,, 63,-, 619, > 0 are
small additional increments. Then relationships between the path variables can be
obtained by considering the triangle PQM in Figure 3.2. If 60,; = PQ, it follows from
(P0)2 = (PM)2 + (MQ)2 that

5c,- 2 sin6i9, 2 3R, , sin(6i9,j/2) 2
— = .— — x as- 2 —— ,
(5.9,) (R1 619.- l +(w. ”2‘8““ ‘/ ) coy/2

52

ea

VI“? = s,

56 = 68,-

PQ = 6c,-

OV = R0,-

OP = R,-

oo = R,- + 6R,
ON = or,-L,-
NV=d
NP=m

N

 

 

 

cs
. 7r/2 C
Q ’ ~19
I M
R,+6R,-

General Path

- - - - Circular Path
(Radius di)

 

Figure 3.2. Geometry of a general absorber path.

where we have divided through by (619,-)2. Then
a 2- L54:
2549,- ‘ 5c,- 30,-
_ as, 2 sin6i9, 2 5R,- . sin(6i9,t/2) 2
— (o—c.) [(2747) +(oo;+28m<52/2>—ot/2—) -

In the limit as Q ——> P (that is, 619,- —> 0) it follows that

(3%)2 = (1)2[<R.<1>>2 + (2?: + R.<0><1))2] .

 

01‘

(48.)2 = (It-c4932 + (4302. (31)

Equation (3.1) gives the fundamental relationship between the path variables 8,, 12,-,

and 19,. By dividing through by (di9,-)2 and (dS',-)2 the equivalent expressions

 

 

dS,-_ 2 dB, 2
di9,-—\/Ri+(di9i)’ (3.2a)
._ as, _ an, 2

easily follow, which are employed in the next section and elsewhere in the thesis. In
light of their frequent appearance in the subsequent analysis, the expressions appear-
ing in Equation (3.2b) are denoted by I}. Then the angle subtended by S,- is given

by the integral

 

_ SzTAX)
l9i(Sil —-/O Ri(x)dx' (33)

Next the angle between the unit vectors éZR and 6? is quantified.

3.3.2 Angle Between Radius Vector and Tangent

In Figure 3.2 consider the ratio

 

 

' 5:9-
tan/3- _ m _ R,sin 50,- __ Rina—ism , (3 4)
1 MQ 512,- + 2n,- sin2(6i9,-/2) gig; + 2 R, Sin(5192' /2)sin5(§§/i2/22 ’

54

where MP = R, sin 619, has been employed and also

MQ = OQ — OM
= R, +6R, — R,cos619,
= 6R, + R,(1— cos 619,)

2 6R, + 2R, 511120519,- /2).

In the limit as Q —-> P (that is, 619, —-—> 0), [3, —+ q,- and it follows from Equation (3.4)

that

tan c,- = 53in] 0 tan [3,-

z-—)

2 am
3% a<o><1>

[(129,-

28%;:

 

= R (3.5)

which is an expression for q, in terms of R, and 19,. Expressions for sin c,- and cos c,,

which will be needed in subsequent sections, can be obtained as follows. Consider

d8,- dSi 61191 2 dB,- 2 (119,.
= —_ Z ' '— f E . 3.2
dR’l (119, dB, \/R2 + ((1191; dRi ( rom C111 ( 3.))

_ 2 fl 2 ($1.21 dfii )2
“ [ESE-l + co.- 4R.-
dd,- 2
z \fl+ (Rica)
= ‘/1 + tanz g,- (from Eqn. (3.5))

= secq, = l/cosc,.

 

 

 

 

 

 

 

 

From this result, together with Equation (3.5), it follows that

d19,

0119,1113,- _ ._
245,?

isms—R

sin c, = tan q, cos c, = R

55

To summarize, and in light of Equation (3.2b),

 

. d29- dR, 2
r. = . -= -—2 = 1— .
, SIIIC, dS, (dS,) , (3 7a)
dR,

cos c, — —dS,-’ (3.7b)

d19-
tan c, = Ric—ifi’ (3.7c)

2

which relate the path variables R,, 8,, and 19, to the angle q, between the unit vectors

éiR and 3,5.
We now turn to a derivation of the governing equations of motion for a bladed
disk assembly fitted with general-path absorbers, from which a number of specific

models are distilled.

3.4 Bladed Disk Assembly Fitted with General-Path Ab-

sorbers

In this section an idealized mathematical model of a bladed disk assembly under en-
gine order excitation is systematically developed. Each blade on the rotating assembly
is fitted with a centrifugally-driven, order-tuned vibration absorber and the governing
nonlinear equations of motion for the overall coupled system are derived. The anal-
ysis to follow in subsequent chapters is carried out for the case of perfect symmetry
among the sectors, implying identical blade and absorber models, and for a specific
family of absorber paths. However, the equations of motion are derived for the gen—
eral case of nominal cyclicity and arbitrary-path absorbers, a model that is amenable
to ongoing work on, for example, the effects of parameter mistuning and it allows
for the eventual investigation of various path geometries. The model is described in
Section 3.4.1, followed by the development of the system kinetic and potential energy
in Section 3.4.2. The general nonlinear equations of motion are subsequently derived

in Section 3.4.3 by employing the method of Lagrange and a particular two-parameter

56

i+1

 

Figure 3.3. Lumped parameter model of a rotating bladed disk assembly.

family of paths, which are employed in Chapter 5, is described in Section 3.4.4.

3.4.1 Lumped-Parameter Model

An idealized, lumped-parameter model of a rotating bladed disk assembly is shown
schematically in Figure 3.3. It consists of a nominally-cyclic array of N blades, and
each is modeled by a simple pendulum of length L,- and mass M,. These are uniformly
attached around the periphery of a rigid disk of radius H, which rotates at a constant
speed 9 about a fixed axis through C. The single-mode fiexural stiffness of blade i
(the ith primary system) is modeled with a linear torsional spring of stiffness kg’, and
the elastic inter-blade coupling (due to shrouds, aerodynamic effects, and so on) is
captured by linear springs of stiffness 105. As indicated in the sector model shown
in Figure 3.4, the coupling springs connect adjacent blades at a distance b (radially
along the blade lengths) relative to their attachment points to the rotor. It is assumed
that the springs are unstressed when the blades are in a purely radial configuration,
that is, when 0,- = 0 for each i E N.

The blades are fitted with nominally identical vibration absorbers, which essen-

tially consist of particle masses m,- (typically each m, << M,) riding on user-specified

57

 

Figure 3.4. Sector model of a bladed disk assembly fitted with a general-path absorber.
The mathematical details of the absorber path are given in Section 3.3.

paths. In what follows the equations of motion are derived for arbitrary paths, and
then specific paths are chosen for the analyses in subsequent chapters. Figure 3.4
shows a schematic of the it“ blade fitted with a general-path absorber, which to-
gether with a portion of the rigid disk composes the it“ fundamental sector. If we
require that point O in Figure 3.2 coincides with the attachment point of blade i to
the rotor, and that the unit vectors éz-L and 6? are aligned and rotate with the blade as
shown in Figure 3.4, then the mathematical details for the it“ absorber path are given
in Section 3.3. Loosely speaking, the absorbers are said to be “centered” a distance
a,L,- radially along the blade pendulums. Then oz,- is the dimensionless distance from
the blade base point 0 to the absorber “attachment” or “base” point N.

There are a number of ways to model the system damping, but in light of the
inherently small levels encountered in practice the details are not crucial. (It is ac-
knowledged, however, that modeling and quantifying these details in actual structural
systems and absorber implementations can be quite challenging.) For the purposes
of this study it will suffice to employ simple linear viscous damping models; the blade

and inter-blade damping is captured by linear torsional and translational dampers

58

(not shown in Figure 3.3 or Figure 3.4) with constants c]? and CE, respectively. The
effective translational absorber damping constant is denoted by cf}, where a bar has
been added to the superscript a to distinguish it from the torsional damping constant
of to be employed in the linearized model in Section 3.5. Further comments on damp-
ing are given in Section 3.6, including estimates of realistic damping constants and
an approximate (resp. exact) relationship between c? and cf for the case of general
(resp. circular) paths.

Finally, the primary systems (blades) are harmonically excited in the transverse
sense by engine order excitation of order n and the model described in Section 2.3, that
is, Equation (2.14) is adopted for this purpose. Consideration of all possible engine
orders can be somewhat cumbersome, since one must not only distinguish between
odd and even n in the analysis, but also how certain order-dependent features alias
relative to N [91-94]. These additional complications were revealed in the analysis
of the generic cyclic system in Section 2.4. In order to eliminate some of these
details, and to focus on a particular absorber tuning strategy, the engine order is
restricted such that 0 < n < N throughout the remainder of this work, though it
is possible for n 2 N in practice. This does not qualitatively affect the approach
nor the conclusions.2 The case of noninteger n E R+ is non-physical for bladed disk
assemblies under engine order excitation, but it is of academic interest and may be
possible in other systems. This situation and its implications in cyclic systems is

treated briefly in Appendix C.

 

2The reader should note that the effect of large engine orders, specifically those greater than N,
can be directly inferred from the results of Section 2.4.

59

3.4.2 Kinetic and Potential Energy

The total system kinetic energy is that of the N blades and their attendant absorbers

and is given by

1 N 1 N
T = 5 Z Mpllupn2 + 5 Zmpnvpr. (3.8)
where H - I] denotes the vector norm and

u, =Hné§2+L,(o+é,-)é§ ZEN

v, = He o? + 12,-(0 + ago? + 3,422?
are the absolute velocities of the it“ blade and absorber masses, respectively. The unit
vectors 6? and 6? are mutually orthogonal, as are the vectors é? and (3,5, and these
are defined in Figure 3.2 and Figure 3.3. (See also Figure 3.4.) Physically, H126? is
the velocity of the it“ blade basepoint relative to the hub center C and L,(Q + 6,)é?
(resp. R,(Q + 9,)6? + 3,132? ) is the velocity of the ith blade mass M, (resp. absorber

mass m,) relative to O. The corresponding speeds are given by

|]u,|| =H292+L12(Q+g,‘)2+2L,HQ(Q+9,)COSQ,, 'lEN
“V,” = H202 + 12,2(0 + 9,)2 + 3,2
+ ZHQR, (n + d,)cos(9, + 19,-)

+ 2H (25‘, (1‘,- cos(6,~ + 19,) + :11? 8111(91 + 191))

 

2

+2&(Q,+é,)f‘,, ’lEN

where the inner products

s? «of = cos 9,
oil 49 r
e,- e2 = cos(6, + 1),) . E N
, i
s? of = r, cos(9, + 19,-) + ——f- 3111(1),- + 19,)
7.
of - of = r,

60

have been employed, as well as the expressions given in Equation (3.7).
Ignoring gravitational effects, the system potential energy arises only from the
fiexural stiffness of the blades (linear torsional springs) and elastic coupling (linear

coupling springs) among the sectors. It is given by

N N
1 1
v = 5} :kgog + 5 E :kgb2(s,,+, — 9p)2, (3.9)
p=1 p=1

where 6N+1 = 61. The coupling elements 10f are meant to capture only the basic
pliancy between adjacent blades and hence their nonlinear kinematic contributions
have been neglected in Equation (3.9). This approximation is done independently of
any assumptions on the blade amplitudes and does not imply small (linearized) blade
motions.

Next the governing nonlinear equations of motion are derived.
3.4.3 Equations of Motion

DIMENSIONAL FORM

The equations of motion are derived by employing Lagrange’s method with the gen-
(3) (b)

eralized coordinates q,- = S, and q,- = 6, for each i E N. They follow from
T .
%( all/1))+ all)’ 8(1)=Qll), z'eN, k=a,b (3.10)
aqi aqt 391

where the kinetic and potential energy terms T and V are defined by Equation (3.8)
and Equation (3.9). The it“ set of generalized forces arise from the engine order

excitation and linear viscous damping. They are

®2=—#$
Q?” = —c§’é,- — c,¢_,b(0,- — 91—1) — cfb(é, — 0,“) , ie N (3.11)

+ C?R,F,S,‘ + FoLiejcbieant

61

where F0 and n are the strength and order of the excitation. Then the governing
equations of motion for the it“ sector follow from Equation (3.10) and take the form
.. .. _ . dR- .
m,S, + m,R,F,6, + CEIS, - ”MHZ-#62 + 9,)2
'l

d .
+ m,H§22 (F, Sin(6, + 19,-) — (1,—1; cos(6, + 1%)) = 0, i E N (3.12a)
Z

M,L,29, + c979,- — glans,- + 196,- + M,L,Hn2 sin 9,-
R226, + R,F,‘S,‘ + 213,325, (9 + 6,)
. ’l
+ m, 411.11.)

+ W522 + H0212, sin(d, + 192')

+ C$_lb(é, - 0,-” + Cfbw,‘ — 9,41)

+ k§_,b2(s, — 91—1) + k§b2(0, — 9,11) = Ragnar“, 2' e N (3.12b)

which describe the absorber and blade dynamics, respectively. The subscripts on the
blade angles are taken mod N here and in all subsequent sections such that 6N+1 = 191
and 60 = 6N, which are cyclic boundary conditions implying that the N m blade iS

coupled to the first.

DIMENSIONLESS FORM

It is desirable to work with a dimensionless form of the governing equations. This is

done by restricting L, = L, 114, = M, and k? 2 lab for all i E N and rescaling time

_ Heb/L2
UJO -— M (3.13)

is the undamped natural frequency of a single isolated blade (without an absorber)

according to r = wot, where

 

with zero coupling (kf = 0) and zero rotor Speed (Q = 0). Then if s,- = S,/L
and r, = R, / L denote the it“ nondimensional arc and radial lengths, respectively, a
dimensionless form of the equations of motion follows by dividing Equation (3.12a)

(resp. Equation (3.12b)) through by the inertia term Mng (resp. MLng). They

62

are

11,3? + 1i1.,r,F,f)[’ +6513; — #1442..ch + 6;)2
Z
d .
+ 11,602 (F, sin(6, + 19,) — i3 cos(6, + 1%)) = 0, i E N (3.14a)
l

62,-, + Elbe; — gfl'l’,F,.9; + 6., + 602 S111 6,

dr,

”Fl-29;, + 7311,82, + 27‘,E;;S£(0 + 9:)
+“2 (10131) x z 2 -
+ _ds- 32'51' + 6o 7', s1n(6, + '19,)
'l
C 9’. _ g'. C 9’. _ 9’.
+£2—1( Z 'l—I)+€Z( 2 1+1)
+ 242.116. — a.-.) + 12.26. — 2+1) = Porter“, 2'6 N (3.14.)

where ()' = d( - )/dr and it follows from Equation (3.3) and Equation (3.7a) that

_9_ = sirdx)
4.1.) [0 on)“ (3.15)

F,(s,) = \/1— (3%)? (3.16)

The dimensionless parameters appearing in Equation (3.14) are defined in Table 3.1

 

 

and the nondimensional distance from the blade basepoint O to the path vertex V is
denoted by r0, E r,(0) = a, + 'y, (dimensionally R0,- : a,L + d,). It Should be noted
that actual selection of parameter values is application-specific. The reader is referred
to [34] for an example discussion on how to map physical experimental parameters
onto these nondimensional parameters.

There are no assumptions pertaining to the absorber paths in the nonlinear models
described above, other than their gross placement relative to the blades. In the
next section a specific two-parameter family of paths is derived and these are used
to investigate the basic effects of nonlinearity on the absorber performance. The

equations of motion are derived for the case of circular absorber paths in Section 3.5.

63

Table 3.1. Selected list of dimensionless variables and parameters.

 

 

Parameter Description
F FOL/kb Strength of the engine order excitation (nonlinear models)
f FOL/ 1‘;wa Strength of the engine order excitation (linearized model)
“rm- — 73(0) = az- + 72- Radial length from blade base point 0 to path vertex V
7*,- R2- / L Radial length from blade base point 0 to ith absorber at P
32- S,- / L Arc length from path vertex V to it” absorber mass at P
xi 01/1/20 Normalized blade angle
91' 131/Ibo Normalized absorber angle (circular path)
02- Distance from blade base point 0 to absorber base point N
k /L2 . _ . .
v V i):— Distance from blade base to couphng spring attachment pt.
7,- di / L Length of ith absorber pendulum (circular path)
6 H / L Radius of the rotor disk
,uz- m2- / 1W ith absorber mass
k? ,

1/2' wf/wo = W Strength of coupling between blade 2' and blade 1 + 1
C51 27r7’6(2' — 1) it" inter-blade phase angle
cpl- 21963—12 Angle subtended from sector 1 to sector 13
'r wot Time
192- Angle subtended by St-

CC-l/L2 'th . .
{f ——l—— 2 absorber (torsmnal) damping constant

(kb/L2)M

- Gil/[42 'th . . .

{:1 ' z absorber (effective translational) damping constant

(kb/L2)M

1? 2

{If C /L ith' blade damping constant

(kb/L2)M

b 2 Cc 't} . .
{f (I) 2 ‘ coupling damping constant

(kb/LQW

0 - Q/wo Angular speed of the rotor

 

64

3.4.4 A Generalized Two—Parameter Family of Paths

In what follows, a two-parameter family of paths is derived in terms of linear and
nonlinear tuning parameters, which will serve as the fundamental absorber design
variables in the subsequent chapters. The basic idea is to assume an expanded form
of each absorber position relative to its basepoint, which is captured by the radial
lengths 7‘,(s,-). In doing so, only even terms are included such that the paths are
symmetric about their vertices. The expansions are introduced to the full nonlinear
equations of motion and, by restricting zero blade motions relative to the rotating hub
and appropriately truncating nonlinear terms, a set of well-known nonlinear systems
results. These reduced systems depend only on the absorber dynamics and they
motivate the selection of two tuning parameters. The first parameter sets the linear
absorber tuning order (a topic that is more fully described in Chapter 4) by setting the
path curvature at its vertex. The second parameter prescribes the nonlinear tuning
by varying the curvature along the path, and can be thought of as the strength of the
path nonlinearity. Proper selection of this parameter is motivated in Chapter 5.

In what follows, each absorber path is assumed to be identical and identically

fitted to the blades by imposing a,- = a and '7,- = '7 for all 2' E N. Then
roisr0=a+m Vie/V (3.17)

represents the dimensionless distance from the blade basepoint to the path vertex.
By restricting 6, = 6; = 65' = 0, Equation (3.14a) reduce to

d... A.
s;' — 40%,f + 602 (1‘,- sin 19,- — J1 cos 29,-) = 0, 2' e N (3.18)

3, (1.9,

which describes the nonlinear absorber dynamics for the desired case of zero blade
motions relative to the rotating hub. Next the dimensionless radial length r,(s,-) is

expanded according to
73%,) = 60 + bzs? + (>48? + 0(s?). (3.19)

65

where only even terms are considered since ri(—s,-) = ri(s,~) is even by assumption.
Each has the same constant coefficients, implying that all of the paths are identical.
Since r0 = 0+7 = 72,-(0) = \/b_0_ the first parameter in Equation (3.19) is automatically
prescribed and is given by b0 = r3 = (oz+7)2, and the remaining parameters b2 and b4
are to be specified. Substituting Equation (3.19) into Equation (3.18) and expanding
in 3,: yields

5;, + 73.2023,- + 0028? + (We?) 2 0, (3.20)

_ 2
T] = —2b4 _ ((1)2 1) +12b0b4)5
(box/lb

are defined to be the linear absorber tuning order and the nonlinear absorber tuning

where

 

 

 

parameter, respectively.

Equation (3.20) is recognized to be a standard undamped and unforced Duffing os-
cillator, a comprehensive treatment of which can be found in most texts on nonlinear
systems [95—97]. For small amplitudes the nonlinear term can be neglected and the os-
cillator exhibits free harmonic motions with frequency no. It is well-known that linear
tuning of the centrifugally-driven absorbers under consideration can be accomplished
by setting the absorber tuning order a, which depends on the absorber placement and
system geometry, relative to the order of the excitation n [10]. When these match
identically, and in the absence of damping, a complete elimination of vibrations of
the primary system is possible, which is Shown systematically in Chapter 4 for the
models described above. For larger motions the nonlinearity in Equation (3.20) be-
comes important and the oscillations become amplitude-dependent. In the context of
absorber path design, therefore, the nonlinear tuning parameter r] is used to modify
the absorber behavior without compromising the small-motion linear tuning a z n.

When 77 > 0 the response is hardening, and it is softening for 77 < 0. Finally, the

66

1.2 ,

 

 

1 .

0.8 ,
\——20

—10

100

0.6 » 20

0.4 »

—0.25 O 0.25 0.5

Figure 3.5. A generalized family of absorber paths defined by Equation (3.19) for a = 0.84,
6 = 0.67, a = 3 ('y = 0.168 and r0 = 1.008), —0.47 S 31- S 0.47, and for a range of nonlinear
tuning from softening to hardening: 77 = —20, —-10,0, 10, 20.

coefficient terms in front of (5 in Equation (3.21) appear since the absorber paths are
measured from the base of the blade, and not from the center of the rotor as it is
done in the CPVA work by Shaw and coworkers [14].

The remaining expansion coefficients b2 and b4 can be obtained from Equa-
tion (3.21) in terms of the system geometry and the linear and nonlinear tuning

parameters a and 7]. To summarize,

b0 = r0
6 — 7227};
b = ——
2 6 + To , (3.22)
du+ah2 m
4 :

—m — M"
where r0 is defined by Equation (3.17). An example plot of the generalized family of
absorber paths defined by Equation (3.19) is shown in Figure 3.5 for a particular blade
geometry and absorber design and for a range of nonlinear tuning from softening to

hardening.

Next we consider a special case of the equations of motion in which circular-path

67

absorbers are employed.

3.5 Bladed Disk Assembly Fitted with Circular-Path Ab-

sorbers

In what follows, the nonlinear equations of motion for a nominally cyclic bladed disk
assembly fitted with circular-path absorbers are deduced from the general system
given by Equation (3.12). These are linearized for small blade/absorber motions
in Section 3.5.2 and subsequently modified to account for physical rattling space
limitations of the absorbers. The resulting mathematical model is employed in the

linear analysis of Chapter 4.

3.5.1 Equations of Motion

Consider the sector model shown in Figure 3.6, which features the same pendulum-
like blade model described in Section 3.4. In this case the blades are fitted with
circular-path absorber pendulums of mass m, and radius (1,, the motions of which are
described by the angles 2b,. (The angles 2b? correspond to motion-limiting stops, which
are incorporated into the equations of motion in the next section.) The governing
equations of motion for the overall system could be derived from this model in the
usual manner via the method of Lagrange. However, it will be more convenient to
deduce them from the general results of Section 3.4 by restricting the arbitrary path
shown in Figure 3.2 to be circular.

By restricting each ,0, = d,- to be constant the general path shown in Figure 3.2

(solid line) reduces to a circular path (dashed line). Then S,- = (flab,- for all 2' E N and

68

 

 

Figure 3.6. Sector model for a bladed disk assembly fitted with circular-path absorbers
with limited rattling space.

it can be shown that

‘

R22 = (1,-2L? + d? + 2diaz-Lz- COS 1,02
R1 sin 19¢ = d1 sin wi

R,- c0819,- 2 (1,-L,- + dz', COS 11%

 

 

> , i e N (3.23)
Ril‘, = aiLz- COS 7,02“ + (12'
d(Rz'Fz‘) _ _aiLisin'</2i
d8,- _ d,-
d ' .
lat-Elsi: = -—a,-L,- Sln ’t/Ji J

which relate the general—path variables R1, 8,, and 19,- to the circular-path angle 2b,.
The expressions given in Equation (3.23) can be employed to deduce the circular-
path equations of motion term-by-term from Equation (3.12), the results of which
are summarized in Table 3.2. For the proposed circular-path model it is natural
to express the absorber damping in terms of a torsional viscous damping constant
instead of the effective translational representation of Section 3.4. The generalized

forces are re-formulated to account for this and are given by
an) = —C§l?l11'
Q?” = —c§’é, — c§_1b(é, — 9H) — cfb(é, — 9m) , i e N (3.24)

+ cgzbi + FoLz-ej‘piejnnt

69

where c? is the torsional damping constant for the ith absorber and is not to be con-
fused with cf”. (These are implicitly related in Section 3.6.) The desired equations of
motion follow from Equation (3.12) by performing the substitutions given in Table 3.2
and by employing the re—formulated generalized forces given by Equation (3.24). For

i E N they are

mid? (6,- + 16,-) + 0616., + m,d,-HQ2 sin(6,- + '16,)

+ m,d,a,~L,-6i cos 162' + nzidioziLz- (Q + 6,)2 sin 11),; = 0, (3.25a)

M,L,26, + cfé, — egg/5,. + 1.536,- + M,L,-Ho2 sin 9,-

- 022L226, +6122 (61; + 16;)

+diaiLi (163‘ + 262') C08 161‘

—diaiLi (16,,-2 + 2(9 + 6i)16i) sin 162'
+HQ2(a,-L,- sin 6,- + d,- sin(6,- + 22.))

 

 

+ C$_1b(6i—6i_1)+ Czcb(6z' - 624.1)

+ k§_,b2(a, — 6,-_1) + kfb2(9,- — em) = FOL,eJ'¢iej"Qt. (3.25b)

In the next section these are linearized for small blade/ absorber motions and are
modified to account for limited absorber amplitudes imposed by the geometry of the

blades.

3.5.2 Linearized Model with Restrictions on the Absorber Amplitudes

In any realistic physical implementation the absorber amplitudes will be restricted by
the blade geometry and this is captured by the motion-limiting stops in Figure 3.6,
where 16? represents the limiting angle of the ith absorber. Impacts occur whenever
Wil = 2629, the dynamics of which have been investigated in [34,98] for the case of
a single isolated blade/ absorber combination. This feature is included for generality
but will not be directly exploited in the analysis of Chapter 4, where it is assumed

that I162”! < 16;? throughout.

70

Table 3.2. Selected terms in the general-path equations of motion given by Equation (3.12)
and their counterparts for the case of circular-path absorbers.

 

 

General Path Term Circular Path Term

miSi midflh

m-z'Rin'éz' 71121011»; COS 162' + 0106

—m,Ri%%(Q + 62-)2 miaiLz-(Q + 6,)251n 1!),

mz-HQ2 (F,- sin(6, + 19,-) — %% cos(6,- + 1%)) miHQ2 sin(6i + 162‘)

miRz-26i rn,-(oz?L,2 + d? + 2d,;oiiL, cos 16,-)6,
mz’R/irigi midi(di + aiLi COS 16016
2miR4%%Si (Q + 6,) —2m,-d,a,L,~(Q + 6,)16, sin 26,-
miflg—g-‘ilS? —m,-d,-o:,-L,~z6é2 sin 16,-

mz-HQ2R, sin(6,- + 6,) miHQ2(a,-L,- sin 6,- + d,- sin(6,; + 1621))

 

Equation (3.25) is linearized for small blade and absorber motions and is made
dimensionless in the same way as it was done in Section 3.4.3. The absorber and blade
motions are subsequently scaled according to 2:, 2 61/160 and y,- = «1),/7,120, where each
16,9 = 160 has been assumed to be identical, and thus impacts correspond to Iy,| = 1.

Then by dividing through by 260 the dynamics of the ith sector are governed by
rim-2632' + af') + 63y; + uni-602m + yi) + uni-am? + 02y.) = 0, 2' E N (326a)

113$, + £65”; — £39; + 1‘,- + (502110,-
a? 1C? +7362? + yg’) + aim-(31,” + any) i
+ Hi 2 2
+065” “Ci +7260 (3% + it)
C I I C I I
+ €1—1($i— 2315—1) + £2 (1131- — $24.1)

+ v3.16. — x._1>+ v.26.- - x...) = fejaef’m. ie N (3.261))

where the dimensionless parameters are defined in Table 3.1. Equation (3.26) forms
the basis for the linear analysis of Chapter 4.
Estimates for the blade and absorber damping constants are developed next in

addition to a relationship between the nondimensional torsional and effective transla—

71

tional absorber damping constants {a and 55,.

3.6 Estimates of the Dimensionless Damping Constants

As a guide to sensical parameter selection in numerical simulations, approximate
expressions for the blade and absorber damping constants 5b and {a are formulated
in terms of their respective damping ratios. These can be obtained by considering the
free vibration of an isolated blade / absorber combination when the absorber is locked
in place and from the free response of an absorber when the blade is locked relative
to the rotating disk. The linearized model of Section 3.5.2 is used for this purpose.
A relationship between the torsional absorber damping constant {a and the effective
translational damping constant {a is also derived. In all of what follows identical
sectors are assumed.

Consider the special case when the absorbers are locked in their zero positions
relative to the blades. In the absence of coupling (i.e., only an isolated sector is consid-
ered) the corresponding linearized equation of motion follows from Equation (3.26b)

by setting y = y' = y” = 0 and it can be expressed as
(1+ a(a + 71)?)2” + 5,319+ (1 + (1 +)1(a + 7))602)r = faint". (3.27)

In practice the absorber-to—blade mass ratio is very small and centrifugal stiffening
has a negligible effect on the blade natural frequencies. By restricting a = 0 and

o = 0 it thus follows from Equation (3.27) that

63 '5 2,01), (3.28)

where pb is the blade damping ratio. Damping levels in the blades can be 0.1% or
less relative to critical, which gives rise to the approximation 5b g 0.002.
A similar calculation can be carried out for the absorbers. When the blades

are locked in their zero positions relative to the rotating disk the absorbers become

72

dynamically isolated and their linearized dynamics are governed by

#7231” + {ay’ + #761 + (5)0231 = 0, (329)

which follows from Equation (3.26a) with :1: = :13’ = :L'” = 0. By employing the

equivalent mass, damping, and stiffness terms in Equation (3.29), one can obtain
a = 21112pra. (3.30)

where a = W is the linear absorber tuning order (this is discussed more fully
in Chapter 4) and ,0a is the absorber damping ratio. As discussed in the forthcoming
chapters, we will be interested in the system dynamics for absorber tuning a close
to the engine order n and also near resonance conditions, which correspond approxi—
mately to a = 1 / m ’5 1 / n. Under these conditions no g 1 and Equation (3.30)
can be approximated by

5a 9: 2H72Pa- (3.31)

Given the absorber mass, pendulum length (circular path) and critical damping level,
Equation (3.31) can be employed to approximate the torsional damping constant Ea.

The damping constants 5a and 55 can be (approximately) related by comparing the
absorber damping term in Equation (3.14a) (nonlinear system with general-path ab-
sorbers) to that in Equation (3.26a) (linearized system with circular-path absorbers).
By multiplying the latter through by the stopper angle 160 and dividing by 7 (this
makes it possible to compare the equations directly) the absorber damping term for
the linearized system can be written as 5016: / 'y = Easg/vz, where s, = 716,- has been
employed, and the corresponding damping term for the nonlinear system is 5513;. An
exact (resp. approximate) relationship between the torsional and effective transla-
tional absorber damping constants for a circular (resp. general) path can be obtained

by equating these expressions and is given by
_ 2 _
a — 7 5a, (3.32)

73

le.o.

 

 

 

 

 

25 1 /
“to 20 1
(a) 3'2 15 »
cs 10 .
w ..——2e.o.
5 ) m
4: f0
0.005 0.01 0.015 0.02 0.025
Mass Ratio a
718.0.
25 . /
“1; 20 »
(b) :3 15
«533 10 >
5 .
0.005 0.01 0.015 0.02 0.025
Mass Ratio 11

Figure 3.7. Approximate (a) torsional and (b) translational absorber damping constant
versus mass ratio 6 based on Equation (3.31) and Equation (3.33) for pa = 0.005, oz = 0.84,
6 = 0.67, and engine orders (e.o.) n = 1, . . . , 10.

where 7 is the dimensionless length of the absorber pendulum (resp. the dimen-
sionless curvature of the absorber path at its vertex). When the path is general
Equation (3.32) gives a reasonable relationship between the absorber damping con-
stants, depending on the strength of the path nonlinearity 77 and the amplitude of
the absorber motions. For linear absorber tuning a at n and for rotor speeds close to

resonance

which follows from Equation (3.32) together with Equation (3.31).
Figure 3.7 shows example plots of 5a and €51 in terms of the mass ratio ,u for an
absorber damping ratio pa = 0.001 and for various engine orders. These charts can

be used to obtain an appropriate order of magnitude for the damping constants to

74

be used for the frequency response loci and in numerical simulations. In Chapter 5
the frequency response curves are generated for a model with a = 0.84, 6 = 0.67,
a = 0.0035, n = 3, and pa = 0.005, which gives rise to 5a 2 2 x 10‘6. The values
51) = 2 x 10‘3 and £5, = 2 x 10‘6 are used when damping is included, and also in the

simulations.

3.7 Concluding Remarks

A lumpted-parameter mathematical model of a bladed disk assembly fitted with
centrifugally-driven, general-path vibration absorbers has been systematically devel-
oped, which serves as the basis for all of the analysis to follow. We shall be interested

in three specific cases of this general nonlinear system:

1. The linearized system with motion-limiting stops, which is given by Equa-
tion (3.26);

2. The fully nonlinear system given by Equation (3.14) with zero inter-blade cou-
pling (1/ = 0), together with the two—parameter family of paths defined by
Equation (3.19); and

3. The fully-coupled nonlinear system (V 74 0) given by Equation (3.14), together
with the two-parameter family of paths defined by Equation (3.19).

These three systems are systematically analyzed in the next two chapters, and through-
out the remainder of this work they shall be referred to as (1) the coupled linear or
linearized system, (2) the isolated nonlinear system, and (3) the coupled nonlinear
system.

We begin in the next chapter with an analysis based on the coupled linearized

system.

75

CHAPTER 4

Forced Response of the Coupled Linear

System

4. 1 Introduction

In this chapter the fundamental linearized dynamics of a cyclically-coupled bladed
disk assembly fitted with circular-path vibration absorbers are investigate in detail.
The aim is twofold: to quantify and understand the underlying linear resonance
structure of the coupled linear system under engine order excitation and, based on
these findings, to design the absorbers to eliminate or otherwise reduce blade motions
relative to the rotating hub. (The basic effects of nonlinearity are investigated in
Chapter 5 for an isolated blade/ absorber combination and also for the fully coupled
cyclic system.) An auxiliary, but very important topic includes a decoupling strategy
based on the cyclic symmetry of the system whereby the fully-coupled linear model
can be simplified to a set of reduced-order models, from which analytical results
easily follow. The analysis is based on the well-known theory of Circulants, which
is summarized in Section 2.2 and covered in detail in Appendix B, and it is also
employed in Chapter 5 to handle block diagonalization of Jacobian matrices.

As we shall see, the underlying linear resonance structure is surprisingly subtle and

complicated, a feature that arises from the order-nature of the absorbers. However,

76

the (block) decoupled models to be formulated give rise to a set of tractable analytical
expressions for the system eigenfrequencies which, when represented in a Campbell
diagram, clarify the nature of the resonance structure for various absorber designs and
for any number of sectors, engine order, or coupling strength. A particular absorber
design strategy is motivated from these eigenfrequency versus rotor speed plots in
terms of a detuning parameter, which essentially assigns the absorber tuning order
(which depends on the system geometry) relative to the order of the excitation. It
will be shown that ideal (exact) tuning, where the absorber tuning order is chosen to
identically match the excitation order, completely eliminates the system resonances
and (in the absence of damping) results in zero-amplitude steady-state blade motions
over all rotor speeds. Such a tuning scheme, however, is susceptible to the effects
of parameter uncertainties. An important contribution of this chapter is that, in
addition to the exact tuning, there exists a range of absorber undertuning values for
which there are no system resonances—independent of the rotor speed. Therefore, a
practical tuning strategy involves intentionally detuning the absorbers within this “no-
resonance zone.” The approach offers a more robust design against system resonances,
but at the expense of some residual steady—state blade vibrations.

The chapter is organized as follows. The linearized system to be considered is
described in Section 4.2, where the dimensionless equations of motion are formulated
for a single sector and subsequently for the overall coupled system. Two special
cases of these governing equations are considered: the case when the blades (resp.
absorbers) are locked in their zero positions relative to the rotating hub (resp. blades)
in Section 4.2.3 (resp. Section 4.2.4). The former motivates the absorber tuning order,
which is employed in subsequent sections to tune the absorbers to a given order of
the excitation, and the latter is investigated in detail in Section 4.3.1, with the aim
of providing a benchmark against which the effectiveness of the absorbers can be

evaluated. The forced response of the general system is detailed in Section 4.3.2, and

77

an absorber tuning strategy is motivated in Section 4.4. The effects of damping are
briefly considered in Section 4.5, and the chapter closes with some concluding remarks

Section 4.6.

4.2 Mathematical Model

The bladed disk model to be considered is shown in Figure 4.1a in dimensionless
form. It consists of a rotationally periodic array of N identical, identically-coupled
sector models, one of which is depicted in Figure 4.1b. The disk has radius 6 and
it rotates at a fixed speed a about an axis through C'. Each blade is modeled by a
simple pendulum of unity mass and length, the dynamics of which are captured by
the normalized angles :15,. (See Table 3.1 on page 64.) The blades are attached to the
rotating disk via linear torsional springs with unity stiffness and adjacent blades are
elastically coupled by linear springs with stiffness V. It is assumed that the springs are
unstretched when the blades are in a purely radial configuration, that is, when each
2:,- = 0. As shown in the inset of Figure 4.1b each blade is fitted with a pendulum-
like, circular-path absorber with radius 7 and mass 11 at an effective distance a along
the blade length. The absorber dynamics are captured by the normalized pendulum
angles y,- and they are limited according to ly,| S 1 by stops, which represents the
rattling space limits imposed by the blade geometry.1 Linear viscous damping is also
included (but not indicated in the figure). Blade and inter-blade damping is captured
by linear torsional and translational dampers with constants £1, and (c, respectively,
and the absorber damping is captured by a torsional damper with constant {0. Finally,
the system is subjected to the traveling wave dynamic loading described in Section 2.3,
and Equation (2.14) is employed for this purpose. Throughout the remainder of this

thesis it is assumed that 0 < n < N for simplicity. This does not, however, affect the

 

1This feature is included for generality, but in all of what follows it is assumed that Iy,~| < 1, i.e.,
that impacts do not occur. The impacting dynamics for this system are investigate in [34] for an
isolated blade/ absorber combination.

78

 

 

Figure 4.1. (a) Model of bladed disk assembly and (b) sector model.

results; they can be generalized to account for n 2 N according to the discussions in
Section 2.3 and also Section 2.4.

Next a mathematical model that describes the linear dynamics of the ith sector is
described. The overall system is composed of N such models, and these are cast in a

matrix-vector form with block circulant coefficient matrices in Section 4.2.2.

4.2.1 Sector Model

The governing equations of motion for the ith 2—DOF sector follow from Equa—

tion (3.25) of Chapter 3. They are
#7203? + 31.”) + anf + #760261. + 312') + #701652, + 023/1) = 0, 2' E N (41a)

2’ + 5be — €ay,’- + 2:,- + (50211:,-
+ 522:? +72(:r,;' + yg’) + ma,” + 2mg)
#.
+a'602x, + 7602(27, + y.,')
+ €c(—$i—1 + 217i — $i+ll

+ V2(—:1:,-_1 + 2.1:,- — 513,11) = fej‘biej'laT, i E N (4.1b)

79

where the dimensionless parameters are defined in Table 3.1 on page 64. In Equa—
tion (4.1) the parameter subscripts have been dropped since the sectors are assumed
to be identical and identically coupled, and the remaining subscripts i are taken
modN such that $N+1 = $1 and (1:0 2 :rN. In matrix—vector form, Equation (4.1)

becomes
Mzg’ + CZ; + Kz, + Cc( - Z;_1+ 22; " Zi+1l

. _ , i e N (4.2)
+ K.( — z,-_1 + 22,- — zi+1) = 16119223in

.I'z' 61/160
'1' = = , 4.3
2 [311] [191/160] ( )

captures the sector dynamics and the elements of the sector mass, damping, and

where the vector

stiffness matrices are given in Table 4.1. The matrices

C 0 2 0

capture the inter-blade coupling and vanish when {0 = 0 and l/ = 0, respectively, in

CC:

 

which case Equation (4.2) describes the forced motion of N isolated blade/ absorber
systems. (Equation (4.2) is studied in detail in [34, 98] for the case when N = 1,
Kc = 0, and Cc = 0, including the impact dynamics that occur when [yz] = 1, that

is, [16,-] = 1190.) The sector forcing vector is given by

_ f
f _ [0], (4.5)

where f is defined in Table 3.1. Finally, the parameter we 2 %\/kc/M (see u in
Table 3.1) is the undamped natural frequency of a single isolated blade (with no
absorber) with kb 2 0 and Q = 0 and with a single coupling stiffness element kc

connected to an adjacent, stationary blade.

80

Table 4.1. Elements of the sector mass, damping, and stiffness matrices M, C, and K.

 

M11= 1+#(Cr+7)2 011:6; K11: 1+(1+#(01+7))502
M12 = 67(0 + 1') C12 = -€a K 12 = #7602

M21: Mi2 C21: 0 K21: K12

M22 = #72 C22 = {a K22 = m (a + 5) 02

 

4.2.2 System Model

By stacking each z,- into the configuration vector q = (21,22, . . . ,zN)T, the governing

matrix equation of motion for the overall 2N —DOF system takes the form
Mq” + Cq' + Kq = fej'wT, (4.6)

where M is block diagonal with diagonal blocks M and K E fl‘fifiyz N has generat-
ing matrices K + 2K0, —KC, 0, . . . , 0, —KC. The matrix C E fi‘ffljfi, N is similarly
defined by replacing K with C and Kc with Cc in K. In terms of the circulant

operator the system mass, damping, and stiffness matrices are given by

M = circ (M,0,0, . . .,0, 0) = diag (M)
. teN
C = circ (C + 2C0, -—CC, 0, . . . ,0, —CC) , (4.7)

A

K = circ (K + 2K6, —KC, 0, . . . ,0, —KC)

where the circ ( - ) operation is defined in Section 2.2 and also in Appendix B.2 Finally,

the system forcing vector is
. . . ,. T
f: (feJ¢l,fe]¢2,...,feJ¢N) , (4.8)

where f is given by Equation (4.5) and ([5,- is defined by Equation (2.15).

 

2See [62] for a comprehensive treatment of circulant matrices and their properties. A brief review
of such matrices is given in Section 2.2 on page 11 and a more exhaustive treatment of the theory,
including many proofs, is given in Appendix B. In all of what follows and whenever reference is
made to Section 2.2 it is understood that, in most cases, more details can be found in Appendix B.

81

4.2.3 Special Case: The Blades Locked

Consider the special case when the blades are locked in their zero positions relative
to the rotating disk. This leads to a system of dynamically isolated absorbers that

oscillate freely under the influence of centrifugal effects. The governing equations

ll:

follow from Equation (4.2) by setting 2:,- = 1:: = 371‘ 0, and are given by

11122y[’+ 0221/; + K223}, = 0, i E N (4.9)

where the mass, damping, and stiffness terms 0122, 022, and K22 are defined in
Table 4.1. Equation (4.9) is a set of N uncoupled and unforced single-DOF harmonic

oscillators. Their dimensionless undamped natural frequencies are given by

 

1.1/'22 : — = I 0' E 7710', (4.10)

or 11222 = an in dimensional form, where 010 is defined by Equation (3.13) and

(1+6
7

 

:31
||

(4.11)

is defined to be the absorber tuning order. Since the absorbers are restrained only
through centrifugal effects, £222 scales directly with 0 [32,61]. This feature is exploited
in Section 4.4 to tune the absorbers to a given order of the excitation, rather than to a
fixed frequency, as is done in the classical sense [99]. The tuning parameter a is used
for this purpose and is determined by selecting the dimensionless curvature of the
pendulum absorber '7' (dimensionally d) and the distance of its effective attachment

point from the center of rotation of the rotor, that is, a + 6 (dimensionally aL + H).

4.2.4 Special Case: The Absorbers Locked

Here we consider a model in which the absorbers are locked in their zero positions

relative to the blades. By setting y,- = y: = y;’ E 0, the equations of motion for the

82

it" sector follow from Equation (4.2) and are given by

Mum? + 0111?; + Kiwi + €c( — 552—1 + 21”; — $f+1l
_ . , ie N
+ 1/2( — 1131—1 + 2:17,- " WM) 2 femiejnm
(4.12)

where 56, u, and f are defined in Table 3.1 and also :1: N+1 = 2:1 and 51:0 = :1: N- The

governing matrix equation of motion for the overall N —DOF system is given by

M11XII+C11XI+K11X =f11€jnOT, (4.13)
where x = (171, 1'2, . . . ,xN)T is a configuration vector and
f1, = (feJ¢1,feJ'¢2, . . . , feJ¢N)T (4.14)

is the system forcing vector. The system matrices are both symmetric and circulant,

and they can be represented by
M11 = circ(ll«111,0,0,...,0,0) = diag(lv[11)
. ieN
C11 = circ (C11 + 260, —€C, 0, . . . , 0, -£C) . (4.15)
K11 = circ(K11+ 2V2, —1/2,0, . . . ,0, —I/2)
In the absence of coupling, that is, when U = 5C E 0, the system matrices given
by Equation (4.15) are all diagonal, and Equation (4.13) is a decoupled set of N
harmonically forced, single-DOF oscillators.

We now detail the steady—state forced response of Equation (4.13) (with the ab—
sorbers locked) and, subsequently, that of Equation (4.6) (with the absorbers free to

move). In both cases a coordinate transformation is employed in order to significantly

uncouple the governing matrix equations.

4.3 Forced Response

The forced response of the overall system is governed by Equation (4.6), which can

be handled using standard techniques [65]. Its solution in the steady-state follows in

83

the usual way and is given by
qSS(r) = Z—lfejmfl, (4.16)

where Z = K—n202M+]'noC is the system impedance matrix of dimension 2N >< 2N .
However, Equation (4.16) does not offer any insight into the design and effectiveness
of the proposed vibration absorbers, and it also requires computation of 2‘1, which
can be quite involved for many bladed disk models. We thus turn to a decoupling
strategy that exploits the system symmetry and it is systematically shown how to
reduce the governing matrix equation of motion to a set of reduced-order models.

It is well-known that, due to its cyclic symmetry (and in particular due to the
circulant structure of the system matrices [62]), Equation (4.6) can be decoupled
via a modal (unitary) transformation to a set of N reduced-order models, each with
two DOF [36,92,100—102]. (The reduced-order models have the same number of
DOF as an individual sector, in this case two.) Similar statements can be made for
Equation (4.13), which captures the system dynamics for the special case when the
absorbers are locked in their zero positions relative to the blades. Since the system
matrices are circulant for this special case, one can fully decouple the N -DOF model
to a set of N, single—DOF systems. A special feature of the uncoupled systems
described above is that only mode n + 1 is excited, provided that n is an integer (as
shown subsequently). Hence the steady-state response of the overall 2N —-DOF (resp.
N —DOF ) system described above reduces to the solution of a single, harmonically
forced, 2—DOF (resp. single—DOF) system.

In order to provide a benchmark against which the effectiveness of the absorbers
can be evaluated, we first consider the forced response of the system when the ab-
sorbers are locked relative to the blades. It should be noted that the corresponding
analysis could be obtained directly from the more general analysis of Section 4.3.2.
However, it is instructive to introduce the modal transformation in this simpler set-

ting, which clearly demonstrates the essential features of the approach. The forced

84

response of the general system, where the absorbers are free to move, is investigated
in Section 4.3.2 and employs the same methodology. An absorber tuning strategy is

motivated in Section 4.4 based on these results.

4.3.1 Response with the Absorbers Locked

The purpose of this section is twofold: to demonstrate the essential features of the
analysis, a generalization of which is employed in Section 4.3.2 for the case when
the absorbers are free to move, and to review some of the vibration characteristics
of linear cyclic systems. Some specific topics include: decoupling the equations of
motion; orthogonality of the modal forcing vector; the steady-state system response;
characteristics of the natural frequencies and attendant normal modes (see [9, 92, 94]
for further characteristics); and conditions for resonance. The results will also be use-
ful for comparisons when evaluating the effectiveness of the absorbers in subsequent
sections. The reader who is familiar with these topics can proceed, with minimal loss

of continuity, to Section 4.3.2.

MODAL ANALYSIS

Consider the forced response of the system in Figure 4.1a for the special case when the
absorbers are locked in their zero positions relative to the blades. Due to the cyclicity
of the model and the corresponding circulant structure of the system matrices given
by Equation (4.15), one can employ a standard unitary (similarity) transformation
to decouple the governing equations of motion. In particular, we wish to apply the

result given by Equation (2.9) of Section 2.2.4 to each of the system matrices. This

85

can be achieved by introducing the change of coordinates3’4

x = 15322, or $1, = e351, 1) E N (4.17)

where E is the N x N complex Fourier matrix and ep is its pth column (these are
defined by Equation (2.2) and Equation (2.4)), and 51 = (5:1, 5:2, . . . ,2N)T is a vector
of modal, or cyclic coordinates. Substituting Equation (4.17) into Equation (4.13),
multiplying from the left by the unitary matrix EH, and invoking Equation (2.9)

yields a system of N decoupled scalar equations. They are

11138;; + 6318;, + 118%,, = gillej’w", p e N (4.18)

_. T ' " .
where (- )H = ( ) denotes the conjugate transpose and eyfu IS the pth element of

EHfll, and is discussed subsequently. The modal mass, damping, and stiffness terms

follow from Equation (2.10) and are given by

19119;) = Mn

6]?) = 011 +2§C(1—Cos<,op) , pEN (4.19)

1?]? = K11+ 2V2(1- coscpp)
where cpp is defined by Equation (2.3) and the elements M11, Cu, and K11 are
defined in Table 4.1. Note that the identity raw—1) + HAN—1X64) = 2cos app has
been employed, where 11W 2 w = 82677: is the primitive Nth root of unity.5 The
transformation of the single N —DOF system given by Equation (4.13) to the system of
N decoupled single~DOF systems given by Equation (4.18) is illustrated in Figure 4.2.

Assuming harmonic motion, the pm steady-state modal response follows easily

 

3The reader who is not familiar with transformations of this type should regard Equation (4.17)
as the usual modal transformation employed in elementary linear vibration theory. (See Figure 4.2.)
The columns e, of the fourier matrix E are, in fact, the eigenvectors of any circulant matrix, and
hence they define the system mode shapes for all linear cyclic systems with a single DOF per sector.

4The index p corresponds to the p“ mode of vibration and shall be referred to as the mode
number.

5See Section B.4.1 of Appendix B for a derivation of the N‘" roots of unity. The distinct N ‘h
roots are plotted in Figure 8.1 on page 191 for N = 1,. . . ,9.

86

Blade
® Rotor

.__. .3

Modal (Cyclic)

 

Transformation (p E N )
(a) A Single Coupled N—DOF System (b) N Decoupled 1—DOF Systems
in Physical Space in Modal Space

Figure 4.2. The tOpology of a bladed disk assembly in (a) physical space and (b) modal
space. The modal transformation x(r) = Ex(r) reduces the cyclic array of N single—DOF
coupled models 8, which together form a N —DOF coupled system, to a set of N single—DOF
decoupled models HP.

from Equation (4.18) and is given by

- 1 . -
wish) = meyfllejnm, p 6 N (4.20)
I111
where
I‘M?) _f{(p) _ 2 2M(p) - 61(1)) N 421
11— 11 ”011+?”0111 116' (-l

Under the assumption that 0 < n < N is an integer, the pth modal forcing term

simplifies considerably and is given by [36,92]

N
H“ _ L k—1)n+1— )
p fll — WEI!” ( p

{Wfl p=n+1

0, otherwise

e

(4.22)

which follows from Theorem B3 on page 192. Equation (4.22) shows that only mode

87

n + 1 is excited and, therefore,6

 

- \/Nf '
$2.11”) = -(n+1)€]nm (423)
1111

is the only non-zero modal response in the steady-state. The response of sector i (in

physical coordinates) follows from the transformation given by Equation (4.17) and

is given byr " — —eTxSS, or
$:S(T) = Xejf’iejnm, i E N (4.24)
where xSS(r) = (0,.. ,,0 :rn+1(r r), 0, ...,0)T and 10716-1) 2 e361' have been employed

and X: f / F] (n+1) is the steady-state amplitude of the blades. Equation (4.24)
shows that each blade behaves identically except for a constant phase Shift from one
sector to another, which is captured by the inter-blade phase angle (0,. This approach
offers a significant computational advantage over the brute force solution of the full
N —DOF system, and it is employed in Section 4.3.2 for the general case when the

absorbers are free to move.

EIGENFREQUENCY CHARACTERISTICS AND CONDITIONS FOR RESONANCE

Since the transformation given by Equation (4.17) is unitary, the (dimensionless)

natural frequencies 02(1)) are preserved and they follow in the usual way from If [21) and

MS). In terms of the parameters defined in Table 3.1, they are given by

 

(4.25)

 

_(p)_ (4.1-ii) 1+ 602(1 + a(a + 7)) + 2V2(1— cos app) _
“’11— _ = 2 , z E N
“’0 l+a(a+7)

where 020 is given by Equation (3.13) and 1,0,, is defined by Equation (2.3). For zero

inter-blade coupling (V = 0) all of the natural frequencies are identical, and they

 

6It is customary in the rotordynamics literature to designate the system modes in terms of
their “diamatral components,” or number of “nodal diamters.” (These can be clearly visualized
in the modal configurations shown in Figure 2.7 on page 35.) Specifically, if 0 < n S N/2 (or
0 < n S (N — 1)/2 if N is odd) then an n e.o excitation can only excite modes with n nodal
diameters. However, such a designation is slightly more cumbersome if one considers larger values
of n (in this chapter and the next we consider 0 < n < N) and hence we shall say instead that an
engine order n excites only mode p = n + 1.

88

increase with increasing rotor speed a due to centrifugal effects. The presence Of
the absorber masses (,u 74 0) slightly lowers the natural frequencies. For very small
absorber masses relative to the mass of the blades, that is, 0 < u << 1, the natural

frequencies can be approximated by

 

a]? ’3.“ \/1+ 602 + 2V2 (1 — cos 1,919), i E N (4.26)

which clearly exhibits the effects of centrifugal stiffening and coupling. Finally, if
a = V— — a _=_ 0 we recover w]l[)— — 1, or w[fi)— — wo, which was used in Section 4. 2. 1 to
nondimensionalize the model. By comparing Equation (4.26) to Equation (2.38), it
is clear that the eigenfrequency characteristics discussed in Section 2.4.3 apply here
as well.

111 the turbomachinery literature it is common to plot the natural frequencies
in terms of the “diametral components,” that is, the number of “nodal diameters”
(n.d.) in their attendant mode shapes [103]. (These can be visualized in Figure 2.7
on page 35.) However, in light of the crucial role centrifugal stiffening plays in the
absorber performance (this is investigated in detail in Section 4.4), we shall opt instead
for an interference, or Campbell diagram representation of the natural frequencies.
Such a diagram is shown in Figure 4.3a for N = 10 blades and for a particular
sector model. In this figure, the natural frequency loci are plotted in terms of the
dimensionless rotor speed, and they are seen to increase for increasing a due to
centrifugal effects. The diametral component of each frequency locus is also indicated.

or, equiva-

(16(0)

In general, there may be a system resonance whenever no — -w
lently, nQ— — to]? (Q), and these possible resonances can be identified in a Campbell
diagram by the intersections of the natural frequency loci with an engine order line no.
(An example of such a frequency versus rotor Speed diagram is Shown in Figure 4.3a,
where several order lines are indicated.) Such resonances can arise, for example, from

excitations with multiple dominant orders, mistuning [9], nonlinear effects, or non-

integer n. (Appendix C briefly discusses the case of n E R+.) However, for linear

89

Table 4.2. Data to accompany Fig. 4.3 and Figs. 4.5—4.7 for a model with N = 10,
n = 3, a = 0.84, 6 = 0.67, ,a = 0.015, and £0 = (3;, = {c = 0. Then 6“. =

—0.00701

 

 

(from Equation (D2) of Appendix D) and 633p —0.00706 (from Equation (4.46) of
Section 4.4.2).

Figure Tuning 6 'y a a aapp V ores
4.3 — — 0.169 - - — 0.50 0.442
4.5a Over +0.15 0.127 3.45 3.473 3.487 0.01 0.338
4.51) Over +0.15 0.127 3.45 3.473 3.487 0.25 0.364
4.5C Over +0.15 0.127 3.45 3.473 3.487 0.50 0.443
4.6a Under —0.07 0.194 2.790 2.811 2.820 0.50 0.468
4.61) Over +0.07 0.147 3.210 3.232 3.244 0.50 0.423
4.7a Under —0.00351 0.169 2.989 3.011 3.021 0.50 -
4.7b Exact 0 0.168 3 3.021 3.032 0.50 -

 

cyclic systems, and in the absence of parameter mistuning, an integer engine order
1 g n < N excites only mode p = n + 1, which is clear from Equation (4.22). (See
Section 2.4.5 for a description of the resonance structure for the case when n 2 N.)
For a given engine order, the corresponding actual resonance, that is, the value of a

for which

710' : Q]?+l)(0’),

or, equivalently nil = w[7]+1)(Q) (4.27)

is indicated by a circle in Figure 4.3a for each of the engine orders considered, and the
corresponding frequency response curves are shown in Figure 4.3b for f = 0.01 and
£1, 2 6C = 0. The resonant frequency for the n = 3 case is indicated in Table 4.2, along
with other data corresponding to Figures 4.5—4.7; these are explained in Section 4.3.2

and Section 4.4.2.

4.3.2 Response with the Absorbers Free

We now turn to the forced dynamics of the overall 2N —DOF system, which are gov-

erned by Equation (4.6), and employ an approach similar to that of Section 4.3.1. In

90

0711 987 6/5 4 38.0.

 

 

 

 

 

 

2[ // / / (16.0, d
5 n. .
%§
7 \—-—-2
’,J x],
16.0 0
A A n A A U
0.2 0.4 0.6 0.8 1
(a) Campbell Diagram
longl
21 9876 5 4 3 2e.o.
1 .
0» 1 1‘
1 e.o.
j l u
-2 éyeag‘
. _ . 0.
0.2 0.4 0.6 0.8 1

(b) Frequency Response Curves

Figure 4.3. (a) Campbell diagram for N = 10, or = 0.84, 6 = 0.67, :7 = 0.169, a = 0.015,
V = 0.5, and engine orders (e.o.) n = 1, 2, . . . , N — 1 and (b) the corresponding frequency
response curves with f = 0.01, and £1, = {c = 0.

91

the present case the system matrices M, C, and K are block circulant, and one can
(block) decouple these equations to a set of N, 2—DOF forced oscillators by employing

the result given by Equation (2.11) on page 15.7

MODAL ANALYSIS

We introduce the change of coordinates
q = (E 18> I)u, or 2,; = (egg) I)u, p E N (4.28)

where E is the N x N complex Fourier matrix and ep is its pth column, <8) is the
Kronecker product (these are defined in Section 2.2.2 and Section 2.2.1), I is the
2 x 2 identity matrix (the dimension of I corresponds to the number of DOF in
each sector), and u = (111, 112, . . . , uN)T is a vector of modal, or cyclic coordinates
with each up = (52p,gp)T Substituting Equation (4.28) into Equation (4.6) and
multiplying from the left by the unitary matrix (E (X) I)” = (E718) I) yields a system

of N block decoupled equations, each with two DOF. They are given by
Mpug + Cpu], + Kpup = (e338) I)fej"m, p E N (4.29)

where (e;f® I)f is the pm 2 x 1 block of (EH69 I)f. Figure 4.4 illustrates the trans-
formation of the single 2N —DOF system given by Equation (4.6) to a system of N
block decoupled 2—DOF forced oscillators given by Equation (4.29).
The 2 x 2 mass, damping, and stiffness matrices associated with the pth mode
follow from Equation (2.12) of Section 2.2.4 and are given by
191,, = M
(3;, = C + 2Cc(1 — cos (pp) , p E N (4.30)
KP = K + 2KC(1 — cos app)
where (,0,, is defined by Equation (2.3), the elements of M, C, and K are defined in

Table 4.1 and their attendant parameters are given in Table 3.1, and the coupling

 

7The number of DOF in each decoupled system is that of an individual sector, in this case two.

92

® Absorber

Blade
® Rotor

H .1842
()pEN

     

t
,, \Sec or
I

Modal (Cyclic)
Transformation

(a) A Single Coupled 2N—DOF System (b) N Decoupled 2—DOF Systems

Figure 4.4. The topology of a bladed disk assembly fitted with absorbers in (a) physical
Space and (b) modal space. The modal transformation q(r) = (E (8 I)u(r) reduces the
cyclic array of N, 2—DOF sector models (B,A), which together form a 2N—DOF coupled
system, to a set of N, 2—DOF block decoupled models (8p, AP).

matrices Cc and Kc are defined by Equation (4.4). In light of Equation (4.22), the

pth modal forcing vector takes the form

Hf”
H ‘_ ep 11
(ep®I)f—[ 0]

{Wfl p=n+1 (4.31)

0, otherwise

where f11 is the system forcing vector for the case when the absorbers are locked in
their zero positions relative to the blades, f = (f, 0)T is the sector forcing vector, and
= (0, 0)T. Since only mode p = n + 1 is excited, un+1(r) is the only nonzero modal
response in the steady-state.
Assuming harmonic motion, and in light of Equation (4.31), the pth steady-state
modal response follows easily from Equation (4.29) and is given by

(r) [WWI fem” p=n+1

(4.32)
0, otherwise

93

where
2,, = Kp — n202Mp + jnon, p E N (4.33)
is the pm modal impedance matrix. The response of sector i (in physical coor-

dinates) follows from the transformation given by Equation (4.28) with uSS(r) =

(0, . . . ,0, uff+1(r), O, . . . ,0)T and is given by

ZS-S(T) = 2641f emiejnm, i E N (4.34)

’l

where 1117194) 2 ej‘f’i has been employed. From Equation (4.34) it is clear that each
blade / absorber combination behaves identically except for a constant phase shift from
one sector to another, which is captured by the inter-blade phase angle (1),. This
approach offers a significant computational advantage over the brute-force solution

to the full 2N —DOF system, as given by Equation (4.16).

EIGENFREQUENCY STRUCTURE AND CONDITIONS FOR RESONANCE

The 2N dimensionless natural frequencies of the system are defined implicitly by the

characteristic polynomial
det (K — 02M) = 0,

the solution of which can be quite involved for any reasonable bladed disk model.
This effort can be significantly reduced, however, by instead using the modal matrices
defined by Equation (4.30). We recall that each Mp and Kp follow from a unitary
(similarity) transformation of M and K and hence the system natural frequencies
are preserved. These eigenfrequencies follow from the N, second—order characteristic

polynomials det (Kp — QQMP) = 0, or
det(K — 072M + 2KC(1 — cos $12)) = 0, p e N (4.35)

where the sector mass, stiffness, and coupling matrices are defined in Table 4.1 and by

Equation (4.4). Equation (4.35) features the same cyclic term, i.e., Equation (2.39),

94

that was encountered in Section 2.4.3 and we thus expect Similar eigenfrequency
characteristics to the ones described there and also in Sectioni 4.3.1. If P is the
number of DOF in an individual sector (in the present study P = 2), then there are P
natural frequencies 07(1)) corresponding to each p E N. There are, in this case, N such
groups of P = 2 natural frequencies of the overall system. The multiplicity of these
groups of natural frequencies is identical to that of the individual eigenfrequencies
described in Section 2.4.3.

If vp is an eigenvector of the pt" decoupled modal system, then the corresponding
normal mode Of the overall system is ep ®vp, where ep is the pth modal vector for the
case when the absorbers are locked relative to the blades.8 With the exception of the
p = 1 mode, vp is influenced by the overall system configuration, and in particular
by its elastic coupling, which is clear by inspection of the modal stiffness matrices
Kp. In a particular mode of vibration, the blade and absorber in each sector oscillate
either in phase or out of phase relative to one another with amplitudes that depend on
the strength and nature of the inter—sector coupling, and these features are captured
by vp. The dynamics of each sector are identical, except for a constant difference
in phase from one sector to another, and this is captured by ep. Hence the modal
configuration of the overall system is described by a composite of these two vectors,
which is mathematically given by ep (8) vp.

The 2N dimensionless natural frequencies [0% (p E N) are plotted in Figure 4.5
in terms of the rotor speed a for N = 10, n = 3, for a particular sector model, and

for various levels of the inter-blade coupling V. (The natural frequencies a]? and

17222 are also Shown for reasons discussed below. Also, for quick reference here and
in subsequent chapters, Table 4.3 gives a selected list of commonly used undamped

natural frequencies.) In these Campbell diagrams, the N natural frequencies 122]”

 

8When the absorbers are locked each of the the system matrices is a circulant. Since all circulant
matrices share the same linearly independent eigenvectors, which are the columns of the Fourier
matrix, the system normal modes are ep with p E N.

95

Table 4.3. Selected list of dimensionless undamped natural frequencies.

 

Freq. Eqn. Description

 

(D?) (4.35) Of the coupled system corresponding to in-phase modes

079)) (4.35) Of the coupled system corresponding to out-of-phase modes

0(1)) (4.25) Of the coupled system if the absorbers are locked relative to the blades
([111 — Of an isolated blade without an absorber

(.022 (4.10) Of each absorber if the blades are locked relative to the rotating hub

 

branching from a = 0 (since N is even there are (N — 2) / 2 = 4 repeated pairs) cor-

respond to in—phase modes, wherein the absorber/ blade combination in a particular

— (p)

sector oscillates in phase. The remaining N natural frequencies (.02 have the same
number of repeated pairs and correspond to out—of-phase modes. As Shown in Fig-

[10) and also 1.03))

ure 4.5a, the frequencies (I) are nearly coincident when the inter-blade
coupling is weak, that is, when V is small and they spread out for increasing V, which
is shown in Figure 4.5b and Figure 4.5c. In the absence of inter—blade coupling, 61]!) )
are identically coincident (as are big») and there are exactly P = 2 distinct natural
frequencies, each with multiplicity N.

The frequency loci in Figure 4.5 exhibit the classical eigenvalue veering phe-
nomenon, or mutual repulsion of the eigenfrequencies [104,105], which arises due
to small the dynamic coupling (via the absorber mass) between the blades and ab-

(P)

sorbers. To see this, we focus on Figure 4.5a, where the sets of frequencies ‘31 and
tag”) are mutually nearly coincident. This plot also Shows the natural frequencies a]?
(resp. (.322), corresponding to the case when the absorbers (resp. blades) are locked
relative to the blades (resp. rotor). As the rotor speed or is increased from zero, the

[12) (resp. avg») initially lie close to (.322 = no (resp. 10%)), where

natural frequencies (I)
the absorber tuning order a is defined by Equation (4.11). (For zero rotor speed

15]”) and 02222 are, in fact, coincident and each has the same initial Slope of a.) They

96

3)

Q Q Q
ll
E1
(\3
N

3 :1 §>

II II

“WW
9
01
O

H
U‘

.9
an
\

Weak Coupling
,A
K m

 

 

 

   

 

 

 

7 ‘ ‘ o
0 2 0 4 0.6 0 8 1
(b) u = 0 25
/'— 710 - £022
2 —no (3 e o )
5 n.d
4
ED 1 5 d’gpjffl ’9 - ~ 5: - ,, :11; :f' f 3
Q “7:: ”' ‘ 7 : - i:'\—2
g ———_,___,._.—::’ / ; _ F: I -’ .:"(‘Ii)'--)- " x1
0 1 :::__,_ —:::// ~ (DI 0
1:0
:3
8 0 5
:73 .
o

 

 

0.2 0.4 016 0.8 1
(c)V=0.50

Figure 4.5. Campbell diagrams showing the engine order (e.o.) line no and the dimen-

sionless natural frequencies wi’i’, 0722 = no, and 0]”; versus the dimensionless rotor speed
0 for N = 10, n = 3, a = 0.84, 7 = 0.127, 6 = 0.67, and ,u = 0.015 with: (a) V = 0.01;

(b) V = 0.25; (C) V = 0.5.

97

exhibit veering near the intersections of a]? and Q22, and for large rotor speeds the

eigenfrequencies 15]” asymptotically approach a]? for each p E N. However, the

frequencies wép) nearly track (1222 = no as o becomes increasingly large, but with a
slight offset in slope. This is Shown in the inset of Figure 4.5a. In fact, it can be
shown that

(Jig?) —+ no as o ——> OO, (4.36)

where the critical absorber tuning order 71(6) > a is defined by Equation (D.1) in
Appendix D. This is a crucial observation, one that is exploited in the absorber
tuning of Section 4.4. Finally, note that there is a fixed relationship between a
and a, which is nearly linear for a > 1. Once the absorber mass it and its tuning
order a are prescribed, then the critical tuning order is automatically set and can be
approximated by

mm, = (1 + c1214) 11, (4.37)

which works quite well for a > 1 and for reasonable choices of a and ,u.

Possible resonances can be identified in Figure 4.5 by the intersections of the
eigenfrequency loci 063(0) with the order line no, and they correspond to rotor
Speeds o = ores for which no = Q[,%(o). However, it was shown in Section 4.3.2 that

only mode n + 1 is excited in the steady-state, and hence there is a system resonance

only when

no 2 Q[7l;l)(o), or, equivalently nil = w]"2+1)(f1) (4.38)

is satisfied. These resonances are indicated by circles in Figure 4.5 and they are sum-
marized in Table 4.2 on page 90 along with other relevant data. The main objective
of this chapter is to select the absorber parameters to avoid such resonances over a
range of rotor operating speeds; this is the subject of the next section.

As a final note, the inter-blade coupling V can be quite small—on the order of 1%

or less—but much larger values are also possible. Aerodynamic coupling also exists

98

and can be Significant in terms of both stiffness and damping. In order to show clearly
which modes are excited, and also the effects of absorber (de)tuning, a rather large
(possibly unrealistic) value of the coupling will be employed in the ensuing numerical

analysis; this does not qualitatively affect the approach nor the conclusions.

4.4 Absorber Tuning

Absorber tuning refers to a particular choice of absorber parameters to attenuate,
as much as possible, the response of the primary systems (blades) over a range of
operating speeds, and in particular near resonance. This is done by prescribing the
dimensionless parameters a, 7, and a, which in turn specify the absorber mass m,
the radius r of its path, and it’s placement along the blades, respectively. It is shown
in Section 4.4.1 that, in the absence Of damping, there exists an absorber tuning such
that full annihilation of the blade vibrations is possible, although this may require
large-amplitude vibrations of the absorbers. This tuning is accomplished by matching
the order of the isolated absorbers to that of the excitation, just as it is done with
frequencies in the classical dynamic vibration absorber [99], and also with orders for
the centrifugal pendulum vibration absorber [7]. (See Section 2.5.) In the presence
of small absorber damping, however, it becomes impossible to eliminate the blade
vibrations completely, a topic that is briefly addressed in Section 4.5. The effects of
detuning the absorbers relative to the excitation order is explored in Section 4.4.2.9
It is shown that overtuning the absorbers results in only one system resonance over
all possible rotation speeds, even though there are two DOF per sector and there
are N such sectors, and the same is true for most values of undertuning. However,

there exists a small region of absorber undertuning, bounded on one side by the exact

 

9In this work detuning means that all absorbers are identically over- or under-tuned relative
to n. This is not to be confused with mistuning, which refers to small random uncertainties in
the system parameters. In the turbomachinery literature, detuning and mistuning are often used
interchangeably, but they must be clearly distinguished in this investigation.

99

tuning (zero detuning), for which there are no system resonances. This no-resonance
gap motivates a particular tuning strategy, which offers a Significant reduction of the
blade amplitudes, and it is robust to random perturbations of the system model.

Consider again the Campbell diagrams in Figure 4.5. Whereas n is fixed for a par-
ticular engine order excitation, the tuning order 5 depends on the model parameters
a, 6, and '7 (these are prescribed by design), and the additional choice for the di-
mensionless absorber mass ,a sets the critical tuning order 71(5). The tuning strategy
employed here is to simply choose these parameters to optimally orient the line no
(and hence no) relative to no in the frequency-o plane. This in turn sets the asymp-
totic behavior of the system natural frequencies 07]]? and hence prescribes the system
resonance structure. It is clear that by choosing no 2 no (this corresponds to zero
detuning) there will be no crossings of the order line no and the natural frequency loci
1:2[13(o), and hence there will be no system resonances over the full range of possible
rotor speeds. However, Slight errors in this tuning can introduce a resonance, and
therefore such a design is not robust. One can more generally avoid resonances by
choosing parameters such that a S n < 11(6). This is clear from the large—o asymp-
totic behavior of dig), and specifically from the inset of Figure 4.5a. The existence
Of this finite, but narrow, tuning range allows one to design an absorber system with
some level of robustness to parameter uncertainties.

The arguments described above are developed in detail in the next section using

the steady-state system response of Section 4.3.2, and in the context of absorber

detuning in Section 4.4.2.

4.4.1 Exact Tuning

It is customary to introduce the tuning order a as one Of the absorber parameters,

and this is done in the present study via the substitution

 

(4.39)

thereby replacing 7 with a in the formulation. With zero system damping, i.e., 5a 2
5b = 56 = 0, and after some simplification, the steady-state response described by

Equation (4.34) can be reduced to

 

 

[343] = [if] emiejnm, i E N (4.40)
i
where
X #12022 — 52)
— r
fa? (gm? — a?) + n2(1+ 52)) (441)
Y _

(1 + %)F
are the blade and absorber steady-state response amplitudes and
r = 1152(1 + 52671202 + 52012 — a2)
+ (n.2 — n2) (ua0(1+ a?) — 712(n2 — 6))02
+ 2V2'fi.2(n2 — n2)(1— cos 1,97,“),
where 99, is defined by Equation (2.3). The ideal, or exact absorber tuning follows by

inspection of the first entry of Equation (4.41) and is given by
a = n, or 11722 = no = no. (4.42)

If the system is tuned according to Equation (4.42) the blade and absorber amplitudes

reduce to

fn2 , (ft = n) (4.43)
ua2(1+ %)(1+ n2)o2

which shows that the blade vibrations can be eliminated completely. In this case

 

the absorber amplitudes are inversely proportional to the mass ratio a and also to
a(oz + (5). It is therefore desirable to make the absorber masses large relative to the

blade mass and to place them as close to the end of the blades as possible. In practice,

101

however, there are limits on the size and makeup of the absorber masses (typically )1
is very small, on the order of 10"2 to 103) and on their placement relative to the
blades. The negative Sign in Y implies that the absorbers oscillate out of phase with
respect to the excitation. Physically, this means the absorbers exert forces on the
blades that identically counter the action of the applied loading for all time and for

all rotor Speeds.10

4.4.2 Absorber Detuning and the NO-Resonance Zone

By implementing the absorber tuning given by Equation (4.42) one is simply setting
the natural frequency of the isolated absorbers to the excitation frequency, that is,
1022 = no = no, and the absorbers are said to be exactly tuned. Again, we emphasize
that the said tuning is valid at all rotation speeds, a feature that is made possible by
the structure of 1.7222(o) :2 no. However, any perturbation of the model or absorber
parameters, due to in-service wear, environmental effects, and so on, will invariably
destroy the exact tuning. To account for such effects, and to allow for intentionally

detuned designs, we let

a = n(1+ a), (4.44)

where 6 is a detuning parameter. Perfect, or exact tuning corresponds to 6 = 0, while
undertuning (resp. overtuning) corresponds to 6 < 0 (resp. 6 > 0).

Figure 4.6 and Figure 4.7 depict the blade/ absorber frequency response amplitude
curves and also the natural frequency loci for a set of four representative detuning
values. (The corresponding tuning orders, detuning data, and the resonant rotor
speeds are given in Table 4.2 on page 90.) In these plots we take f = 0.01 and use
the parameters employed in Figure 4.5c. The solid lines in the blade and absorber
response curves Show the response amplitudes as a function of rotor speed. The

dashed lines correspond to the blade/ absorber amplitudes when the absorbers are

 

1(These results remain valid even for varying rotor speeds, that is, o = 0(7), so long as the
variations occur on a much longer time scale than the dynamics of the blades and absorbers.

102

locked in their zero positions relative to the blades; these curves are used for reference
to assess the dynamic effects of the absorbers.

Overtuning the absorbers (i.e., setting 6 > 0) increases the Slope a of11222(o) = no
relative to the engine order n in the frequency-o planes, and it is clear from Figure 4.6b
that a resonance of the iII-phase mode corresponding to @[HU is guaranteed. For
sufficiently large undertuning such that 6 < 6“ < 0 (with 6a defined below), the
out-of—phase mode corresponding to £281“) is excited near resonance; an example of
this situation is Shown in Figure 4.6a for 6 = —0.07. One of the more interesting

findings of this chapter is that there are no system resonances for absorber tuning

values that satisfy

61:1 < 6 S 0, (445)

where 1' cr is the critical absorber undertuning and is given implicitly by Equation (D2)
of Appendix D. Zero (resp. critical) detuning, that is, 6 = 0 (resp. 6 = 6“),
corresponds to a = n (resp. a = n). An example tuning within the no-resonance gap
defined by Equation (4.45) is shown in Figure 4.7a for 6 = 6m / 2 = —0.00351, where
6“ = —0.00701, and the perfectly tuned case is shown in Figure 4.7b. These cases
clearly demonstrate the effectiveness of properly tuned absorbers. The resonance
that occurred at ores = 0.442 when the absorbers are locked is completely eliminated,
and the response amplitudes of the blades are significantly reduced (or eliminated
completely) over the full range of possible rotor Speeds.

Another way to visualize the no-resonance gap defined by Equation (4.45) is to
construct a plot of the rotor speeds corresponding to (possible) resonance(s) versus
the absorber detuning parameter 6. Such a plot is shown in Figure 4.8 for the same
parameters used in Figures 4.5-4.7. In this diagram, the no—resonance gap is identified
by the shaded region between the dotted lines corresponding to 6 = 0 (zero detuning)
and 6 = 6“ (critical detuning), where 6cr = —-0.00701 for this case.

The extent of the no-resonance gap depends on the absorber parameters and the

103

’ Absorbers Locked 2 ’ Absorbers Locked

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. . - - . o . .\ . - . a
0.2 0.4 resO.6 0.8 1 0.2 0.4 0re30.6 0.8 l
eff-l 6019
2 [ 2 .
Resonance /
1.5 » I 1.5 ~ 4’
1 , 1 .
j : Resonance
0.5 . 0.5 . E
= a
. I I\ . . 4 0 1 LJ\ . . . 0-
0.2 0.4 0r930.6 0.8 1 0.2 0.4 Ores 0.6 0.8 1
(a) 6 = —0.07 (b) 6 = +0.07
(Undertuned) (Overtuned)

Figure 4.6. Absorber and blade frequency response curves and Campbell diagrams for
N = 10, n = 3, a = 0.84, 5 = 0.67, a = 0.015, V = 0.5, f = 0.01, and zero damping:
(a) 6 = —0.07 (undertuned); (b) 6 = +0.07 (overtuned); (— -) frequency response with the
absorbers locked. See Table 4.2 on page 90 for the corresponding tuning order data and
resonant rotor speeds ores.

104

 

2
Absorbers Locked
1 .
. I
0 1
II
I\
—1 I \
I \
2 ‘*

 

 

0.2 0.4 0.6 0.8 1

0.5

 

0.2 0.4 0.6 0.8 l

(a) [B : [Ber/2
(Slightly Undertuned)

longl
3 .

Absorbers Locked

OHM

 

 

 

 

0.2 0.4 0.6 0.8 1

 

 

 

 

 

0.5 .

 

 

0.2 0.4 0.6 08 1

(b) 6 = 0
(Perfectly Tuned)

Figure 4.7. Absorber and blade frequency response curves and Campbell diagrams for
N = 10, n = 3, a = 0.84, 6 = 0.67, 11 = 0.015, V = 0.5, f = 0.01, and zero damping:
(a) 6 = 6cr/2 = —0.00351 (slightly undertuned); (b) 6 = 0 (zero, or perfect tuning);
(— —) frequency response with the absorbers locked. The critical absorber detuning is

ficr =

resonant rotor Speeds ores.

—0.00701. See Table 4.2 on page 90 for the corresponding tuning order data and

105

—.1 r—No-Resonance Gap

 

       

1 ' A: '.

:vé
i .13
fol
§N5 — Resonance, ares
g g __ Possible
3:: Resonance
5,.de
. H.q;
2 I'd

b em:
a: :o 4
of” ‘
:nfn,
3°95 .
and;
:o3
;z

 

 

 

 

-030 2 0 0.62 0.04

hi

Figure 4.8. Rotor speeds 0% corresponding to resonance (—) and possible resonant speeds
(— —) in terms of the absorber detuning fl for N = 10, n = 3, a = 0.84, 6 = 0.67, n = 0.015,
and 1/ = 0.50. The no-resonance gap is defined by ,Bcr < 6 S 0, where 3a = —0.00701.

engine order, but is independent of the inter-blade coupling V. The sensitivity of the
gap to variations in these parameters is indicated in Figure 4.9, which follows from
Equation (D2) in Appendix D. (A simpler approximate expression, which works
quite well over a wide range of parameters, is described below.) In all cases, the
sensitivity is most pronounced for small engine orders and it decreases for increasing
n. As shown in Figure 4.9a (resp. Figure 4.9b) the critical detuning ficr exhibits
near-linear (resp. -quadratic) behavior in terms of p (resp. a), and by inspection
of Figure 4.9c it is nearly independent of 6 for most engine orders (n > 2). Note
that ficr vanishes (implying that the no—resonance gap vanishes) in the absence of the
absorbers (u = 0) or when the absorbers are attached to the periphery of the rotor
( = O). This is consistent with intuition since zero-mass absorbers cannot provide
the required loads to counter the action of the excitation on the blades. Also, if
the absorber and blade pendulum attachment points coincide on the circumference

of the rigid rotor, their dynamics become independent. It is clear, therefore, that

106

l e.o. 2 3 4 20 e.o. 1 e.o.
/

 

 

 

 

 

 

 

3 3
2.5 25
8 2 8 2
H 1—1 2
E 15 )515 /3
n n
l 1 I 1 /4
‘20 e.o.
0.5 05
p a
0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1
(a) (b)
O
O
H
X
6 n
04 08 12 16 2 5 10 15 20 25
(C) (d)

Figure 4.9. Critical undertuning *5“ x 100 of the absorbers versus (a) the dimensionless
absorber mass )1 (with a = 0.84 and 6 = 0.67), (b) the dimensionless distance from blade
base to absorber base point a (with 6 = 0.67 and p = 0.015), (c) the dimensionless radius
of rotor disk 6 (with a = 0.84 and p = 0.015), and (d) the engine order n (with a = 0.84,
6 = 0.67, and n = 0.015). The dashed line in ((1) corresponds to the large-n approximation
of flu given by Equation (D3) in Appendix D.

both a (absorber mass) and a (absorber placement relative to its attendant blade)
are coupling parameters in the sense that their departure from zero implies dynamic
coupling between the blades and absorbers.

The parameter trends described above, and in particular those shown in Fig-
ure 4.9a and Figure 4.9b, motivate a two-parameter expansion of the critical detuning

ficr about (,u, a) = (0,0). This results in the simple approximation

C‘ _ —2n2(n2 _ (5)1102 + 003: O3), (446)

107

which works quite well over a large range of realistic parameter values. More im-
portantly, Equation (4.46) clearly shows that the extent of the no—resonance zone
depends primarily on the absorber mass and its placement along the blade. It can be
widened by increasing either ,u or a. For any engine order n > 2, Equation (4.46) also
shows that firm is essentially independent of the rotor diameter, which is consistent
with the exact curves shown in Figure 4.9.

Based on the above analysis it is reasonable to choose [3 = Her / 2, which is simply

the average of the exact and critical detuning. Then Equation (4.44) becomes
a = n(1+ 3r) (4 47)
T o

and the engine order n is very close to (but not exactly) the average of am) and
73.. Such a tuning strategy, which was used in the example shown in Figure 4.7a,
guarantees (in the absence of damping) no system resonances and it offers good
robustness to parameter and model uncertainties.

The effects of system damping are investigated next.

4.5 The Effects of Damping

When an undamped absorber is exactly-tuned according to Equation (4.42), its ac-
tion identically counters that of the engine order excitation on the blade to which
it is attached. This results in a full elimination of blade motions independent of the
blade and inter-blade damping levels. However, any level of absorber damping (or
detuning) will give rise to residual blade motions, in which case coupling and blade
damping levels will also affect the response. The aim of this section is to numerically
characterize the effects of damping on absorber performance, particularly within the
no—resonance zone described in Section 4.4.2. It is shown that the no-resonance gap
persists in the presence of sufficiently small absorber damping and that it is essentially

unaffected by realistic blade and inter-blade damping levels. We begin by describing

108

the effects of the absorber damping for tuning values outside the no-resonance zone.

To understand the effects of absorber damping on the overall system dynamics, it
is instructive to consider two extremes: when the absorbers are free and undamped
and when they are locked relative to the blades, which correspond to the limiting cases
of {a = O and {a ——> 00, respectively. When the absorbers are free, undamped, and
tuned outside of the no—resonance zone (and in the absence of blade and inter-blade
damping) there is a single resonance at a = ares, which corresponds to the intersec-
tion of the engine order line no with 0:71“) or LUSH” , and it is defined implicitly by
Equation (4.38). For infinite absorber damping the absorbers are essentially locked
relative to the blades and hence each blade/ absorber combination has the same ampli-
tude, that is, |a:,-(T)| = |yi('r)| for 2' E N. In this case there is also a single resonance
(denoted in this section by a = 0L), which is defined implicitly by Equation (4.27).
An example of ares and a L is shown in Figure 4.10a and the corresponding frequency
response curves are depicted in Figure 4.10b—c. There must be a continuous spectrum
of frequency responses between the {a = 0 and Ea —+ 00 extremes, which is charac-
terized by a resonance shift toward (IL. This is shown in Figure 4.11 for the same
parameter values used in Figure 4.10 with 5b = {C = 0 and for various levels of the
absorber damping {(1. As the absorber damping is increased from zero, the resonance
point shifts (in this case to the right) toward 0L, and the peak blade/ absorber am—
plitudes initially decreases, which is shown in Figure 4.11a—d. By further increasing
{a the resonance point continues to evolve toward a L and the peak amplitudes begin
to increase, which is shown in Figure 4.11e—f. Finally, in the limit as Ea —> 00 the
frequency response becomes essentially identical to the locked absorber case. This is
shown in Figure 4.11g.

A qualitatively similar trend can be observed when the absorbers are tuned within
the no—resonance zone, except that there are no system resonances for the limiting

case of 5a = 0, and hence if £0, is sufficiently small. This is shown in Figure 4.12a-d

109

 

 

 

 

 

 

 

I

I

I

E
1—1 .
‘io

 

 

  

 

I
0.5 » :
i
. I . . . . 0-
0.2 \Ores 0.4 \aL 0.6 0.8 1
Absorbers Free/Undamped (Ea = 0)
log I Xl Absorbers Locked (fia—y oo)
3 .
2 - |
|
1 - l
(b) o » l
_1 b | //:\\
_2‘. ___,l.. ’ i\‘-“‘
4 i 1 .- T — _ ‘ - 3 0

 

 

0.2 \ares 0.4 \oL 0.6 0.8 1

 

 

 

Figure 4.10. (a) Campbell diagram showing the resonant rotor speeds ores and UL and
the corresponding (b) blade and (c) absorber frequency response curves for a model with
N = 10, n = 3, a = 0.84, B = 0.01, 6 = 0.67, p = 0.05, V = 0.5, f = 0.01, and (b = 5c = O.

110

 

—-— Absorbers Free (50 2 0, 5b = {c = 0)
Absorbers Free (Ea = {b = 5c = 0)
. Absorbers Locked (5a = 5b = {c = 0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13ng“

(a) £a=0

(b) a=10—6

(c) £a=10’5

(d) £a=10‘4

(e) €a=10_3

(f) €a=10_2

(g) {cl-+00

 

 

 

 

 

7576 0.8

Figure 4.11. Blade and absorber (free and locked) frequency response curves for overtuned
absorbers (B = 0.01), for the same parameter values used in Figure 4.10, and for various
levels of the absorber damping 50.

111

(a) £a=0

('0) {(1:10-6
(c) £..--:10‘5
(d) aflo—4
(e) ado—3
(f) 50:10-2
(g) Err—+00

 

— Absorbers Free (£0 2 0, £1, = {c = 0)
Absorbers Free (Ea = 5b = (C = 0)
Absorbers Locked (Ea = 5b = 6c = 0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

log IXl log IYI
3 _ 3
2 2 I
1 i 1 i
0 : O 1
_1 ,H, _1 /\\ *—
_2 HR_ ”2 ~~~~~~ ’,2 \“~—-
- ~ ‘ ' '~ 0 f “““““ - a
0.2 0.4 0.6 08 l 0 2 04 06 0 8 1
log IXI lolel
3 » 3
2 2 .
(15 I l I
z 0 '
._l _1 F In *—
—2_.. ...... ” Nan“
‘ 0' “‘“ 0
1 02 0.4 06 08 1
log WI
3
: 2 '
1) 1 1 i
0 : 0 l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—-_~_‘_..

 

 

0.4 0.6 F 0.8

Figure 4.12. Blade and absorber (free and locked) frequency response curves for absorbers
tuned within the no—resonance zone (B = BC, / 2 = —0.00351), for the same parameter values
used in Figure 4.10, and for various levels of the absorber damping (a.

112

for [3 = flcr / 2, 5b = 5C = 0, and for various levels of the absorber damping 60. A
system resonance is born if the absorber damping is sufficiently increased, which is
shown in Figure 4.12e-g. In these cases the absorber’s performance is so severely
degraded by the presence of damping that they are no longer effective in attenuating
blade motions, even with proper tuning. However, for reasonable absorber damping
levels there are no system resonances over the full range of rotor speeds, and in this
sense the no—resonance zone is seen to persist for sufficiently small 50. Figure 4.12b
shows a representative set of frequency response curves for a typical level of absorber
damping.

Finally, it can be shown that the blade and inter-blade damping has a much less
dramatic effect on the system dynamics. By increasing either Q, or EC the blade and
absorber amplitudes are simply reduced and, for absorber tuning outside of the reso-
nance zone, there is generically no significant shift in the resonance point. Physically,
this is a sensical result; the presence of blade or inter-blade damping actually helps
the absorbers achieve attenuation of blade motions, whereas an increase in {a does

the exact opposite.

4.6 Concluding Remarks

An implementation of order-tuned vibration absorbers to a linearized, cyclically sym-
metric bladed disk assembly has been investigated. A standard change of coordinates
based on the cyclic symmetry of the system was employed to reduce the governing
2N equations of motion to a set of N, reduced—order equations, from which an ab-
sorber tuning strategy was formulated. One of the main findings of this chapter, and
indeed of this entire thesis, is the existence of a no-resonance zone, that is, a range of
absorber undertuning values for which there are no system resonances over the full
range of possible rotor speeds. By tuning each absorber within this generally small

(but finite) range, resonance can be avoided altogether and there is at least some level

113

of robustness to parameter uncertainty. The extent of this no—resonace gap depends
primarily on the mass of the absorber to that of the blade and on its placement along
the blade length, especially for larger engine orders. The gap can be widened by
increasing the absorber mass and / or by placing it further away from the root of the
blade, effectively strengthening its dynamic potential to suppress blade motions.
These fundamental results are now generalized in the next chapter to include

first-order nonlinear effects.

114

CHAPTER 5

Forced Response of the Nonlinear System

5. 1 Introduction

The linear results of Chapter 4 are now generalized to include the basic first-order
effects of nonlinearity, which is introduced via the absorber path geometry. Together
with the linear design recommendations of Section 4.4, the aim of this chapter is
to possibly exploit nonlinearity to further improve the absorber performance, partic-
ularly for linear tuning within the no—resonance zone. It is therefore of particular
importance to determine if the no-resonance zone persists under increasing absorber
path nonlinearity. Also, it is well known that for the lightly damped and weakly
coupled cyclic systems under consideration, there may be a host of solution types
other than the desired traveling wave response [106—108] in which all sectors behave
identically except for a constant shift in phase among adjacent sectors. If nonlinearity
is to be exploited, or if it is otherwise present and unavoidable in the sector models,
these additional instabilities (if they exist) must be addressed as a part of the design
process.

It is convenient to employ the generalized, two-parameter familiy of paths that
were developed in Section 3.4.4 to introduce the nonlinearity. These allow the final
path design to be specified directly by choosing a linear tuning order a (this was

carried out in Section 4.4 in terms of the linear order detuning )3) and a nonlinear

115

tuning parameter 7]. According to the linear theory there is a continuous spectrum
of (under)tuning values for which there are no system resonances over the full range
of possible rotor speeds. Proper linear design therefore involves tuning the absorbers
within this no-resonance zone. The path parameter 17 acts essentially as the strength
of the path nonlinearity, which has a continuous range from softening (77 < O) to
hardening (n > 0). Its selection forms the main focus of this chapter.

The nonlinear sector models from Section 3.4 are employed, along with the two-
parameter family of paths described above. These are systematically reduced via
scaling and averaging to a set of nonlinear averaged sector models, which forms a
basis for the analytical and numerical investigations that follow. In addition to the
desired traveling wave response, the averaged models are general enough to capture
solutions with slowly-varying amplitudes and phases in individual sectors. However,
we focus only on traveling wave responses, where all of the sectors behave identically
but with a fixed phase difference among adjacent sectors, and on instabilities of this
response, which would result in bifurcations to other response types. The analysis
is carried out first for the isolated nonlinear system, consisting of a single linear
blade and nonlinear absorber. This allows for a complete description of the effects of
nonlinearity on the sector dynamics without additional complicating features due to
inter-sector coupling. When coupling is present, and under a traveling wave response,
the nonlinear system qualitatively features these same dynamics on a sector-to-sector
basis, but there may also exist bifurcations to non-traveling wave motions in addition
to the usual jump bifurcations associated with the isolated sector. In fact, it is shown
that the multi—sector traveling wave response corresponds directly to an equivalent
single sector model whose natural frequency is shifted by a specific amount that
depends on the coupling and the mode being excited.

It is shown that the underlying linear resonance structure—and hence the no-

resonance gap and desired linear absorber tuning—qualitatively persists in the pres-

116

ence of nonlinearity, provided that the excitation and path nonlinearity are sufficiently
small. Moreover, given any linear tuning strategy and for zero damping, there exists
a nonlinear tuning that guarantees a branch of solutions for which there are zero
blade motions relative to the rotating hub. However, for proper linear undertuning
this gives rise to an undesirable hardening path with potentially problematic aux-
iliary resonances and it is highly susceptible to parameter uncertainty. Even more
importantly, the nonlinear tuning criterion depends on the rotor speed as well as
the strength of the excitation and is thus effective near a single operating condition,
much like the frequency—tuned DVA of Section 2.5.2. These findings suggest that it is
impossible to exploit nonlinearity to further improve the absorber performance, and
it is therefore desirable to maintain nearly-linear absorber motions. Should nonlin—
earity be unavoidable, the results clearly show that softening characteristics are more
desirable than hardening, where the former simply sets an upper limit on permissible
rotor speeds and the latter involves potentially problematic auxiliary resonances at
low rotor speeds, particularly for light damping. Finally, when inter—sector coupling
is included, no instabilities to non-traveling wave motions could be identified. In
this way, the analysis of an individual sector offers global qualitative results that are
applicable to the fully coupled system, including stability.

The chapter is organized as follows. A mathematical formulation is carried out
in Section 5.2, beginning with a description of the governing nonlinear equations of
motion in Section 5.2.1. These are sealed in Section 5.2.2, the underlying linear reso-
nance structure is shown to persist under the scaling in Section 5.2.3, and the method
of averaging is employed in Section 5.2.4. This gives rise to simplified approximate
sector models that form the basis for all of the analysis that follows. Existence of
the desired traveling wave response is discussed in Section 5.3 in terms of station-
ary points of the averaged system, and its local stability is subsequently addressed.

Features of the forced response of the isolated nonlinear system are highlighted in

117

Section 5.4 with an emphasis on the blade and absorber frequency response and also
criteria for zero blade amplitudes. Finally, the forced response of the fully coupled
nonlinear system is considered in Section 5.5 with the goal of quantifying possible
instabilities to non-traveling wave responses, and the chapter closes in Section 5.6

with some concluding remarks.

5.2 Formulation

In what follows the nonlinear equations of motion given by Equation (3.14), together
with the generalized family of paths described in Section 3.4.4, are systematically
reduced via scaling and averaging to a model that is amenable to the tools from
nonlinear dynamics. As with the analysis of Chapter 4, this is carried out under
the assumption of identical, identically-coupled sectors. The governing equations of
motion are briefly reviewed in Section 5.2.1 and they are scaled in Section 5.2.2 to
capture first-order nonlinearity via the absorber paths. In Section 5.2.3 a compar-
ison is made between the scaled sector models and the linearized sector models of
Chapter 4, and it is shown that the linear resonance structure qualitatively persists
under the scaling. Finally, averaging is carried out in Section 5.2.4 in both polar and
Cartesian forms. The resulting averaged sector models form the basis for all of the

analysis that follows.

5.2.1 Equations of Motion

The cyclically-coupled model to be considered features the same lumped-parameter
arrangement for the bladed disk assembly that was employed in Chapter 4, which
is shown in Figure 5.1a (see Section 4.2 on page 78 or Table 3.1 on page 64 for a
description of it parameters), and the circular-path kinematic model for the absorbers
is replaced by the more general, arbitrary-path description of Section 3.3. This is

shown in the sector model in Figure 5.1b, where a is the ith dimensionless absorber

118

i+1

 

Figure 5.1. (a) Model of bladed disk assembly and (b) sector model.

mass, 3,- is its nondimensional displacement along the arbitrary path, which subtends
an angle 19,-(si) relative to the vertex at V, and r,(s,~) is the dimensionless radius
length to the absorber relative to the blade basepoint 0. Relationships between these
fundamental path variables were derived in Section 3.3.1.

The equations of motion for the ith sector follow from the development in Sec-

tion 3.4 and they are given by

dr-
as? + uriF16;'+ €518; — rim-$2707 + 0;)2
Z
d .
+ p.502 (F,- sin(c9,' + 19,-) — a—E cos(6’,° + 19,-)) = O, i E N (5.1a)
I

0;, + 51,9; — gaTz'I-‘z'sg + 6i + (502 sin 9i

dTi

r226? + rin-sg’ + 2ri-C-Esg(o + 6;)
‘l‘ l1 d 'F-
+ 1513—21255; + 6027‘,- sin(9,' + 29,)
+ u2(—6,-_1 + 26,- — 9,“) = Fcos(nor + (15,-), ie N (5.1b)
where the coupling damping {c has been ignored, the functions F,- = F,(s) and

19,- : 19,-(5) are defined by Equation (3.15) and Equation (3.16), respectively, and

the subscripts i have been dropped from the system parameters (these are defined in

119

Table 3.1 on page 64) since the sectors are taken to be identical. The absorber paths

are assumed to be of the form given by Equation (3.19) on page 65, where

(am? + 1) +6)2
b0 = 752

~2
an

b — — (5-2)

2 a + 6

 

 

b __ 6736 _ a(n2+1)+6
4 ‘ mm + (5)3012 +1) 2(0. + 6m? +1)’7

 

 

are the path coefficients if ”y is written in terms of the absorber tuning order according
to Equation (4.39) of Chapter 4. In this representation, each path depends only on
the linear and nonlinear tuning parameters a = VII—g and 77. Once these are set by
design, and given the disk radius 6, then a and ’7 are automatically prescribed (these
represent the effective placement of the absorbers along the blade lengths and the
curvature at their vertices).

NeXt the full nonlinear system given by Equation (5.1) is systematically reduced
to a set of weakly nonlinear oscillators, and perturbation techniques are subsequently

carried out on these reduced equations.

5.2.2 Scaling

In any realistic physical implementation, the absorber masses will be much smaller
than that of the blades, primarily due to stringent restrictions on the absorber rattling

space and therefore on its dimensions and mass. It is thus reasonable to take

at

x1=€

as the basis for the scaling, where O < 6 << 1 is a small dimensionless parameter and
the constant m is to be determined. The blade and absorber dynamics are assumed

to scale with 5 according to

120

for each i E N in a manner that is to be determined. Blade damping levels relative
to critical are often 0.01% or less and, in order to achieve the desired order tuning,
absorber damping is generally made as small as possible. As discussed in Section 3.6,
the corresponding dimensionless blade and (effective translational) absorber damp—
ing constants are on the order of 10‘3 and 10—6, respectively, and they are scaled

according to
£5 = epéb and Ea 2 5451-1.

It is additionally assumed that the inter-blade elastic coupling is weak, that is,

which together with the assumption of light damping allows for the investigation of
a host of possible instabilities, bifurcations, and multiple interacting modes.1 Finally,

weak forcing is assumed, that is,
F = e’ f ,

since the nonlinear dynamics near resonance are of interest.

The scaling parameters 111, k, I, p, q, n, and r are chosen such that, to leading order
(8 = O), a simplified and solvable system is obtained, one that is used as the basis
for the method of averaging. In order to investigate the potentially rich dynamics
when the system is weakly coupled and lightly damped, the scaling is chosen so that
the nonlinearity, damping, and coupling all appear at (9(5) and it should preserve, as
much as possible, the linear resonance structure described in Chapter 4. To this end,

a suitable choice for the scaling parameters is found to be

m=2, &=%, [2%, p=3, q=1, 11:1, r23. (5.3)

 

1When the coupling is strong (which can occur in shrouded assemblies or via other coupling
mechanisms) and for sufficiently weak forcing, there exists the possibility of only two interacting
modes, the analysis of which is left for future work.

121

h

Upon substitution and simplification the it scaled sector model becomes

92’ + (Didi + €192( - éz'_1 + 261' - displ)
= e( — rosy — 602$,- — Q6; + fcos(nor + (151)) + 0(53/2), (5.4a)

gg’ + eggs, = e( — roég’ — 6026, — gag; — 770%?) + 0(53/2), (5.4b)

where the frequencies 5111 = m and (.7322 = no are summarized in Table 4.3
on page 96 and r0 is defined by Equation (3.17) (where 'y is eliminated according to
Equation (4.39)).

When 5 = 0, Equation (5.4) reduces to a pair of decoupled, undamped, and un-
forced linear oscillators. The first oscillator describes the free vibration of an isolated
blade without an absorber, while the second captures the dynamics of each absorber if
the blades are locked relative to the rotating hub, which was discussed in Section 4.2.3.
The general case of small 5 7.4 0 is simply a perturbation of these uncoupled systems;
Equation (5.4a) is a linear, weakly forced oscillator that approximates the motions of
blade i E N while Equation (5.4b), which captures the ith absorber dynamics, is un-
forced and weakly nonlinear due to the cubic absorber path term. These oscillators are
weakly coupled due to the assumptions of small [1 (representing the blade-to—absorber
coupling) and small inter-blade stiffness coupling.

Absorber design is carried out by choosing the linear absorber tuning order a
(the proper selection of which was discussed in Section 4.4) and the nonlinear tuning
parameter 7), both of which appear only in the ith absorber equation, that is, Equa-
tion (5.4b). This in turn prescribes the absorber paths by setting the constants b0,
b2, and b4 in Equation (5.2) and it fixes the effects of the absorbers on the blade
dynamics via the first two (inertia and stiffness) terms in the parentheses on the right
hand side of Equation (5.4a).

Before proceeding with a further reduction of Equation (5.4) via averaging, it

should be verified that the scaled sector models suitably capture the underlying linear

122

resonance structure that was described in Chapter 4. This is done in the next section,

where it is shown that the no-resonance zone qualitatively persists under the scaling.

5.2.3 Linear Resonance Structure of the Scaled System

If the scaling in Section 5.2.2 is applied to the ith linearized sector model defined
by Equation (4.1), and if the assumption of motion-limiting stops is removed by
multiplying through by the stopper angle rho, then it can be directly compared to the
ith ELIE sector model given by Equation (5.4) with 77 = 0. These two systems (in
this section we refer to them as the linearized model and scaled model for simplicity)
match identically if u = 0 in 11411 and K11 (see Table 4.1 on page 81), which simply
ignores the contribution of the absorber inertia in those terms. In this way, the scaling
is seen to essentially linearize the blade dynamics (with an accompanying additional
loss of some dynamic coupling terms involving a) while at the same time capturing
the basic first-order effects of the absorber path nonlinearity.

Since a = 82 will generally be small, it is expected that the linear resonance struc-
ture qualitatively persists under the scaling of the previous section. This is verified
in Figure 5.2a which shows example plots of the rotor speeds ores corresponding to
resonance for the linearized (solid lines) and scaled (dashed lines) systems described
above versus the linear absorber detuning parameter 6 for zero damping and for var-
ious values of a. (Table 5.1 lists the corresponding values of 5, 6a, and Aer for the
various mass ratios.) To simplify matters, the curves are shown for the special case
of a single isolated blade/ absorber combination, that is, for z/ = 0 (this is equivalent
to considering only the possible resonance corresponding to p = 1), but this does
not preclude a direct comparison of the two models for accuracy. It is clear that the
linear resonance structures of the two systems are in good agreement for sufficiently
small mass ratios, and both feature the no—resonance gap. As a increases, so too does

the percent error in ores between the linearized and scaled systems, which is shown

123

Table 5.1. Data to accompany Figure 5.2.

 

 

,u 5 ficr (X 103) Aer .14 5 Her (X103) /\cr

0 0 0 - 0.005 0.0707 —2.348 —0.597
0.001 0.0316 —0.470 —0.268 0.010 0.1000 —4.685 —0.841
0.002 0.0447 —0.940 -0.378 0.015 0.1225 —7.012 —1.027
0.003 0.0548 —1.410 -0.463 0.020 0.1414 -—9.329 —1.182
0.004 0.0632 —1.879 —0.534 0.025 0.1581 —11.637 —1.317

 

 

in Figure 5.2b. For reasonable mass ratios, however, the error is seen to be small
over a wide range of detuning values, except near the critical detuning 6 = Ber where
the error becomes unbounded (implying that the scaled system predicts a slightly
larger no-resonance gap). The scaled model increasingly overestimates ficr by a fi-
nite amount,2 but otherwise satisfactorily captures the underlying linear resonance
structure.

Next the nonlinear scaled sector models of the previous section are further re-
duced via the method of averaging. The perturbation analysis simply casts them into
a standard and tractable form, from which blade and absorber amplitudes can be

estimated.

5.2.4 Averaging

In what follows averaging is carried out on the scaled sector models of Section (5.2.2).
This is done first in the standard polar form and the results are subsequently converted
to cartesian coordinates. Both forms are used in the analysis, either explicitly or

implicitely—whichever is most convenient.

 

2This could be obtained by deriving the counterpart to 6C, for the scaled system (see Equa-
tion (D2) of Appendix D) and comparing the two critical detuning values.

124

1 n—No—Resonance Gap
(Predicted by Linearized Eqns.)

   
 
 
  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
 

-— Linearized Eqns.
—--- Scaled Eqns.
(with n = 0)
(a) 5“"
o ——4 E . . .
f; Increasmg p g g
(b) m ; §
so —6 ~ ii
E
-—8 . § E

 

 

 

 

 

 

—0.02 0 0.02 0.04
3

Figure 5.2. (a) The rotor speed om corresponding to resonance for the linearized system
of Chapter 4 and the scaled system formed by Equation (5.4) (with 17 = 0) versus the
linear absorber detuning parameter [3 for a model with N = 1, n = 3, a = 0.84, 6 = 0.67,
v = 0, and for mass ratios 0.005 S u _<_ 0.025; (b) the corresponding percent error. The no-
resonance gap (predicted by the linearized system of Chapter 4) is defined by 6" < 6 S 0,
where the values of the critical undertuning 6a are given in Table 5.1 for the various mass
ratios.

125

POLAR FORM

The weakly nonlinear set of oscillators defined by Equation (5.4) can be cast into a

form that is suitable for averaging via the transformation3

A

6,-(7) = uz-(r) cos (nor + 96,- + gi(r))

A

6’.(T) = —nou,~(7’) Sin (nUT + (bi + 921(7))

’1.

A

3,,(7) = 1),-(r) cos(nor + (,6,- + 9(7))

A

sém = wow) smear + 4.- + 9(7))

along with the usual constraint equations. Equation (5.5) represents a standard
variation of parameters to transform the dependent variables from 6,- (resp. 3,) to u,-
and g,- (resp. v, and g), which allows for solutions with slowly-varying amplitudes
and phases in individual sector responses, and it also serves to capture the desired
traveling wave response among the sectors by including the inter-blade phase angle
96,-. In this way, the transformation additionally carries the continuous time and
discretized space duality that was systematically described in Section 2.3.2 in the
context of engine order excitation. Depending on the value n relative to N, it can
therefore capture BTW, FTW, and SW responses. It should be noted from the
onset, however, that there could be a multitude of other response types and that
the averaged models to be developed based on Equation (5.5) are general enough to
capture them. However, the desired and most basic response is that of a traveling
wave, the existence, stability and bifurcation of which forms the main focus of this
chapter.

Upon substitution of Equation (5.5) into Equation (5.4) and elimination of terms

 

3The notation g,- and (,- is recycled in this chapter for convenience, and should not be confused
with Equation (B.15) or the circular-path absorber angle in Figure 3.6.

126

at 0(53/2) and higher we obtain

 

1
ug = no —((w¥1 — n2o2)u,- cos(nor + ¢i + 9,) + eff; )) sin( nor + (b, + 9,)
I 1 2 2 2
21,9, = (—(w11 — n o )u, cos(nor + <25, + 9,) + Efil ) cos nor + ct, + 9,)
"10' (5.6)
v; = no —((c;,%2 — n2o2)v, cos(nor + p, + q) + 5f;2 ))sin( nor + (25,- + C1)
. l _ 1:2 _ 2 2
11,9 ._ no (£22 71 o )v, cos(nor + d), + c,) + sf”) cos( nor + (b, + <,)
for each i E N, where the functions
(i) - . ‘
f11 = — nogbu, sin(nor + (25, + 9,) — fcos(nor)
— (r0032 — (502)22, cos('n.oT + (25, + c,)
- 2171' 0080107 + 942' + <1) - 174—1 0080107" + 451-1 + <i—1)
+ V2 _ > (5.7)
— 7424100801“ + ¢i+1 + (1+1)
f2? = — noéav, sin(nor + 95, + 9') + 770%? cos3(nor + (f), + 9')
-— (roe—2% — 602)u,- cos(nor + 6,; + 02') 1
capture all of the (9(5) terms in Equation (5.4). The differences
2 2
o —o
(3%1 — 77.202 = ___02_1" (5.8a)
7'
£232 -— n202 = (a? — n2)02, (5.8b)

give a measure of proximity of the rotor speed relative to blade resonance and to the
absorber design relative to ideal linear tuning, respectively, where (1)11 is the natural
frequency of an isolated blade without an absorber, or = m is its resonant
frequency,4 and 022 is the natural frequency of each absorber if the blades are locked

relative to the rotating hub. Equation (5.8) motivates the speed and order detunings

02 = 03(1 + 5A), (5.9a)

a? — n2 = 5A, (5.9b)

 

. . . . . . _. +1 .
4This 1S not to be confused With ores, which corresponds to the Intersection of calf} ) With the

engine order line no.

127

 

 

l ”U
5 nd
at” £3,
\2
\
----- 7"” ' (Dip) \(1)
all) :
i—>A
/ > 0

Figure 5.3. Example Campbell diagram showing the speed and order detuning parameters
A and A.

where A plays the role of the rotor speed and A is the counterpart to the linear order
detuning from Chapter 4. These can be visualized in Figure 5.3, where it is observed
that 6211 2’ 6291) for small mass ratios )1. (See Equation (4.26) on page 89.) Under the
detuning scheme defined by Equation (5.9) the right hand side of Equation (5.8a) and
Equation (5.8b) reduce to —5A and er2, respectively, and Equation (5.6) becomes

suitable for averaging. Finally, the two order detuning parameters A and 6 are related

by
= mm + 2)
fl ’

which follows from Equation (5.9b) and Equation (4.44). The no-resonance zone is

A (5.10)

such that 60, < 6 S 0, to which there corresponds a range Aer < A S 0 that can be
computed using Equation (5.10).

After the appropriate substitutions are made Equation (5.6) is averaged over one
period T = 27r/no. The result is partitioned into two vector functions, one that
defines the stationary points of the ith averaged sector model (this is discussed more
fully in Section 5.3.1) and the other inherits any remaining terms. To 0(83/2) and

for each i E N, the desired form is

_ _ _ _ _ a 5
(14.1492, ”02', 112'le = §EG(V¢—1,Vi,w+1l+ '2n—03(Vi—1vviavi+1)v (5-11)

128

where v, = (21,, 6,, 27,, 6,)T. The elements of the 4 x 1 vector G are given by

‘

G1 = — cr(n2 +1)0217,sin(§,— ,) — noébu, —— fsin 6,-
G2 = - a(n2 +1)o2i),cos( ,-— 6,) — Au, — fees 6,-
}, i e N (5.12)

2

Cg = + 0(n2 +1)o a,- sin(@, -— 6,) — noédii,

3

z J

 

G4 = — a'(n2 + 1)o2u, cos(@, — 6,) + A0227, + $7026
where the identity (1),-3:1 — a, = :l: 27rfi = :t 4,0,,“ has been employed, and the first

element of g is

91 = I9211i—1sin(@i—1 - @2; - 9071+1) + l7211241sin(éz'+1 - 61+ <Pn+1), 2'6 N
(5.13)
with the remaining entries being equal to zero.

As a final and important note, it is customary at this point to expand all ap—
pearances of 0 according to Equation (5.9a) and to keep only 0(5) terms in Equa-
tion (5.11), which amounts to simply replacing a with the constant 0,. However,
we opt instead to keep a in the analysis which, as we shall see, gives much more
satisfactory results.5 Benchmark results in which the substitution is made are given
in Appendix E; these are to be compared with the analysis and results to follow,
particularly the blade and absorber frequency response curves of Section 5.4.

The averaged sector models defined by Equation (5.11) serve as the basis for the
analysis in the rest of this chapter. The corresponding Cartesian form is also useful,

which is given next.

 

5Note that Guckenheimer and Holmes take the same approach in their analysis of a simple Duffing
oscillator in Chapter 4 of [96]. (In particular, see Equation (4.2.14) on page 174.)

129

CARTESIAN FORM

The averaged systems given by Equation (5.11) can be converted to Cartesian form

via the transformation

 

W) = (Mama-26>. tam-(T) = —B.<r>/A.<r>

 

 

, i E N.
m7) = (63(7) + 0.2m. tanler) = wan/0.47)
Then if w, = (A,, B,, C,, D,)T it can be shown that
Wi = 5%; lei—1»Wie W241) + % P(Wz'—1,Wz'aWi+1)+ 0(53/2), i6 N (5-14)
where the elements of P are given by
P1 = —a(n2 + 1)o2D, — AB, — noébA, l
+ 192(232' — (Bi—1 + Bi+1) COS 901141)
P2 = +a(n2 +1)02C,+ AA, — noébB, + f
+ 1920- 2141' + (Al—1 + Ai+1l COS 99n+1) >' (515)
P3 = —o'(n2 +1)oQB,+ Ao2D, — noéaC, + g77072(D,‘-3 + CED,)
P4 = +cr(n2 + 1)02A, — A020, — noéaD, — E43-'r)o2(C,D,-2 + 0,3) 1
The first two elements of the vector p are
.2 ‘.
P1 = -V (Ar—1 - 142405111 9911+1 , ie N (5.16)

.2 .
P2 = —V (Bi—1 — Bi+1)31n‘Pn+1
and the remaining two elements are zero.
Existence and local stability of the desired traveling wave response of the coupled

system is discussed next.

5.3 Traveling Wave Response

The desired system response is that Of a traveling wave (TW), where each sector

behaves identically, except for a fixed phase difference among its nearest neighbors.

130

Such a response is characterized by the fullest degree of (cyclic) symmetry that is
possible, which can be visualized in Figure 2.7 on page 35. Then the absorbers become
entrained with the (discrete BTW, FTW, or SW) applied dynamic loading among the
blades (see Section 2.3.2) and are hence most effective in addressing their attendant
vibrations. This type Of response is the counterpart to the desired synchronous motion
in the CPVA work by Shaw and coworkers, where any other response type implies
a degradation of absorber performance in which some of the absorbers work against
the others [22, 23].

The averaged models described in Section 5.2.4 were formulated specifically to cap-
ture a TW response, but they are also general enough to identify other solution types
(if they exist) that involve amplitude and phase modulations in individual sectors.
Throughout the remainder of this chapter we focus on TW solutions, the existence,
determination, and stability of which is considered next. Possible bifurcation to other

response types is addressed in Section 5.5.

5.3.1 Existence

A TVV response is characterized by identical dynamics of individual sectors together
with a fixed phase difference in these dynamics among neighboring sectors. If v =

(ii, ,6, 'D, 6)T and w = (A, B, C, D)T, then such a response corresponds to

(v,_1,v,, v,+1) = (v,v,v), Vi E N (Polar Form) (5.17a)

(w,_1,w,, w,+1) = (w,w,w), Vi E N (Cartesian Form) (5.17b)

where the phase difference among adjacent sectors is built into the transformation de—
fined by Equation (5.5) via the inter-blade phase angles 96,-. Since g(v, v, v) = 0 (resp.
h(w,w, w) = 0), which can be verified from Equation (5.13) (resp. Equation (5.16))

by setting @,_1 = 6,41 = @ (resp. A,_1 = A,+1 = A and B,__1 = Bi+1 = B), it thus

131

follows that

O = C(v, v, v), (Polar Form) (5.18a)

O = P(w, w, w), (Cartesian Form) (5.18b)

defines a TW response when the averaged models are in polar (resp. Cartesian) form.
That is, if a stationary point v (or w) can be found that satisfies Equation (5.18a)
(or Equation (5.18b)), then there exists a corresponding TW response. Existence,
therefore, follows from the equilibria of an individual averaged sector model, a sim-
plification that follows from the assumption of identical, identically-coupled sectors,
and expressions for their determination are derived in the next section. In contrast,
local stability of a stationary point involves all N averaged sector models, which is

discussed in Section 5.3.3.

5.3.2 Stationary Points

To each stationary point v (or w) that satisfies Equation (5.18) (when it exists)
corresponds a TW solution. In general, the determination of these equilibrium points
is too formidable to be carried out analytically, but some important insight can be
gleaned from their defining equations. If damping is ignored, however, a simplified set
of implicit expressions can be Obtained, from which blade and absorber amplitudes |a|
and [6| easily follow. These expressions are used to generate representative frequency
response curves in Section 5.4 for the isolated nonlinear system and in Section 5.5 for
the fully coupled nonlinear system. Finally, in all of what follows it is assumed that
n E Z+, o > 0, f # 0, 0 < a < 1, and n2 — 6 > 0. Physically, the last two conditions
imply that the absorber is attached along the length of the blade and, essentially,
that the rotor radius is comparable to the blade length if the engine order is very
small—a restriction that will generally be satisfied for the engine orders of interest

and for practical rotor/ blade geometries.

132

We begin by considering the fully damped system and focus on general insight;

the special undamped case is considered subsequently.

THE DAMPED SYSTEM

When they exist, the stationary points can be Obtained from either Equation (5.18a)
or Equation (5.18b). The former is slightly less cumbersome to work with analytically
and hence it is employed throughout the remainder of this section. It can be written

as

2

0 = —o:(n2 +1)o 63in ‘ — 6) — noébu — fsin Q, (5.19a)

0 = —()z(n2 +1)026cos(§ — 6) — (A — 2192(1— cos cp,,+1))u — fcos @, (5.19b)
0 = +Or(n2 + 1)0221 sin(‘ — 6) — 7206,76, (5.19c)
0 = —a(n2 + 1)02'& cos(@ — 6) + A0217 + 237702173, (5.19d)

from which a number of general insights can be gleaned.

In the absence of nonlinearity, which appears only in Equation (5.19d) via the
cubic absorber path term, Equation (5.19) approximates the amplitudes of the exact
linearized response defined by Equation (4.34) as o is swept from zero. Moreover,
the inter-sector coupling appears only in Equation (5.19b) in combination with the
speed detuning parameter A, and it therefore reflects the same shift in the linear
resonance (associated with coupling) that was observed in Chapter 4. In this way,
the fundamental effects of nonlinearity on the TW response amplitudes, and bifurca-
tions to other traveling wave responses, can be qualitatively captured in the absence
of coupling—that is, for an isolated sector. This is the focus of Section 5.4. The
frequency response amplitudes for the fully coupled nonlinear system are essentially
the same as those of the uncoupled case, except for a shift in the primary resonance
by an amount that is directly proportional to 122; when the coupling is small, this

shift is nearly undetectable. However, there may also be potentially rich instabilities

133

to response types other than the desired TW (i.e., those with a reduced degree of
symmetry) when coupling is present. This possibility is discussed in Section 5.5.
The presence of damping renders Equation (5.19) essentially intractable since
there is no clear way to eliminate phases and solve for the desired amplitudes 1'1 and 27.
Even the corresponding Cartesian form poses much difficulty. Thus when damping is
included the equilibrium points are obtained numerically.
The special case of zero damping is considered next. In this instance, an implicit

pair of expressions for the blade and absorber amplitudes can be derived.

THE UNDAMPED SYSTEM

For zero damping, and when the stationary points exist, they are defined implicitly

by Equation (5.19) with Eb = 55, = 0, that is,

0 = —Oz2(Tl + 1)o2 v sm( 6—)— fsin 6, (5.20a)
0 = —a(n2 + 1)02v cos (6— 6—) (A— 2(1— cos go,,+1))u —fc ocs @, (5.20b)
0 = +a(n2 + 1)02u sin( — 6), (5.200)
0 = —a(n + 1)o 2acos(§— 6) + A0227 + 9,7702273, (5.20d)

from which a simple pair of implicit expressions for the blade and absorber amplitudes
can be distilled. However, the case Of zero damping is highly degenerate and this
system gives rise to nonhyperbolic equilibria. Nonetheless, an arbitrarily small level
of damping removes the degeneracy, which is shown in Section (5.3.3), and thus the
expressions in Equation (5.20) collectively give a good approximation to the blade
and absorber amplitudes in the presence of light damping. Local stability is obtained
accordingly.

It should be pointed out from the onset that, due to the appearance of sin'g‘
and cosg in the forcing terms of Equation (5.20a) and Equation (5.20b), the usual

approach to eliminate phases cannot be employed and, correspondingly, the analysis

134

is somewhat more cumbersome. Equation (5.20c) is used as a guide on how to proceed.
Under the assumptions indicated above, it is satisfied only if sin(é — c“) = 0, 21 = 0, or
both. The first condition gives rise to the most general form of the equilibria and it
is described in detail below. The second condition gives rise to the desired motions in
which the blades remain stationary relative to the spinning rotor. It is simply a special
case of the first and is described in Section 5.4.2. Finally, the third condition can be
obtained from the first two, and it does not give rise to any additional equilibria.

If sin(g — g“) = 0 then Equation (5.20c) is trivial and it follows from Equa-
tion (5.20a) that sin@ 2 0, and together these expressions imply sin(" = 0. The

phases must therefore satisfy
53 = k7r and f = In, 13,16 Z (5.21)
and the system given by Equation (5.20) reduces to

0 = —cr(n2 + 1)0217 cos(k7r) cos(l7r)
— (A —— 2192(1— cos gon+1))fi — fcos(k7r), (5.22a)

0 = —-a(n2 + 1)02flcos(k7r) cos(l7r) + A0227 + 37,0263, (5.22b)

for k,l E Z. There are at most four integer pairs (1:, I) that yield distinct
blade/ absorber amplitudes (11,27), and one such choice is summarized in Table 5.2.
Physically, when (k,l) = (0,0) (resp. (1,1)) each blade and its attendant absorber
oscillate with the same phase and their motions are in phase (resp. out of phase) with
respect to the applied dynamic loading. For the case of (k, l) = (0,1) (resp. (1,0))
their motions are out of phase and the blades (resp. absorbers) responds out of phase
(resp. in phase) relative to the excitation. (Of course, some of these motions may be
unstable.) These details are not belabored, however, since in the absence of damping
every stationary point is nonhyperbolic, which is shown in Section 5.3.3.

The (k, l)-dependence in Equation (5.22) can be eliminated by multiplying Equa-
tion (5.22a) by cos(k7r) and Equation (5.22b) by cos3(l7r) = cos(l7r). Then by defining

135

Table 5.2. Example integer pairs (k, I) that yield distinct stationary points 22 and 13 of the
averaged system. Also indicated are the corresponding values of f) and g" and whether the
resulting blade/ absorber motions are in phase (IP) or out of phase (OP).

 

k l cos(k7r) cos(l7r) cos(k1r) cos(l7r) Q = law 6 = l7r Phase

 

 

0 0 1 l 1 0 0 IP
1 1 —1 —1 1 7r 7r IP
0 l 1 —l —1 0 7T OP
1 0 —1 1 —1 7r 0 OP
1] = a cos(k7r) and 17 = 27 cos(l7r) and simplifying the result it follows that
0 = a(n2 +1)0227 + (A — 2192(1— cos ¢n+1))a + f, (5.23a)
0 = a(n2 +1)a — A73 — 3,7703, (5.2310)
from which the steady-state blade/absorber amplitudes lftl = lfll and liil = |17| can

easily be computed.
There are at most three roots for 5, and these can be obtained by eliminating it

in Equation (5.23). They follow implicitly from

 

 

 

 

3 2 2 1 2 2 2 1 A
-77‘03 + a .(n + ) a + /\ ,~, + ‘2" + )f = 0, (5.24)
4 A — 2122(1— cos 90,1“) A - 2V2(1 — cos 907,“)
where the corresponding values
2 2 A
21 = (1(7). + 1)0 ~ f (5.25)

—A - 2192(1— cos cpn+1)v _ A — 2192(1— cos gen“)
follow from Equation (5.23a). Given the geometry of the bladed disk assembly, the
strength and order of the excitation, a particular linear tuning, the coupling level,
and the strength of the absorber path nonlinearity, one can construct the blade and
absorber amplitude frequency response curves in terms of the rotor speed using Equa-
tion (5.24) and Equation (5.25). In doing so, the reader is reminded that the dimen-
sionless rotor speed a and its detuning parameter A are related by Equation (5.9a),

where e = fl.

Local stability of the TW response is considered next.

136

5.3.3 Local Stability

A hyperbolic fixed point v (or w) implies a unique hyperbolic periodic orbit in the ith
nonlinear sector and together all such orbits form a TW response among the sectors
with the same stability type [96,109]. Local stability of v is considered next for
the fully coupled system. The results are subsequently stated for the corresponding

Cartesian form and also for the special case of a single isolated sector.

THE COUPLED NONLINEAR SYSTEM

Local stability of a stationary point v = (21, g”, 27, CT, the elements of which are defined

implicitly by Equation (5.19), can be obtained by considering the small perturbations
17,; = v,- — v, 2' E N

away from v,- = (11,, £32" 27,, 5,)T. Then for each i E N, Equation (5.11) can be locally
approximated to leading order by the linearized equation

5

= 2nal(A ‘ B)m_1+ U771 + (A + B)m+1] + HOT, 2' e N. (5.26)

I
77:”

In Equation (5.26) the subscripts are taken mod N such that 770 = TIN and "NH = 171

and
_méb 2 Ewe—a) — fCr) “AZVQWW +1) Cilia—e) “Cw-6)
_—&C(§_—) aaS(§_C—) A02 + 27102272 —aaS(§_C—)d

 

 

where the shorthand notation a = a(n2 + 1)a2, SOC) = sin(x) and C(X) = 005(X) has
been introduced. If the matrices A and B are partitioned into four 2 x 2 blocks then

the (1,1) blocks are given by

0 19211 cos 99,,“ 192 sin <pn+1 0
A11 = 3 B11 =

—-192 cos 99n+1 0 0 13217. sin apn+1 ’

and all other entries are zero.

137

The N linear systems defined by Equation (5.26) can be handled in the same

way as the sector models in Section 4.3.2. By stacking each 4 x 1 vector 11, into the

configuration vector 17 = (711,172,. . . ,nN)T it follows that 77’ 2 5J( (Q17, where
1
JS’S) = E—circ (U, A + B, 0, . . . ,0, A — B) (5.27)
no

is the 4N >< 4N J acobian matrix, which belongs to fiffflyzuv. That is, .18? is a block
circulant, block symmetric matrix of type (4, N) and can thus be block diagonalized
via the unitary transformation defined by Equation (2.11) on page 15. This gives rise

to the set of N, 4 x 4 matrices

(P)_
Ir

J U + 2Acos (pk + 2jB sin 90k), k E N (5.28)

_2n0 _(

where, recall, (pk = 2V1“ — 1). The 4N system eigenvalues are preserved under this
transformation and hence they can be obtained by solving N reduced-order eigensys-
tems involving 4 x 4 matrices instead of just one potentially formidable eigenvalue
problem involving a 4N x 4N matrix. However, the transformation renders the re-
duced Jacobians complex, which makes their interpretation more difficult. This is
discussed more fully in Section 5.5.

A similar analysis can be carried out for the averaged sector models in Cartesian

(P)

form. In this case the Jacobian matrix J ( S) (the counterpart to J CS when Cartesian

coordinates are used) has generating matrices P, Q + R, 0, . . . ,0, Q — R, where
_ —no€b —A + 2192 0 —&02 .
P _ A — 2192 —n0€b 5202 0
_ 0 —€1'02 gnazCD — 71.065 3T]0’ 2(02 + 3D?) + A02
L 5102 0 «917702602 + D2) — A02 —%nazCD — 71065 _

 

and the (1, 1) blocks of Q and R are given by

0 *2
Q112 [.2

_V COS Son-H
V COS $011+1 0

with all other elements being equal to zero. This gives rise to the set of N, 4 x 4

block decoupled Jacobian matrices

1
2710

Jim: —-——(P + 2Qcos 99k + 2szin (pk),

138

.2 .
V Slnson+1
] , R11 = [

0

0 02 sin ¢n+1

keN

 

(5.29)

which are the Cartesian counterparts to the matrices defined by Equation (5.28).
Some local stability results are given next for the special case of an isolated sector,

including a polynomial expression for the eigenvalues of its Jacobian matrix.

THE ISOLATED NONLINEAR SYSTEM

The Jacobian matrix corresponding to an isolated sector can be obtained from Equa-
tion (5.28) or Equation (5.29) by setting I? = 0, in which case the matrices A, B, Q,
and R (and hence the lat-dependence) all vanish. In polar and Cartesian forms, the

Jacobians are

 

 

 

 

l- —n0£b —C—H:’C(§_<-) — {C(é) —OS(§_<-) 5170(é_§-)-
J(P) : 1 —A OUS(é_<-) + fS(§) —QC(§:C) —C¥’US(§_C-)
15 2720 (SS(§_C) C—YflC(§_—g‘) —n0€a _éfiCQ—‘C—l ,
-_6‘C(§-<‘) ("ism—6) ”\02 + 917702772 _G'flSw—s’)-
(5.30a)
—naéb —A 0 —a02 -
J(C) _ _1__ A —naéb 6202 0
IS — 2nd 0 —c‘w2 3170200 — nag}, 3,00,2(02 + 3D2) + A02 ’
_ (102 0 —gna2(302 + D2) — A02 —%n02C'D — 17.055
(5.30b)

which have the same elements as the matrices U / 2710 and P / 2na, respectively, in the

absence of inter—sector coupling.

Stability results are more transparent and tractable for the isolated sector case,

especially in Cartesian form. The eigenvalues C of Jg) follow implicitly from the

fourth-order polynomial det (J(SC) — (I) = 0, which can be written as

s4 + (1333 + (a2 + d2)32 + dls + (a0 + do) = 0, (5.31)

139

where s = 2noC. The coefficients are
a2 = A2 + 113 (32022012 +1)2 + (377(C'2 + D2) + 4A) (977(C'2 + Dz) + 4)\))o4
50 = 115010202 +1)2o2 + 377(02 + 02m + 4AA)
>< (402(712 + 1)2o2 + 977(C2 + 02)A + 4AA)o4
d3 = 2710(55 + 55)
d2 = ”202K221 + 45555 +61?) l’
d1 = 272.05 ((12012 + 1)2(€5 + 51,) + 1% (372((32 + D2) + 4).) (971(02 + D?) + 4A)éb)
+2noA2éa + 2n3a3éa£b(€a + 55)
d0 = 116n2o6éb(3202(n2 + mg}. + (377((32 + D2) + 4/\)(917(C2 + D2) + 4,05,)

+ 71202552, (A2 + n2o2éfb2)

 

 

1

where each dk (k = 1, 2,3,4) vanishes when 561 = 5b = 0. In the absence of damp-
ing Equation (5.31) is quadratic in 32 and it thus features quadrantal symmetry in
the complex plane. This in turn implies nonhyperbolic or otherwise unstable station-
ary points v, depending on the details of a2 and (10. An arbitrarily small level of
blade and/ or absorber damping destroys the quadrantal symmetry, thus removing
the possibility of nonhyperbolicity.

Next the averaged sector equations are employed to detail some global features
of the forced response. This is done first for the isolated nonlinear system in Sec—
tion 5.4 and subsequently for the fully coupled nonlinear system in Section 5.5. For
convenience, and where appropriate, the analysis is carried out using both polar and

cartesian forms.

5.4 Forced Response of the Isolated Nonlinear System

In order to focus 011 the nonlinear dynamic performance of the absorbers without the
potentially complicating effects of inter-sector coupling (in particular, the possibility

of additional symmetry-breaking instabilities) we focus on an isolated sector in this

140

section, which consists of a single blade and nonlinear absorber. (A treatment of the
fully coupled nonlinear system is postponed to Section 5.5.) The results are presented
in Section 5.4.1 in terms of blade and absorber (amplitude) frequency response curves.
A criterion that guarantees a branch corresponding to zero blade amplitudes is sub-
sequently derived in Section 5.4.2, which is the nonlinear counterpart to the ideal, or

exact linear tuning.

5.4.1 Frequency Response

Example plots of the blade and absorber TW amplitudes lfll and I'DI are shown in F ig-
ure 5.4 versus the rotor speed o for a hardening absorber path (17 > 0), zero damping,
and for various levels of the order detuning fl, and a corresponding set of plots is Shown
in Figure 5.5 for a softening path (7) < 0). In these figures, the undamped absorber
and blade amplitudes were generated using Equation (5.24) and Equation (5.25) with
I? = 0, and stability results were numerically determined according to Equation (5.30)
with the addition of very light damping (approximately 0.01% absorber damping rel-
ative to critical and zero blade damping). For comparison, the linearized frequency
response curves of Chapter 4 are also included. Figure 5.6 features the same frequency
response loci shown in Figure 5.5 for softening absorbers, but with nonzero blade and
absorber damping éb = 2 x 10”3 and 56 = 2 x 10—6. These curves were obtained
numerically according to Equation (5.19) (stationary points) and Equation (5.30a)
(local stability). A set of frequency response curves showing more resolution with
respect to 6 within and near the nqresonance zone is shown in Figure 5.7 for a
hardening absorber path and in Figure 5.8 for a softening absorber path; these are
meant to accompany Figure 5.4 and Figure 5.5, respectively. Finally, it was shown in
Section 5.2.2 that the linear resonance structure described in Chapter 4 is preserved
under the scaling (see Figure 5.2 on page 125), and it is clear from Figures 5.4-5.8

that this structure qualitatively persists in the averaged system as well. However, the

141

nonlinearity gives rise to some additional features, and we describe them first for the
case of hardening absorber paths.

In Figure 5.4 either one or two nonlinear resonances can be observed, depending
on the order detuning /3, and there is interplay/duality — between the two. For
convenience, these are defined as primary and auxiliary (nonlinear) resonances and
are denoted by 72151” and REL, respectively. The primary resonance is simply the
nonlinear counterpart to the linear resonance (denoted by ’RL) and, for a hardening
absorber path, it bends toward the direction of increasing rotor speed. When it exists,
R?” is a secondary resonance that arises due to the presence of the nonlinearity.

For large undertuning values, REL and RL are nearly coincident close to o = or :
0.346, which is shown in Figure 5.4a, but the hardening nonlinearity sharply bends
the primary resonance in the direction of increasing o. In this figure, the nonlinearity
gives rise to an additional resonance RyL, which appears from zero rotor speed. For
the blade, it increases gradually for increasing o, whereas for the absorber it begins
at nearly constant amplitude. For both, the auxiliary resonant amplitudes increase
sharply at o = or which, recall, is the resonant speed of an isolated blade without an
absorber and it also corresponds to zero speed detuning, i.e., A = 0.

As 6 is increased REL and ’RL move together to the right6 (Figure 5.4b) toward
infinite o, leaving behind the same set of linear amplitude branches that were observed
in Chapter 4. This situation, which is shown in Figure 5.4c, corresponds to the no-
resonance zone predicted by the linear theory. However, the auxiliary resonance REL
persists, which is an artifact of the absorber path nonlinearity. The case of perfect
linear tuning (13 = 0) is similar, except that the linear amplitude branch for the blade
vanishes (this is shown in Figure 5.4d) and Ram and 72131” become coincident—that
is to say, as [3 is swept through exact linear tuning 721:1” essentially switches roles to

become the primary resonance. By further increasing the linear detuning (that is, for

 

6The error between the location of these resonances can be approximated by the curves in F ig-
ure 5.21) on page 125.

142

 

— Nonlinear (stable)
---- Nonlinear (unstable)
-~--—~ Linearized

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1,0 lfglfil NL
0. 0 Ra>l
_ _ —l _1
(C) fi— 0.001 _2 _2 ~\
/\ = -0.304 _3 _3
(NO-Res. Zone) -4 .4

   

 

 

 

log |11| lo
I l ..
0 0 ‘
NL_ NL
(d) e=x=0 j 73p 43a :12
(No-Res. Zone) 43 ’ .3
-4 ’ 41
2\ o

   

 

 

 

 

 

 

 

lo
1
0
(e) a = +0.001 :12
/\ = +0.304 .3
.4
‘ 0
lo
1
0
(f) 13 = +0.005 j;
/\ = +1.525 .3 ,
41 ,
0'
lo
1
0
(a) fi = +0.1 :12
,\ = +31.947 4; ,»
.4
0' 0'

 

 

 

 

6.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Figure 5.4. Blade/ absorber frequency response curves for the essentially undamped iso-
lated nonlinear system with a hardening absorber path (17 = 1), for various detuning values
6, and for n = 3, a = 0.84, 6 = 0.67, p = 0.0035 (5 = 0.0592), and F = 0.0001 (f = 0.117).

143

 

-— Nonlinear (stable)
---- Nonlinear (unstable)
—— Linearized

o o 0 Simulation

 

 

 

lo 11 lo “
lgll 1glvl
0 0 RNL
a)e=—01 j 3 p
A=—28904 _3 .3
—4 «1.
‘0’

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

loglfil log [0|
1 , 1 , RNL L
0 0 9 LR
(b) 13: 41.004 3 3k
A = —1.215 4 _3
«4. 4
0.2 0.4 0.6 0.8 i U
10S WI log (5|
1 . 1 RNL
0 > RNL 0 DL
P
a);%=—0001 ‘g 3 .---:----
_ . W
A = ——0.304 _3 _3 “I
(NO-Res. Zone) 411 -4 .
U - o

 

m)fi=x=o 3
(No-Res. Zone) .3
.4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0' 0'
(e) fi=+0.001
A=+0.304
. o o
(0 fi==+0005
A=+1.525
U 0’
@)fi=+01
,\=+31.947
0.2 6.4 0.6 0.8 1" 0.2 0.4 0.6 0.8 1"

Figure 5.5. Blade/ absorber frequency response curves for the essentially undamped iso-
lated nonlinear system with a softening absorber path (7} = —1), for various detuning values
5, and for n = 3, a = 0.84, (5 = 0.67, a = 0.0035 (5 = 0.0592), and F = 0.0001 (f = 0.117).

144

(a) ,8 = —0.1
A = —28.904
(b) (3 = —0.004
A = —1.215
(c) ,‘3 = —-0.001
A = -—0.304

(NO-Res. Zone)

((1) [3 = )\ = 0
(NO-Res. Zone)

(e) )3 = +0.001
A==+0304
(f) B = +0.005
A==+1525
@)fi=+01
A = +31.947

 

 

 

 

 

 

 

 

 

 

 

 

 

 

02 04 06 08 1”
log lfll
1
0
—1
—2
—3 \
—4 : : - - A - - ‘ - - A
' ' J ' T“ o
0 2 0.4 0 6 0 8 1
10g 1141
1 .
0
_1 ,
_2 .
_3 "x
411%
0'.
0.2 0.4 0.6 0.8 1
log lfll
1 .
0
—l
_2 , a".
_3
414M
- V o
0.2 0.4 0.6 0.8 l
loglfil
1
0
—1
_2 ‘
_3 \
‘4 N
- o
0.2 0.4 0.6 0.8 1
log (a)
1
0
-—1
_2 , }
a3 r
1 o

 

 

0.2 0.4 0.6 0.8 l

 

-------- Nonlinear (stable)
-------- Nonlinear (unstable)
— Linearized

o o 0 Simulation

 

 

 

+fioq
g

:vtb—v

 

Q

 

O
t0
0
4:.
O
C)
O
(X)
t—i

l
/

10 If)
1g l

.LclmeLo

 

0‘:
E

L&b¢o~g
W—
‘1
l
1
Q

00
g

 

 

0‘:
g

 

09
g

 

§

- vfioq

 

 

Figure 5.6. Blade and absorber frequency response curves for the same conditions in
Figure 5.5 except with nonzero damping: {b = 2 x 10‘3 and {a = 2 x 10‘6.

145

(a) ,8 = +1501}cr

(C) l3 = ficr
(N(.)-RBS. Zone)

(m g=+e5mnr
(NO-Res. Zone)

(8) l3 = 0
(NO-lies. Zone)

(1‘) (3 = 4.2513“

(g) ,3 = ‘0-503cr

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.6 01.8 1
0.8 1
0.8 1

 

 

0'

 

— Nonlinear (stable)
---- Nonlinear (unstable)
Linearized

 

 

 

(JG
3

11t1.-.

 

 

y—o
O
0‘?
E
O
[\D
O
.1;
_o
C)
O
(I)
g—a

l

 

LdarlaJ—oh-

 

 

0.2 0.4 0.6 0.8 i

 

 

 

(1.4 026 0.8 1

09

E
O
N

 

 

00
E

 

 

0':
g

iattoH;

 

 

 

 

0

 

0.2 0.4 0.6 6.8 1

Figure 5.7. Blade and absorber frequency response curves to accompany those shown in
Figure 5.4 for a hardening absorber path showing various levels of the order detuning 6
relative to ficr = —0.00164 inside and near the no—resonance zone.

146

 

—- Nonlinear (stable)
---- Nonlinear (unstable)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Linearized
loglfi]
1’ 1
0.
(a) e=+1.50,3€, 3 A.) '
_3.
—4.
U
02 04 06 08 1
loglfjl
1 .
0 K\
(b) e=+1.25;3c, 3 A.)
_3
.4.
0'
0.2 0.4 0.6 0.8 1
lloglfil
(C) 5:13“ :% -- "“22
(NO-Res. Zone) _3
_4
.0 U
02 04 06 08 1 0.2 0.4 0.6 0.8 1
lloglfil lfglfil
ol 0. l
, __ f _1 _1
(d) 13'- +0.5OJcr -2) 4-\
(ho-Res.Zone) _3 _3,
41.—\ 41
A o A A o
04 06 08 1 0.2 0.4 0.6 0.8 1
lloglfil
0 . l
_ —1
(e) ,0.—0 _2\
(NO—Res. Zone) .3
.4
0.4 0.6 0.8 1" 0.2 0.4 0.6 0.8 1"
l{)glfll
0: l K
I." _3
- 41
N A A0 A A A o
. 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
lfgl‘vl
0 l l
(g) ,8 : —000 301' 3 —\
51
. /\ —4
U 1 '0'
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Figure 5.8. Blade and absorber frequency response curves to accompany those shown in
Figure 5.5 for a softening absorber path showing various levels of the order detuning 6
relative to fie, = —0.00164 inside and near the no—resonance zone.

147

6 > 0) the linear resonance appears from the left at zero rotor speed and REL moves
with it to the right, which is shown in Figure 5.4e—f. AS 6 becomes large the primary
and linear resonances become nearly identical. This is shown in Figure 5.4g.

In summary, the primary nonlinear resonance behaves in the same way as the
linearized resonance when the linear detuning is swept from negative to positive, and
it observes the no—resonance zone by vanishing when Ba— < 3 < 0. It also features
a hardening bend in the direction of increasing o that extends out to infinite rotor
speed. Finally, the nonlinearity gives rise to an additional auxiliary resonance that
exists for all undertuning values, that is, for any ,8 < 0.

A similar trend can be observed in Figure 5.5 for the case of a softening absorber
path, except that the auxiliary resonance exists when the absorber is overtuned and,
of course, the primary and auxiliary resonances bend in the direction of decreasing
rotor speed. The primary nonlinear resonance does exist for detuning values within
the no—resonance zone, but its effective location corresponds to large rotor speeds
and it approaches infinite o as the linear order tuning approaches zero (ideal tuning).
In this way, the no—resonance zone essentially persists, where the primary nonlinear
resonance Simply places an upper limit on the effective range of permissible rotor
speeds. As ”RISK” moves with RL toward infinite o for increasing 3, it leaves its
remnants behind which, when the tuning such that ,6 > 0, is regarded as the auxiliary
resonance RSTL. A key difference for the softening case is that the linear branches
within the no-resonance zone are (for sufficiently small forcing levels and absorber
path nonlinearity) isolated from nonlinear response branches for rotor speeds to the
left of primary resonance. (Compare Figure 5.5c-d with Figure 5.4c-d.) For a softening
absorber path, it is therefore possible to spin the rotor up from zero speed to some
(sufficiently small) steady operating point without passing through resonance. This
is generally not possible for a hardening path, in which case there are potentially

unavoidable auxiliary or primary resonances for all order detuning values—including

148

those within the no—resonance zone. It can thus be said that absorber designs involving
hardening paths are generally not acceptable. Softening absorber paths are clearly
desired, but they do set an upper limit on the rotor speed, the value of which depends
to a large extent on the excitation strength and the strength of the absorber path
nonlinearity.

Figure 5.5 also shows simulation data corresponding to the full nonlinear model
for blade and absorber damping levels of {b = 2 x 10‘3 and £5 = 2 x 10’6, respectively.
The predicted frequency response amplitudes (which correspond to zero damping) are
seen to be in reasonably good agreement with these data. A corresponding set of fre-
quency response loci were numerically generated with the same damping levels used
in the Simulations. These follow from Equation (5.19) (stationary points) with I? = 0
and Equation (5.30a) (stability) and are shown in Figure 5.6. The results are seen to
be in very good agreement with the simulation data, thus validating the accuracy of
the averaged sector models. If only qualitative features of the forced response are de-
sired, however—response amplitudes in particular—the simple analytical expressions
defined by Equation (5.24) and Equation (5.25) are quite sufficient.

In both Figure 5.5 and Figure 5.6, simulation data could not be obtained for
the nonlinear auxiliary resonance branches (for the particular parameter values used)
due to their small domains of attraction. However, when the excitation strength is
increased, these domains widen and the branches of REL can be captured. This is
Shown in Figure 5.9, which depicts the blade and absorber frequency response loci
for a softening absorber path, linear tuning within the no-resonance zone, and for
various levels of the dimensionless force amplitude F. The figure also indicates an
upper limit on the excitation strength, where a bifurcation destroys the no—resonance
structure.

The frequency response results described above are in overall good agreement with

the actual nonlinear response (based on the full nonlinear equations and indicated

149

(a) 520.0001
f: 0.117

(b) 520.0002
f: 0.235

(c) Ff: 0.0003
f: 0.352

(d) E: 0.0004
f = 0.470

(e) F: 0.0005
f = 0.587

(f) F=0.0006
f = 0.705

(g) F: 0.0007
f = 0.822

—1
—2
.3

41M

 

 

log (11]
l

0
.1 r
—2

 

 

411W

 

 

 

 

 

 

 

 

 

 

10

1i

0)

—l

—2

$1M

‘4 -FPI‘V “ \
0.2 0.4 0.6 0.8 1"

l() ‘11

181!

0

—l

‘27 1‘

—31L—_-"J ------

—4 -\

Ao

 

 

0.2 0.4 0.6 0.8 1

 

 

—-— Nonlinear (stable)
---- Nonlinear (unstable)
._ Linearized

o o 0 Simulation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1103 I13!
01 km
—1‘.
_2 M
_‘30 '
—4 .
A o
0.2 0.4 0.6 0.8 1
log If)!
1 1
O W
.1 ,---
.2 M
-30
.4
A A - o
0.2 0.4 0.6 0.8 1
110ng
0 g
—1
_2 \
.3
.4
o
0 2 0 4 0.6 0 8 1
log lfil
l .
0r
—1~:
-2P
all
-41
A o
0.2 0.4 0.6 0.8 1
lo 13'
If! I
0’ k\r-vvwv--
-1 ' ‘7‘ K
—2
—3.
‘41
o
0 2 0 4 0.6 0 8 1
lo 0
Isl I
0.
—1'-u--r'ifl"‘.fl—”’ 4‘ I -
_2
45"
.4
- A A o
0 2 0 4 0.6 0 8 1
lfglfil
0» ‘ L
—1 ——"" A . g. ----- j.
_2
—3.
.4
o

 

 

0.2 0.4 0.6 0.8 1

Figure 5.9. Blade and absorber frequency response curves for the essentially undamped
isolated nonlinear system with a softening absorber path, for linear tuning within the no-
resonance zone (6 = BC, / 2 = ——0.822 x 10‘3), for various levels of the force amplitude F,
and for n = 3, a = 0.84, (5 = 0.67, and ,u = 0.0035 (5 = 0.0592).

150

by the simulation data), particularly when damping is included and if the parameter
values are reasonable. The representative examples depicted in Figure 5.6 Show that
the averaged sector models can be very accurate, even for very large positive or
negative order detuning 6 (values of n relative to n) or speed detuning A (rotor
Speed o relative to or). The same cannot be said, however, had the usual approach
to averaging been carried out. Recall from Section 5.2.4 that an alternative approach
was adopted in which o is retained in the averaged equations, essentially keeping key
higher-order terms in the model. This is a crucial observation in the methodology
since the usual approach to averaging (which amounts to simply replacing o with the
constant or) gives at best mediocre results, and it fails completely to capture the
no-resonance zone. The reader can verify these claims by comparing the analysis and
results presented thus far with those given in Appendix E, where the usual approach
to averaging is employed.

A criterion for zero blade motions relative to the rotating hub is derived next,
which is the nonlinear counterpart to the ideal, or exact linear tuning described in

Section 4.4.1.

5.4.2 Criteria for Zero Blade Amplitudes

The desired system response is one in which the blades remain stationary relative to
the spinning rotor and the absorbers move accordingly in a TW configuration. Such a
response can be achieved for a linear system (in the absence of damping) by the exact
linear tuning described in Section 4.4.1. However, a more realistic linear tuning strat-
egy is one in which the absorbers are detuned within the no-resonance zone, which
offers robustness to parameter uncertainties but it comes at the expense of residual
blade vibrations. The aim of this section is to possibly exploit the absorber path non-
linearity to address these vibrations and further improve the absorber performance.

The desired response is, in fact, possible (but not necessarily stable) for zero

151

 

damping and the corresponding requirements on the system parameters follow from

Equation (5.20) by setting 11 = 0. Then

0 = —-a'(n2 +1)o203in(@ — E) — fsin @, (5.32a)
0 = —cz(n2 + 1)o217 cos(§ — c“) — fcos Q, (5.32b)
0 = Ao2'17 + $20203. (5.32c)

However, the phase @ is undefined if 11 = 0 and hence Equations (5.32a) and Equa-
tion (5.32b) should be independent of §. To see this, the latter is solved for 17 and is

introduced to the first. Upon simplification the result is
0 = f sin c",

which is indeed @independent and it implies c" = ln for l E Z. Then Equation (5.32a)
and Equation (5.32b) both reduce to a single expression and, together with Equa-

tion (5.32c), the required conditions become

A

0 = c1(n2 + 1)o217 cos(lir) + f, (5.33a)

0 = (A + %n«52)020. (5.33b)

Since 21 = 0 by assumption, the case of '1’) = 0 gives rise to the trivial response
(11,17) = (0, 0) and mathematically it corresponds to a nonhyperbolic stationary point.
By restricting 17 > 0 (meaning that the absorber must assume nonzero motions in
order to achieve zero blade vibrations) and eliminating the absorber amplitude in
Equation (5.33), one can solve for the critical nonlinear tuning parameter. The result
is

4)\(12(n2 + 1)2o4

3 P

which is seen to depend on the engine order, the placement of the absorber relative

 

, (5.34)

Tlcr = —

to the blade, and the linear tuning order. However, it also depends on both the

excitation strength and the rotor speed, which implies that the critical nonlinear

152

tuning is valid only for a single set of operating conditions. While it may be possible
to design an active absorber, where the nonlinear tuning is adjusted on-the—fly for
varying f and o, effective implementation in the harsh operating environments (e.g.,
rotation, extreme temperatures, and so on) is likely to by impractical, if not entirely
impossible. Moreover, since proper linear tuning is negative (Aer < A S 0) and all
other parameters in the right hand side of Equation (5.34) are positive, the nonlinear
tuning requires hardening absorber paths (77 > 0). As discussed in Section 5.4.1 this is
an undesirable path type, one that gives rise to potentially problematic resonances—
even for proper linear tuning.

These findings clearly show that the nonlinear tuning defined by Equation (5.34)
is unsatisfactory for the passive absorbers under consideration. More generally, they
also suggest that nonlinearity cannot be exploited to improve the absorber perfor-
mance. However, we do offer some general recommendations for the critical nonlinear
tuning next with the understanding that they are feasible only for an active absorber
implementation, which may have applications in other settings.

Example plots of blade and absorber frequency response curves are shown in
Figure 5.10 for linear tuning within the no—resonance zone, for perfect nonlinear tuning
according to 77 = 77”, and also for slight over- and undertuning with respect to ncr. In
addition to the host of other issues described above, this figure highlights sensitivity to
parameter uncertainties. As shown in Figure 5.10c, any level of nonlinear overtuning
is accompanied by a jump instability to the nonlinear auxiliary resonance. Slight
nonlinear undertuning relative to ncr is therefore desirable. However, it is again
stressed that this is feasible only for an active absorber implementation, and there
still remains a potentially problematic auxiliary resonance.

Finally, while the nonlinear tuning scheme described above is not acceptable for
applications involving passive absorbers, all is not lost; some insight can be gleaned

and reinforced from the analysis. Equation (5.33a) implies that the absorber ampli-

153

 

-— Nonlinear (stable)
---- Nonlinear (unstable)
~—— Linearized

 

 

 

(a) 7) : 0-90’7cr

 

 

 

 

(b) 77 = 77a

 

 

 

 

(C) 77 = 1~10'77cr

 

 

 

A o A o
0.8 1 0.2 0.4 0.6 0.8 1

 

 

Figure 5.10. Example blade and absorber frequency response curves for linear tuning
within the no—resonance zone (B = Ber/2), for several nonlinear tuning values relative to
Her, and for a model with n = 3, a = 0.84, 6 = 0.67, p = 0.0035 (5 = 0.0592), f = 0.0001
(f = 0.117), and g}, = g}, = 0.

154

tude increases linearly with the strength of the excitation in a way that depends on
the engine order and placement of the absorber along the extent of the blade. Consis-
tent with intuition, the absorber should be placed as close to the end of the blade as
possible to achieve the smallest absorber motions given a specific excitation strength.
Equation (5.33a) also indicates that, for a given forcing level, higher engine order
excitations will give rise to lower absorber amplitudes.

The nonlinear results discussed up to this point for the special case of an isolated
sector qualitatively embody all of the fundamental features of the fully coupled system,
except for the possibility of additional instabilities to response types other than the

desired TW variety. The existence of such bifurcations is considered next.

5.5 Forced Response of the Coupled Nonlinear System

The forced response of the isolated nonlinear system, which consists of a single linear
blade and nonlinear absorber, were described in detail in Section 5.4. These funda—
mental results can be used to qualitatively predict many of the corresponding features
of the fully coupled nonlinear system, including its TW response amplitudes and jump

bifurcations to other traveling wave solutions. The frequency response amplitudes of

 

the coupled system qualitatively match those of the uncoupled case, except for a shift
in the primary resonance (when it exists) according to the third term on the right

hand side of Equation (5.19b), that is,

A — 2122(1 — cos 997,“) E A, (5.35)

2 = 522. Hence the shift will be small

by an amount that is directly proportional to V
if the inter-sector elastic coupling is weak, in which case amplitude predictions for the
fully coupled system based on the isolated sector model are quite accurate. In what

follows, it will be shown that stability results associated with the isolated nonlinear

system qualitatively apply to the coupled system as well, where any bifurcation iden-

155

tified in the former Simply corresponds to a jump in blade/ absorber amplitudes in
the latter to another traveling wave response. These bifurcations are said to preserve
the symmetry of the response. However, the coupled nonlinear system may feature
additional symmetry-breaking instabilities that the isolated system cannot predict,
which involve bifurcation to response types other than the desired TW. These possi-
ble instabilities are addressed in the next section, where the findings strongly suggest
that symmetry-breaking bifurcations do not occur.

It should be noted that, while closed form analytical expressions are available for
the prediction of response amplitudes, determination of local stability in the pres-
ence of coupling is quite a bit more formidable. The set of block decoupled Jacobian
matrices from Section 5.3.3 (for the coupled case) do offer a substantial savings in
computation (which is quite useful for numerical studies), but even these 4 x 4 re-
duced matrices are analytically unaccommodating and hence essentially intractable.
(Stability results for the simplified case of zero coupling follow from Equation (5.31),
which features complicated coefficients. The addition of coupling gives rise to N poly-
nomials of the same form with coefficients that are many times more complicated.)
At least some insight can be gleaned from the reduced Jacobians, however, and this
is done in Section 5.5.1, but they are otherwise handled numerically. In what follows
we offer a sampling of results based on extensive case studies and numerical investiga-
tions. These are briefly summarized in Section 5.5.2 using examples of models with

N = 5 (odd) and N = 6 (even) sectors.

5.5.1 Local Stability of the Traveling Wave Response
For the special case of zero coupling (1? = 0), local stability of a stationary point v
follows from the 4 x 4 Jacobian matrix J (SC ) , which is defined by Equation (5.30b).7

For a particular equilibrium point, and if this matrix is Hurwitz (meaning that all

 

7In what follows, we restrict the discussion to Cartesian forms for convenience, where it is un-
derstood that the same arguments hold for the corresponding polar forms as well.

156

four of its eigenvalues lie in the open left-half complex plane), there corresponds a
stable periodic orbit in the isolated nonlinear system [96,97,109]. If any eigenvalue
of J (SC ) crosses the imaginary axis into the right-half complex plane as the rotor speed
(or any other parameter) is varied, a jump bifurcation occurs in which all of the
blade and absorber amplitudes spontaneously assume different values, and the TW
symmetry is preserved.

A similar type of (symmetry-preserving) instability can be observed in the fully
coupled nonlinear system in which there is a jump in blade and absorber amplitudes
and the response maintains its TW configuration. This is found to occur when an
eigenvalue of one particular reduced Jacobian crosses the imaginary axis. Thus any
crossings by an eigenvalue of any other reduced Jacobian matrix generically corre-
sponds to a symmetry-breaking bifurcation in the coupled nonlinear system to a
response type other than the desired TW. However, based on extensive numerical
investigations, no such instabilities could be identified.

To see these features more clearly, we consider in more detail the reduced J acobian
matrices that were derived in Section 5.3.3, which are stated again here (in Cartesian

form) for convenience. They are
2noJ)CC) = P + 2Qcos 90k + 2jR sin (0k, 1: E N (5.36)

where the factor 2no has been moved to the left of the equality for convenience. The
matrices Q and R vanish when 1? = 0 and the matrix P is the same as Jgg) (isolated
sector), except for the addition of 21?2 (resp. —2192) in the (1,2) (resp. (2, 1)) element.
Since Equation (5.36) was obtained by way of a unitary transformation of the full
Jacobian matrix J (3%) E .‘flfffiy4, N (coupled sector), the eigenvalues persist in these
N, 4 x 4 reduced matrices. Note that they feature a cyclic structure with respect to

the index k E N, which arises from the sine and cosine terms involving (pk. Since

cos 99N+2—k = cos {pk and sin 9910424, = —sin We the reduced Jacobians generally

157

appear as complex conjugates of the form
P + 2Q cos (pk i 2jR sin «pk, (5.37)

except for k = 1 when the form is P + 2Q or if N is Odd, in which the k = (N + 2)/2

case gives rise to P — 2Q in addition.8

Based 011 extensive numerical evidence it is found that the real matrix

C c 1
Jiadp Jl ) = —(P + 2Q) (5.38)

2no

is the only one of the N reduced Jacobians that gives rise to instabilities. It corre-

sponds to Equation (5.36) with k = 1 and is such that

 

 

l—noéb —A 0 —C_YU2 .
A ——noéb do2 0
P 2 = -
+ Q 0 —do2 gno2C'D — noga gno2(C’2 + 3D2) + Ao2 ’
_ do? 0 —%no2(302 + D2) - /\o2 -—gno2C'D — nofia J

where A is the effective resonance shift defined by Equation (5.35) and, recall,
d = a(n2 + 1)o2. Equation (5.38) is the same as the Jacobian matrix JES) of the
isolated nonlinear system, but it additionally incorporates the resonance shift asso-
ciated with coupling, which is reflected in A. In fact, it is the Jacobian matrix of
an efiective isolated nonlinear system consisting of a single blade, a nonlinear ab-
sorber, plus a single elastic coupling element, which follows from the averaged sector
model corresponding to i = 1 rather than the model obtained by setting 1? = 0.
In this way, instabilities observed in .1 $31; directly correspond to those predicted by
J (g ). Moreover, since instabilities were detected only by the effective J acobian matrix

J (3%)}; = J EC) and none of the other reduced Jacobians, it thus follows (based on nu-

merical evidence) that there are no symmetry-breaking bifurcations to response types

other than the desired TW. Again, this observation is based on extensive numerical

 

8Analogous features that arise due to cyclicity can be observed in the multiplicity of eigenfre-
quencies and normal modes of a generic cyclic system (Figure 2.5) and also in the BTW, FTW, and
SW characteristics Of engine order excitation (Figure 2.3).

158

evidence and case studies. Therefore, while it cannot be said with certainty that such
bifurcations absolutely cannot occur, these findings strongly suggest that they do not.

A sampling of results is given next for specific values of N and with an emphasis on
blade/ absorber frequency response and traveling wave characteristics of the coupled

system.

5.5.2 Case Studies

In this section we briefly highlight some case studies for models with a specific number
of sectors. It is clear that N = 1 is a trivial case since it features only one sector.
The next simplest systems with N = 2 and N = 3 sectors are special cases where, in
addition to nearest-neighbor coupling, they also feature all-to-all coupling in which
each sector is coupled to all other sectors. Moreover, it is well-known that the case of
N = 4 sectors gives rise to additional rich dynamics that are not generically observed
for general N [106—108]. Finally, it is recalled from Section 2.4 that the traveling
wave nature (BTW, F TW, or SW) of the system can be different for an odd or even
number of sectors. The case studies are thus summarized for models with N = 5
(odd) and N = 6 (even) sectors. We begin with the case of N = 5.

A representative set of blade and absorber frequency response curves are is shown
in Figure 5.11 for softening absorber paths, undertuned absorbers, and for a number
of coupling levels that increase from zero. In fact, Figure 5.11a corresponds to the
blade and absorber amplitude responses shown in in Figure 5.5b and Figure 5.6b for
the isolated (zero coupling) nonlinear system, and Figures 5.11b-g Simiply show how
this picture changes as the coupling in increased from zero. For small coupling the
frequency response loci are nearly the same as the isolated sector case, where the
resonance Shift associated with coupling is essentially imperceptible. As the coupling
is increased the primary nonlinear resonance (linear resonance) moves to the right.

This resonance shift is expected, and is manifested by the term defined by Equa-

159

tion (5.35). Finally, simulation data is included near primary resonance in several of
the plots, which indicates that the results are valid even for large coupling strengths.
The sparsity Of data points can be attributed to very long simulation run times; even
with today’s computers it can take many hours for a numerical solver to settle into a
steady-state.

Figure 5.12 shows the blade and absorber time responses for the same parameters
used in Figure 5.11e and for a rotor speed of o = 0.55. Since the absorbers are
undertuned outside of the no-resonance zone, the mode shape associated with an
individual sector is (1, —1) and hence the blade and its attendant absorber feature
out—of—phase motions with respect to one another. This can be confirmed, for example,
by comparing the dashed line in Figure 5.12a to that in Figure 5.12b. Finally, the
engine order n = 3 excites mode p = n + 1 = 4 which, according to Table 2.2 on
page 32, is of the FTW variety. This can also be identified in the figure, where the
periodic motions of, for example, blade 1 are followed by the same motions in blade 2,
3, and so 011 until the pattern repeats itself.

A corresponding set of blade and absorber frequency response curves are shown
in Figure 5.13 for the same parameters use in Figure 5.11, except with N = 6 sectors.
Nothing qualitatively different is expected nor observed in these figures. However, in
this case the engine order n = 3 excites a SW mode corresponding to p = n + 1 = 4,
which can be verified by Table 2.2 and clearly observed in the blade and absorber time

responses depicted in Figure 5.14, which corresponds to Figure 5.13e with o = 0.55.

160

(a) V2=0
0:0
(b) u2=0.0001
0:0.041
(Qiflzoml
0:0.130
(d)zfl==001
0:0.411
(e) V2=0.1
o: 1.300
u)u2=05
o: 2.907
(g)b3=1
0:4.111

Figure 5.11.

lO

Ov—

_1,

set

L&&Lo~g

testesg

 

 

 

 

 

 

 

 

 

 
  
 

24 Ill!
02 04 06 08 1”
mm
1
02 04 06 as Si“
8 lfll
g WI
1
0.2 0.4 0.6 0.8 1 a
g Ifll
TW time response
shown in Fig. 5.12a
0’

 

 

0.2 0.4 0.6 0.8 1

0‘:
g

 

 

 

 

02
g lfll

 

 

U

 

02 04 as a8 1

 

— Nonlinear (stable)
---- Nonlinear (unstable)
— Linearized

o o 0 Simulation

 

 

 

. 0‘:
§

 

 

- (N
g

 

 

1.4.1.1.:

 

 

09
g

laticeg

 

 

 

02 04 06 08 1‘I
lo 17
ls! l .
0 . l
—1 'l
_2 ,
—3 . TW time response
‘4 shown in Fig. 5.12b
o

 

02 04 06 08 1

Act:
g

 

 

0’:
g

tsetse;

 

 

0.2 0.4 0.6 0.8 l

Blade/absorber frequency response curves for the essentially undamped

coupled nonlinear system with N = 5 (odd) sectors, softening absorber paths (7] = —1),
linear undertuning [3 = —0.004, various coupling levels u, and for n = 3, a = 0.84, 6 = 0.67,
p = 0.0035 (5 = 0.0592), and F = 0.0001 (f = 0.117).

161

E
Blade Amp. lul (x10'4)

3':
Absorber Amp. |17| (X 104)

 

----- Sector 1
— Sector 2,...,5

 

 

 

 

 

 

 

 

4; l

 

9.9994 9.9995 9.9996 9.9997 9.9998 9.9999 10
Time T (x104)

 

300

200 -

100 »

 

-200

 

  

 

./ ‘ , "\ 0
' n", n

 

—300

9.9994 9.9995 9.9996 9.9997 9.9998 9.9999 10

Time T (x 104)

FTW

FTW

Figure 5.12. (a) Blade and (b) absorber backward traveling wave (BTW) time responses
for the essentially undamped coupled nonlinear system with N = 5 (odd) sectors, softening
absorber paths (1) = —1), linear undertuning fl = —0.004, coupling strength I! = 0.1, and

forn=3,a

o = 0.55.

162

0.84, 6 = 0.67, ,1 = 0.0035 (5 = 0.0592), F = 0.0001 (f = 0.117), and

b
H II
O0

(b) u? = 0.0001

0 = 0.041
(c) u? = 0.001

0 = 0.130
(d) u? = 0.01

o = 0.411
(e) 112 = 0.1

o = 1.300
f) u? = 0.5

o : 2.907
(8) V2 =1

o = 4.111

log |ll|

 

 

0.6 0.8 l

 

 

0.6 0.8 1

 

 

 

 

  
  

 

 

 

 

 

 

 

 

 

10g Ifll
l
0 .
_1 ( TW time response
—2 I shown in Fig. 5.14a
.3
41.__..__./
A o
0.2 0.4 0 6 0.8 1
lo '11
1gl I
0
—1
—2
.3
.4
0.2
log M
1
0
—1
—2
.3
-4 J
o
0.2 0.4 0.6 0.8 1

 

—— Nonlinear (stable)
---- Nonlinear (unstable)
_ Linearized

o o 0 Simulation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

loglfil
l.
o I
_1 .
.2,____..—/
.3
.4.
o
02 04 0.6 08 1
lo 0
155' I ,
O I
.1
42k
.3
.4
A o
02 04 0.6 08 1
lo 1‘)
13| I .
0 l
_1
.2,__.__’/
-3l
.4
A o
02 04 0.6 08 1
lo 17
Isl | .
0 I
—l —_—/
.2
.3.
.4.
A A A o
0.2 0.4 0.6 0.8 1
lo 27
15' I ,
0» I
_1(
—3 TW time response
-4 . shown in Fig. 5.14b
0.2 0.4 0.6 0.8 10
lo 17
1gI I
0 I
_1 A
_2.
.3
.4
A o
02 04 0.6 08
lo 0
1eI I
O.
'1’ >
.2
—3
“11
- o
0.2 0.4 0.6 0.8 1

Figure 5.13. Blade/absorber frequency response curves for the essentially undamped
coupled nonlinear system with N = 6 (even) sectors, softening absorber paths (77 = —1),
linear undertuning fl = —0.004, various coupling levels V, and for n = 3, a = 0.84, 6 = 0.67,
)1 = 0.0035 (5 = 0.0592), and F = 0.0001 (f = 0.117).

163

Blade Amp. IuI (x10'4)

Absorber Amp. lvl (X 10'4)

300

200

100

-100

-200

—300

 

----- Sector 1,3,5
— Sector 2,4,6

 

 

 

 

 

 

 

 

 

 

 

 

 

oif{1,3,5 2': 2,4,6 ~.
“ 'V‘ ' SW
9.9994 9.9995 9.9996 9.9997 9.9998 9.9999 10
Time T (x104)
I ‘/\ ‘ SW
*. I N
9.9994 9.9995 9.9996 9.9997 9.9998 9.9999 10

Time T (x 104)

Figure 5.14. (a) Blade and (b) absorber standing wave (SW) time responses for the es-
sentially undamped coupled nonlinear system with N = 6 (odd) sectors, softening absorber
paths (7] = —1), linear undertuning B = —0.004, coupling strength 11 = 0.1, and for n = 3,
or = 0.84, 6 = 0.67, u = 0.0035 (8 = 0.0592), F = 0.0001 (f = 0.117), and o = 0.55.

164

5.6 Concluding Remarks

This chapter has extended the linear design, theory, and methodology of Chapter 4 to
include the basic, first-order effects of nonlinearity, which was introduced via the ab-
sorber path geometry. A key finding is that, for the passive order—tuned absorbers of
interest, one cannot exploit nonlinearity via the path design to improve absorber per-
formance. However, general conclusions on the effects of system nonlinearity can be
gleaned from the analysis. The results show that hardening characteristics are unde-
sirable since they give rise to a primary resonance, a potentially troublesome auxiliary
resonances at low rotor speeds, or both. If system nonlinearity is unavoidable, it was
shown that softening characteristics are acceptable and that they essentially set an
upper limit on permissible rotor speeds. Finally, for the weakly coupled and lightly
damped system under consideration, no symmetryAbreaking instabilities of the de-
sired TW type solution could be identified. These results, together with the linear
absorber tuning strategy given in Chapter 4, give rise to a final recommendations for

absorber design, which are summarized in the next chapter.

165

CHAPTER 6

Conclusions

This thesis has investigated the use of centrifugally-driven, order-tuned vibration
absorbers to suppress the steady-state motions of a rotating bladed disk assembly
under engine order excitation. For this purpose a simplified, lumped parameter model
was employed. Each sector was assumed to be identical and identically coupled,
consisting of a single-DOF pendulum-like blade model together with a general path,
lumped—mass absorber. At the time of writing, this work reports the first systematic
analytical treatment of systems of order-tuned absorbers applied to cyclic rotating
flexible structures under engine order excitation, and thus such a simplified model is
justified. Just as the generic single-DOF harmonic oscillator can be employed in the
theory of elementary vibrations to capture and understand fundamental properties
such as natural frequency and resonance, so can the model employed in this work to
quantify the rich linear resonance structure, the basic effects of system nonlinearity,
and to clearly motivate both a linear and nonlinear absorber design strategy in terms
of the system and absorber path parameters. The results of this work are fundamental
to future studies of its kind, both analytical and experimental.

The major contributions of this investigation are briefly reviewed in Section 6.1.
These have given rise to a number of detailed recommendations for absorber design,
which are summarized in Section 6.2. Finally, directions for future work are high-

lighted in Section 6.3.

166

6.1 Summary of Contributions

The main results of this study follow from Chapter 4, where the linearized dynamics
of the cyclically-coupled system fitted with order-tuned absorbers were investigated,
and in Chapter 5, where these basic results were generalized to include the first-order
effects of nonlinearity.

According to the linear theory of Chapter 4 there exists an ideal (exact) absorber
design such that, ill the absence of damping, a complete elimination of blade vibrations
(relative to the rotating hub) is possible. The design is accomplished by setting the
(isolated) absorber natural frequency equal to the excitation frequency, just as it
is done with the classical frequency-tuned dynamic vibration absorber due to Den
Hartog [99]. However, since both frequencies scale directly with the rotor speed, this
amounts to an order tuning in which the absorber tuning order identically matches the
order of the excitation. In this way, the linear order tuning is valid independent of the
rotor speed. While exact tuning offers the best possible (linear) absorber performance,
it is susceptible to the effects of parameter uncertainties. Any level of unintentional
absorber overtuning or sufficiently large undertuning gives rise to a linear system
resonance. The most significant finding of Chapter 4, and arguably of this entire
study, is the existence of a no-resonance zone. It consists of a finite range of absorber
undertuning values for which there are no system resonances over the full range of
possible rotor speeds. The upper bound of this spectrum of feasible designs consists
of exact tuning and it is bounded from the bottom by a critical linear detuning.
Proper absorber design involves intentional undertuning within this generally small,
but finite gap. The absorbers profit from such a design in terms of robustness to
parameter uncertainties, but this is accompanied by slightly reduced performance in
the form of residual blade vibrations.

The fundamental linear results described above were generalized in Chapter 5 to

include the basic first-order effects of nonlinearity, which was introduced via the ab-

167

sorber path geometry, and the possibility of exploiting this nonlinearity to improve
the absorber performance was investigated. It was shown that the underlying linear
resonance structure (and hence the no-resonance zone) qualitatively persists, provided
that both the path nonlinearity and excitation levels are sufficiently small. In this
way, the linear tuning from Chapter 4 remains effective. There does exist a critical
nonlinear tuning that guarantees a branch of solutions corresponding to zero blade
motions (which is, in fact, valid for any linear tuning). However, it was shown to
depend on the rotor speed and excitation strength, and is thus effective for only a sin-
gle operating condition—much like the classical frequency-tuned dynamic vibration
absorber. It is therefore impossible to exploit nonlinearity to further improve the
performance of the passive order-tuned absorbers of interest. However, the analysis
does highlight some general conclusions on the effects of nonlinearity (in the absorbers
or otherwise) that can aid in the absorber design process. First, it was shown that
softening characteristics are acceptable, and that they essentially place an upper limit
on permissible rotor speeds. In contrast, hardening characteristics should be avoided
altogether, since they give rise to problematic auxiliary and/or primary resonances.
Finally, for the weakly coupled and lightly damped system under consideration, no

symmetry-breaking instabilities of the desired TW response could be identified.

6.2 Recommendations for Absorber Design

In what follows the major results of this thesis pertaining to absorber design are
consolidated for quick reference. In Section 6.2.1 the linear tuning strategy described
above, together with conclusions based on the nonlinear analysis, are summarized
in a parameter Space that involves the linear order detuning and nonlinear tuning
parameters. Section 6.2.2 comments on system damping, and suggestions for the
absorber Sizing and placement are give in Section 6.2.3. Finally, a particular class of

absorber path types is recommended in Section 6.2.4.

168

6.2.1 Linear and Nonlinear Tuning

Proper absorber design can be summarized in the parameter space depicted in F ig-
ure 6.1. This design chart indicates ideal (exact) absorber tuning and also qualitative
regions of desired, acceptable, possibly poor, and poor absorber performance due to
primary resonance (prim. res.) and / or auxiliary resonance (aux. res.) in terms of the
linear and nonlinear tuning parameters 6 and 77, the values of which span the range
from (linear) under- to overtuning and (nonlinear) softening to hardening, respec-
tively. A nonlinear absorber tuning scheme with 77 > 0, that is, if the absorber paths
are hardening, involves a potentially problematic auxiliary resonance at low rotor
Speeds, an unavoidable primary resonance, or both. Thus any design in quadrants
I and region II of the (5,77) parameter Space gives rise to poor absorber perfor-
mance. Designs ill region [In are generally undesirable, especially for light damping,
in which case the auxiliary resonance is more problematic. Similar statements can
be made for any linear overtuning (6 > 0) or for sufficiently large linear undertuning
(13 < Ber), in which case a linear (primary nonlinear) resonance is guaranteed. Hence
any design in quadrant I V of Figure 6.1 yields poor absorber performance, as do
all designs in quadrant III to the left of ficr. This leaves only the shaded region in
quadrant three (denoted by I I Id) where absorber designs are feasible, though the
performance degrades as the paths are made more softening. (This has the effect of
moving the primary resonance in the direction of lower rotor speeds, thus limiting the
effective operating range of the bladed assembly.) Ideal absorber tuning corresponds
to (,8, 77) = (0,0), but in order to incorporate robustness to parameter uncertainty,
a tuning scheme in which the absorbers are slightly softening and tuned within the
no—resonance zone is recommended, that is, in the region indicated by “Acceptable”

in Figure 6.1.

169

[ Hardening Path ]

 

 

7?
I ll
I
II . I I u I
I
Poor I Possible Poor Poor
Performance : Performance Performance
(prirn./aux. res.) I (aux. res.) (prim. res.)
I
ficr I Desired Ideal
I
[ Undertuning ] r... / ._ > [B [ Overtuning ]
III : Acceptable [V
l I I I d
I
Poor I Decreased Poor
Performance : Performance Performance
(prim. res.) I (aux. res) (prim./aux. res.)
I
' I
I
I
[ Softening Path ]

Figure 6.1. Absorber design chart showing ideal tuning and qualitative regions of desired,
acceptable, possibly poor, and poor absorber performance due to primary resonance (prim.
res.) and/or auxiliary resonance (aux. res.) in terms of the linear and nonlinear tuning
parameters 6 and 17, the values of which span the range from under- to overtuning and
softening to hardening paths, respectively.

170

6.2.2 Damping

The basic effects of absorber, blade, and inter—blade, damping were discussed in Sec-
tion 4.5, where it was indicated that the latter two forms of damping actually help
the absorbers address blade vibrations. However, the presence of absorber damping
essentially weakens their action on the blades and, hence lessens their ability to op-
erate properly. For sufficiently large absorber damping, this can lead to a system
resonance—Aeven for proper linear undertuning—thereby destroying the no—resonance
zone. Thus in addition to the linear and nonlinear design recommendations given
in Section 6.2.1, the absorber damping should be kept sufficiently small so that the
no—resonance zone persists. The response plots shown in Figure 4.12 on page 112 can

be used to estimate maximum permissible absorber damping levels.

6.2.3 Absorber Sizing and Placement

Throughout this work the results have shown that large absorber inertia (relative to
the blades) is highly desirable. This is quite clear in physical terms Since a larger
absorber mass is able to exert increased dynamic loads on its attendant blade and is
hence more effective (if properly tuned) in addressing blade vibrations. It was also
shown in Section 4.4.2 that the extent of the no—resonance gap depends (nearly lin-
early) on the absorber-to—blade mass ratio and (nearly quadratically) on the absorber
placement along the blade length. By increasing either, the gap can be widened. Thus
the absorber mass should be made as large as possible, where it is understood that
this is limited fundamentally by the blade geometry, and it should be placed close to

the end of the blade.

6.2.4 Path Type

A key result from Chapter 5 is that nonlinearity cannot be exploited to improve the

absorber performance, and it is therefore desirable for the absorber motions to be lin-

171

ear. This can be achieved by selecting a tautochronic path, the geometry of which has
been systematically described by Denman [17] for a bifilar pendulum absorber config-
uration. Such a path could be implemented by restricting 77 = 0 in Equation (3.20)
on page 66 and then backing out the appropriate expansion coefficients b0, b2 and 04

in Equation (3.19).

6.3 Directions for Future Work

This investigation has laid the fundamental groundwork for future analytical and
experimental studies involving order-tuned vibration absorbers applied to nominally
cyclic rotating flexible structures under engine order excitation. This is essentially
a new line of study, one that unites the individually mature bodies of research on
absorber systems and cyclic systems, and hence there is considerable work left to be
done. Some major topics that remain to be addressed are briefly considered below.

Many of the individual subtopics could form the basis of an MS. or PhD. level thesis.

MISTUNING STUDIES

The models considered for this study were perfectly cyclic, consisting of identical,
identically-coupled sectors. However, there will always exist mistuning among the
sectors, that is, small random uncertainties in system parameters (due to in—service
wear, machine tolerances, and so on) that break the cyclic symmetry [37—40]. This
can lead to localization of vibration energy to a subgroup of sectors, giving rise to
higher vibration amplitudes than what is predicted by the perfectly cyclic system
[41—45]. With our current knowledge of localization in nominally cyclical systems, it
is expected that their responses will also exhibit localized behavior when absorbers
are attached to the substructures. There are many questions pertaining to absorber

design that must be addressed in this context, which include the following.

172

0 Will the response be localized in the blades, in the absorbers, or both? If so, to
what degree?

0 How is the response affected by system parameters, in particular by key param-
eters that govern localization?

0 In the presence of mistuning, how does one account for nonlinearity (including
impacts) that become unavoidable for the small absorber masses required by
the blade geometry?

0 How does localization affect the linear and nonlinear absorber design? Does the
no—resonance zone persist? Is it possible to exploit nonlinearity when mistuning
is present?

0 Can intentional patters of mistuning of the blades and / or absorbers be employed
to enhance the operation of the overall system?

The answers to these questions are crucial if absorbers are to be implemented in

practical systems.

HIGHER-FIDELITY MODELS

Future analytical studies should consider higher-fidelity blade models, including those
with lumped parameters and many degrees of freedom [36], continuous beam and shell—
type elements [9], and full finite element representations [3]. It may also be possible
to employ systems of vibration absorbers, including multiple absorbers applied to
a single structural element or multiple absorber implementations in a single system,

each tuned to address a specific problematic resonance.

METHODOLOGY

The desired traveling wave response exhibits the highest possible degree of symmetry
and it is said to belong to the cyclic group Z N [110—112]. In addition to the usual
jump bifurcations that preserve this symmetry (these are predicted by the isolated
sector model), there could be a host of other instabilities when coupling is present,

including those with reduced symmetry (the so-called isotropy subgroups of Z N) or

173

no symmetry at all [106—108]. In this work, however, numerical evidence has strongly
suggested that no such symmetry-breaking bifurcations occur. The mathematical
machinery of group theory, which offers a tremendously powerful and systematic way
to catalogue these bifurcations, could possibly be used to prove this claim [113,114].
The theory of groups has also been employed in mistuning studies [57, 115, 116], and is

likely to be very useful to investigate the effects of mistuning on absorber performance.

EXPERIMENTS

Lastly, experimental validation of the results of this thesis (and of future analyti-
cal studies) is critical. In the context of this work, it must be verified that the
small, but finite no-resonance zone is physically realizable in order for the tuning
recommendations to be of use. Moreover, even if it can be analytically proved that
there exist no symmetry-breaking bifurcations, this must be also be observed in a
carefully-controlled experimental setting before the results can be seriously consid-

ered for practical implementation.

174

APPENDIX A

Selected Topics from Linear Algebra

A.1 Introduction

In what follows some selected topics from linear algebra are reviewed. Most of the
basic results are included either as a quick reference or to support theoretical develop-
ments elsewhere in this thesis. Two matrix operations are introduced in Section A2
and some special matrices are described in Section A.3. Similarity transformations

and their basic properties are discussed in Section A.4.

A.2 The Direct Sum and Direct Product

Definition A.1 (Direct Sum) Fori = 1, . . . , N let A,- e CpixPi with each p,- 6 2+.
Then the direct sum of A,- is denoted by

A1€BA2€B...®AN=®£:1A7

and results in the block diagonal square matrix

[A1 0 0‘
0 A 0
A: . .2 , .
_0 0 AN‘

 

 

of order p1 +732 +. . .+pN, where each zero matrix 0 has the appropriate dimensionA

175

In this work the direct sum of N matrices A,- is denoted by the block diagonal matrix

diag(A1, A2, . . . ,AN) = diag (A7).
i=1,...,N
For the case when each A,- = a,- is a scalar (1 X 1), the direct sum of a,- will be denoted
by the diagonal matrix
diag (a1,a2, . . . ,aN) = diag (a7).
i=1,...,N

The direct, or Kronecker product is defined next.

Definition A.2 (Kronecker Product) Let a, b E C". Then the direct product ( or

Kronecker product) of a and bT is the square matrix

-a1b1 0102 albn-

a2b1 0202 - - - a2 bn
a®bT= . . .

Lanbl aan ' ’ ’ anbnd

 

 

Let the matrices A E men and B E Cqu. Then the direct product ofA and B is

the mp >< nq matrix

PallB a12B - - - 0.1an
a B a B a B

A®B= 2% 2? , 2:7 . A
_amlB 'am2B amnB‘

 

 

Some important properties of the direct product are as follows.

1. The direct product is a bilinear operator. If a is a scalar and A, B are square
matrices, then

07(A (X) B) = (0A) (8) B = A (8) (dB). (A.1)

2. The direct product distributes over addition. If A, B and C are square matrices,
such that A and B (resp. B and C) are of the same dimension, then

(A+m®C=A®O+B®O (An)
A®(B+C)=A®B+A®C. (A.2b)

176

3. The direct product is associative. If A, B, and C are square matrices, then

A®(B®C)=(A®B)®C. (A.3)

4. If A, B, C, and D are square matrices such that AC and BD exist, then

(A a B)(C a D) = (AC) 8 (BD). (A4)

5. If A and B are invertible matrices, then

(AaB)—1 = A—1 ®B‘1. (A.5)

6. If A and B are square matrices, then

(A a B)T = AT a BT, (A.6a)
(A a B)H .—_ A“ a B7“, (A.6b)

where ( - )T denotes transposition and (-)H is the conjugate transpose.

7. If A and B are square matrices with dimensions n and m, respectively, then

det(A (8 B) = (det A)m(det B)", (A.7a)
tr(A <87 B) = trA trB. (A.7b)

A.3 Special Matrices

There are a number of special matrices employed in this work, and their definitions
and pertinent properties are outlined here. Hermitian, and unitary matrices are
defined first (a summary of these special matrices is given in Table A.1), followed by
a brief treatment of two important permutation matrices. The details of the Fourier
matrix and circulant matrices, which are employed throughout this work, are deferred

to Appendix B.

177

Table A.1. Some types of special matrices.

 

Type Condition

Symmetric A = AT

Hermitian A = AH

Orthogonal ATA = I or AT = A—1
Unitary AHA = I or AH = A-1

 

 

A.3.1 Hermitian and Unitary Matrices

Definition A.3 (Hermitian Matrix) A matrix H e chN is said to be Hermi-
MnmnzHfl A

The elements of a Hermitian matrix H satisfy hik = hk, for all 1 S i, k S N. Thus
the diagonal elements hi,- of a Hermitian matrix must be real, while the off—diagonal

elements may be complex. If H = HT then H is said to be symmetric.

Definition A.4 (Unitary Matrix) A matrix U E CNXN is said to be unitary if
UHU = I. A

Real unitary matrices are orthogonal matrices. If a matrix U is unitary then so
too is UH. If in addition it is nonsingular then UH = U‘l.
A.3.2 Permutation Matrices

A general permutation matrix is formed from the identity matrix by reordering its
columns or its rows. Here we introduce two such matrices: the cyclic forward shift
matrix and the flip matrix.

THE CYCLIC FORWARD SHIFT MATRIX

The N X N cyclic forward shift matrix plays an important role ill the theory of

Circulants. It is populated with one’s along its superdiagonal and in the (N, 1) position

178

and its remaining elements are set to zero, that is,

— -I

 

 

010---00
001...00
N 000...10 ()
000--.01
_100---00‘NxN

It. will be shown subsequently that Equation (A8) is a circulant matrix, and hence

we defer a treatment of its properties to Appendix B.

TIIE FLIP MATRIX

The N x N flip matrix has one’s in the (1,1) position and along the subantidiagonal,

with all other elements equal to zero. It is given by

 

 

1 0 0 ~-- 0
0 0 ~-- 0 1
0 0 0 ~-- 1 0
"AN = . . . , . . (A.9)
0 0 1 0 0
L0 1 0 0 0‘ NxN
and is such that
xi, = I N, (A.10a)
5716 = reg = EN = rel—VI, (A.10b)

where I N is the N x N identity matrix.

A.4 Similarity Transformations

Definition A.5 (Similarity Transformation) Let Q be an arbitrary nonsingular
matrix. Then B = Q-IAQ is a similarity transformation and B is said to be similar
to A. A

179

Table A.2. Some types of linear transformations.

 

 

Type Condition Transformation
Equivalence P, Q are nonsingular B = PAQ

Congruence Q is nonsingular B = QTAQ

Similarity Q is nonsingular B = Q‘lAQ
Orthogonal Q is nonsingular and orthogonal B = QTAQ = Q‘IAQ
Unitary Q is nonsingular and unitary B = QHAQ = Q—IAQ

 

If B is similar to A, then A = (Q‘1)—1B(Q_1) is similar to B. It therefore
suffices to say that A and B are similar matrices. A summary of some other linear
transformations is given in Table A2. By inspection of this table, it also follows that
if B is orthogonally (resp. unitarily) similar to A, then A and B are orthogonally

(resp. unitarily) similar matrices.

Theorem A.1 If A and B are similar matrices, then they have the same character-

istic equation and hence the same eigenvalues. C]

PROOF. Let p A and p3 denote the characteristic polynomials of A and B, respec-
tively, and let B be similar to A. That is, let B be any matrix such that B = Q-IAQ
for some nonsingular matrix Q. Then the characteristic polynomial of B is
793 ( )= det(B- 1)

=det(Q’1AQ- AQ—IIQI
=det(Q‘1(A- AIIQI
.. —det(Q 1)det(A —AI)det(Q)

: pA()‘)I
where we have used the fact that det(Q"1)det(Q) = det(Q’lQ)= det(I )- — 1. Thus

A and B have the same characteristic polynomial and share the same eigenvalues. I

Theorem A.1 guarantees that the eigenvalues of a matrix are preserved under a
similarity transformation; the same is true for orthogonal and unitary transforma—

tions.

180

Next we Show that if A and B are similar matrices, and if p is an arbitrary finite

polynomial, then p(A) is similar to p(B).

Theorem A.2 Let p be an arbitrary finite polynomial and B = Q“1AQ. Then
73(3) = Q_1P(A)QA 1:1

PROOF. Let Q be an arbitrary nonsingular matrix and let

N

79(75) = Z thk

k=0

be a polynomial of degree N with arbitrary constant coeflicients ck. Then
AB) = p(Q”1AQ)
N
= Z Ck(Q—1AQ)k
k=0

= cor + ClQ—IAQ + c2Q_1AQQ"1AQ + . . . + cNQ—IAQ- . -Q_1AQ
= col + ClQ—IAQ + c2Q_1A2Q + . . . + cNQ‘lANQ

= Q-1(c01+clA+c2A2 +...+cNAN) Q
= Q‘1p(A)Q.

which completes the proof. I

If one chooses p(t) = tk with k > 0, then we have the following.
Corollary A.1 IfB = o-IAQ, then Bk = o-lAkQ for any k e 3.. [:1
Diagonalizability of a matrix is defined next.

Definition A.6 (Diagonalizable Matrix) A square matrix A is diagonalizable if
there exists a nonsigular matrix Q and a diagonal matrix D such that Q"1AQ = D.A

Thus a matrix is diagonalizable if it is similar to a diagonal matrix. If A is
diagonalizable by Q, we say that Q diagonalizes A and that Q is the diagonalizing

matrix.

181

Theorem A.3 An N x N matrix A is diagonalizable if it has N linearly independent

eigenvectors. CI

PROOF. Suppose A has N linearly independent eigenvectors and denoted them by
q1,q2, . . . ,qN. Let A,- be the eigenvalue of A corresponding to q,- for each i =

1, . . . , N. Then if Q is the matrix that has as its ith column the vector q,-, it follows

that

AQ = (AQIIAQ2IMIAQN)
=(Q1A1IQ2A2r-~.QNAN)

=(ql,Q2.....qzv) diag (Al)
i=1,...,N
EQD.

Since Q is nonsingular by hypothesis, D = Q—IAQ. I

182

 

 

APPENDIX B

The Theory of Circulants

B.1 Introduction

This appendix gives a more exhaustive treatment of the theory of circulants and is
meant to complement the overview given in Section 2.2. It is distilled from the seminal
work by Davis [62] and follows the presentation style of Ottarsson [36], one that should
be familiar to the vibrations engineer. The sections that follow act simultaneously
as a detailed reference and tutorial. Thus, in addition to a detailed treatment of the
theory (including many of the proofs), some worked examples are also included.
The appendix is organized as follows. Circulant and block circulant matrices are
defined in Section B2 and Section B.3, respectively, and some of their more relevant
properties are given. Diagonalization of (block) circulants is discussed at length in
Section 8.4, which begins with a treatment of the N M roots of unity in Section B.4.1
and the Fourier matrix in Section 8.4.2. It is subsequently shown how to diagonalize
the cyclic forward shift matrix in Section B.4.3, a circulant in Section B44, and a
block circulant in Section B.4.5. The appendix closes in Section B.4.6 with some
generalizations of the theory, including the diagonalization of block circulants with

circulant blocks.

183

B.2 Circulant Matrices

We begin with a definition.

Definition B.1 (Circulant Matrix) An N x N circulant matrix (or circulant) is

formed from an N -vector by cyclically permuting its entries and is of the form

[‘61 C2 .o. CN .-
CN C1 CN—l

C = . . , . A
_C2 C3 . o 0 C1 d

 

 

Thus a circulant matrix is defined completely by an ordered set of generating
elements cl, c2, . . . , c N in its first row. These are cyclically shifted to the right by one
position per row to form the subsequent rows. The set of all such matrices will be
designated by (5N, and are said to be circulant matrices of type N.

It is convenient to define the circulant operator circ ( - ) that takes as its argument
the generating elements c1, c2, . . . ,c N and results in the array given in Definition B.1,
that is,

C=circ(c1,02,...,cN). (B.1)

An N x N circulant can also be characterized in terms of its (i, k) entry by (C),-)c =

Ck—i+1(mod N) With 1 S i, k S N'

Example B. 1

circ (a, b, c, d) =

9053.9
OREG-
$1.9 o~n
Qty-0821..

 

 

I- .I

If a matrix is both circulant and symmetric it can be written as

circ(c1,c2,...,cN,cN+2,cN,...,C3,c2), Neven
'2' 2 '2'
C: (B2)
CirC(C1)C27" ' ICN—IICN-l-licN-l-IICN—li' "763102)7 N Odd
2 2 2 _2“

 

184

 

and necessarily has repeated generating elements; only (N + 2) / 2 are distinct if N is
even and (N -I- 1)/ 2 are distinct if N is odd. The set of all N x N symmetric circulants
will be denoted by Y‘gN.

 

 

Example B.2
Fa b c b”
b b
N = 4: circ (a,b, c, b) = a C E 54604
c b a b
b c b a
. Ell
fa b c c b
b a b c c
N25: circ(a,b,c,c,b)= c b a b c €y<g5
c c b a b
_b c c b a_

 

 

It is clear from Equation (A8) of Section A.3.2 that the cyclic forward shift matrix
is a circulant with generating elements 0,1,0,. . .,0,0. Hence the integer powers of

0N can be written as

(79,, =circ(1,0,0,0,0,...,0,0) =1N
(7],. = circ(0,1,0,0,0,...,0,0)

0%; =circ(0,0,1,0,0,...,0,0)

oN—l =circ(0,0,0,0,0,...,0,1)

 

0% =circ(1,0,0,0,0,...,0,0) =5?V =INJ

Next we give (without proof) a necessary and sufficient condition for a square

matrix to be circulant.

Theorem B.1 Let oN be the cyclic forward shift matrix. Then a N x N matrix C
is circulant if and only if CoN 2 UNC. CI

185

Any matrix that commutes with the cyclic forward shift matrix is, therefore, a
circulant. Theorem B.1 also says that circulant matrices are invariant under similarity

transformations involving the cyclic forward shift matrix.

Example B.3 Consider the 3 x 3 matrix

a b c
A: ca b
b c a
Since
abc 010 cab 010 abc
cab001=bca=001cab,
bca 100 abc 100 bca
it follows that A = circ (a, b, c) E ‘53. EU

An important feature of circulants is that they can be represented by a finite ma-
trix polynomial involving the cyclic forward shift matrix and its powers. In particular,
by inspection of the structure of the matrices OR, in Equation (B.3), it is clear that

a circulant matrix with generating elements c1, c2, . . . ,c N can be represented by the

matrix sum

Circ (611021 - - - ’CN) = CIIN + C20N + 030117 + . . . + cNolNd1
N
k_
= Z CkO'N 1- (BA)
k=l

This property is exploited in Section B.4.4 to diagonalize a general circulant matrix.

Example BA The matrix A = circ (a, b, c) from Example B.3 can be represented by

the matrix sum

A 2 a1;; + bo3 +60%

100 010 001 abc
=a010+b001+0100 = cab. (217,1
001 100 010 bca

Next we introduce block circulant matrices, which is a natural generalization of

ordinary circulants.

B.3 Block Circulant Matrices

Suppose each entry ck of the circulant array in Definition B.1 is replaced by the
M x M matrix Ck for k = 1,. . . , N. Then the resulting NM >< NIVI array is a block

circulant matrix of type (M, N) and is written as
C=Cer (C1,C2,...,CN), (B.5)

where C1, C2, . . . , C N are its generating matrices. The set of all such matrices will
be denoted by fi‘d’Mw. A matrix C E QT)”, N is not necessarily a circulant, as the

following example demonstrates.

Example B.5 Let

 

 

 

 

 

2 —1 —-1
A: , B: 0
—1 2 0 —1
Then
’2 —1§—1 0§0 05—10'
F ‘ “120‘10 ..... 00—1
ABOB —102—1§-10§00
c- B A B 0 - 0—1—120-10 ..... 9..
OBAB 005—10g2—13—10
B 0 B A -9 ..... 9-.i..Q..-.-:.1.§:.1..--2.-i.-Q---:.l.
- - ——10§0 0§—10;2—1
_0 -—1§0 0:0 —1§—1 2,
is a block circulant of type (2,4), but it is not a circulant. ED

Next we give (without proof) a necessary and sufficient condition for a matrix to

be a block circulant.

Theorem B.2 Let 0N be the cyclic forward shift matrix of dimension N and IM be
the identity matrix of dimension M. Then a NM x NM matrix C is a block circulant
of type (1M, N) if and only if C(a’N (8) 1M): (0N <8) IM)C. D

187

 

The reader can verify that the matrix C in Example B.5 satisfies the condition in
Theorem B.2, but not that in Theorem B.1.

A block circulant, block symmetric matrix of type (M, N) has the same form as
Equation (B.2), and is obtained by replacing each entry ck by the M x M matrix
Ck for h = 1, . . . , N. The set of all such matrices will be denoted by @‘fiflVM’N.
The matrix C iii Example B.5 is recognized to be a block symmetric, block circulant
matrix of type (2, 4), that is, it is contained in .93‘633Y2A.

A block circulant with generating matrices C1, C2, . . . ,C N can be represented by

the mariX sum

Cil‘C(C1,C2,...,CN)=IN®C1+UN®C2+...+O’%—1®CN

where the integer powers of the cyclic forward shift matrix are given by Equa-

tion (B.3).

B.4 Diagonalization of Circulants

Any circulant matrix can be represented in terms of the cyclic forward shift matrix,
which is clear from Equation (B4). The diagonalization of a general circulant begins,
therefore, by finding a matrix that diagonalizes oN. Together with some basic results
from linear algebra (these are summarized in Appendix A) this leads naturally to the
diagonalization of an arbitrary circulant. Regarding a suitable diagonalizing matrix,
there are a number of candidates [91,92,94,100,102,117], but all seem to feature
powers of the N m roots of unity or their real/imaginary parts. In this work we
employ the complex Fourier matrix, which has as its elements the distinct N m roots
of unity and their integer powers; these are defined in Section B.4.1. The Fourier
matrix is introduced in Section 34.2, whereupon its relevant features are detailed.

The diagonalization of the cyclic forward shift matrix is carried out in Section B.4.3

188

via a unitary transformation involving the Fourier matrix, and the diagonalization of
a general circulant matrix is subsequently described in Section B.4.4. These results
are generalized to handle general block circulant matrices in Section B45 and some

special block circulants in Section B.4.6.

B.4.1 Nth Roots of Unity

Here we follow the presentation in [118]. The N M roots of a complex number 20 =
roelgo are given by a nonzero number 2 = rejg such that zN = 20 with N E Z+, or
A?

upon substitution, r ejNO = 7.06790. This equality holds if and only if rN = r0 and

N0 = 90 + 2771: with k: E Z. Therefore,

r=N\/f‘;

, keZ (BO
6 _ 60 + 277k
" N
and the N M roots are
6 2 k
z: Wexp<j LL13”), kEZ. (B8)

It is clear from this exponential form that the roots all lie on a circle of radius N r0
centered at the origin in the complex plane, and that they are equally distributed every
2rr/N radians. Hence all of the distinct roots correspond to k = 0, 1,2, . . . , N — 1.
The distinct N m roots of unity follow from Equation (B8) by setting r0 = 1 and

2.1/N Xp( N A) , , ’ 7. . I, ' ( ' )

The primitive N th root of unity corresponds to k: = 1 and is denoted by
27);:
to N = e . (B.10)

Note that the integer powers w N -(e_ll7')k— — {Ark of the primitive N m root of unity

are equivalent to the distinct N th roots of unity, i.e., those given by Equation (39).

189

 

They are

1, 71W, 10%;, . . . ,wfi—l,

example plots of which are shown in Figure B.1 for N = 1, 2, . . . ,9.

B.4.2 The Fourier Matrix

2 '71'
Definition B.2 Let row 2 e72 be the primitive Nth root of unity with N E Z+ and
j = \/—1. Then the N X N complex Fourier matrix is defined as

 

 

f1 1 1 1 1
1 1 wN 11712V log—1
2(N—1)
EN = — 1 10%, wjlv w . A
W . . . N.
_1 ”mg—1 wiéN—l) - - - wng—lxN—UJ NXN

Clearly the Fourier matrix is symmetric, but generally it is not Hermitian. It can

be written element-wise as

1 (i-1)(k—1)
E - = ——
( N)lk r—NwN

___ Lem—0w.
\/N
1 BRIG-User

W I

where tp, is the angle subtended from the positive real axis in the complex plane to

i,k=1,...,N (B.11)

the ith‘ power of wN. It will be shown in Section B.4.4 that all circulant matrices
share the same linearly independent eigenvectors, the elements of which compose the

N columns (or rows) of EN. They are denoted by the column vectors

1 (-'—1) 27—1 N—l i—l T
fi(1,u71:, ,wA] )’°"’le )( ))

. . . T
2 _1_ (1, 67991, (232992} , _ , . BAN-0902‘)

W

A very important feature of E N is that it is unitary. To see this, we first consider

82':

, i=1,...,N. (13.12)

the finite geometric series identity and subsequently a result involving a summation

on powers of the primitive N m roots of unity.

190

<5
LABELING 1 _ 200
KEY _ 5
ID?

 

 

 

 

 

N=7 N=8 N=9

Figure B.1. The distinct Nth roots of unity wfv (k: = 0, 1, 2, . . . , N — 1) arranged on the

unit circle in the complex plane (centered at the origin) for N = 1,2, . . .,9. Note that
log, = 1 is real, as is 20X,” 2 -1 if N is even. The remaining roots appear in complex

conjugate pairs.

191

Lemma B.1 (Finite Geometric Series Identity) Let N E Z+ and q E C. Then
foranysEZ andqaél,
s+N— 1
8(1 _
Z qr _ ‘1 _____)_N C]
1 - q

7:8

PROOF. Consider the finite geometric series

s+N-—1

Z qr:qs+qs+1+qs+2+.uqs+N—1
_ s 2

—q (1+q+q +...+q

l\=‘lultiplying from the left by q yields
s+N—1
q 2 c1" =qs(q+72+q3+m+q

r=s
Subtraction of the second equation from the first results in

s+N—l

(1-Q) Z qr=qs(1-qN),
T=8
from which the proof is established since q 74 1 by restriction. I
Lemma B.1 is now used to establish the following theorem, which is needed to

Show that the Fourier matrix is unitary. The result will also aid in the diagonalization

of circulants in subsequent sections.

277
Theorem B.3 Let i, k E Z and wN = e72- be the primitive Nth root of unity with
N E Z+. Then for any 3, m E Z,

3+N—1 . N, i—k=mN
r(i—k) _ E]
Z “’N — ,
7:3 ' 0, otherWISe

° . 2’71' .
PROOF. Let q : nix—k) = eNu—H and note that qN = 1. If i — k = mN, then

 

q = erm" = 1 for any integer m and it follows that
s+N—1 (_ k) s+N—1 N
r 2— -1
E “IN = 2,17 = (1)8 +(1)S+1+(1)S+2 + . . . + (1)8+ J = N.

N terms

192

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

——> ——> ———>
1 2 3 1 2 3 1 2 3

1 l 0 0 1 1 0 0 1 0 1 0

l 2 0 1 0 Z 2 0 0 1 2 2 0 0 1
3 0 0 1 3 0 1 0 3 1 0 0

(a) (Ishk (b) (53M (C) (03%

Figure B.2. Arrays showing the (i,k) elements of the (a) identity, (b) flip, and (c) cyclic
forward shift matrices of dimension N = 3 for i, k = 1, 2, 3.

For the case when i — k 75 mN it follows from Lemma B.1 that

s+N—1

81—1
qu=———q( )=0,
l-q

7‘23

which completes the proof. I

Theorem B.3 allows for a representation of the N X N identity, flip, and cyclic

forward shift matrices in terms of certain conditions on their indices relative to N.

For i, k = 1, . . . , N and for any integer m the (i, k) element of these matrices is given
by
N—l
1 7’(i-k) 1, i — k = mN
.. : _ = 013
(INN: N r20 wN { 0, otherwise (B a)
N—l .
1 r(i+k—2) 1, i + k — 2 = mN
. = — : B.1 b
“$ka N r—O wN { 0, otherwise ( 3 )

N—l _ ._ _
T(2—k+1) : { 1a Z k +1 — mN (B.13C)

1
(UNMC = N wN 0, otherwise
respectively. The reader can check these by verifying the arrays in Figure B.2 for the
special case of N = 3.
We are now ready to state the main result of this section, and indeed one of the

most important results of this appendix.
Theorem BA The Fourier matrix EN is unitary. [:1

193

PROOF. For 1 S i, k S N the (i, k) entry of ENE}; is given by

N
(ENE/hill: = Z<EN)ir(EN)rk
1

‘1
ll

Lari—wr-ULw—
1W N x/I—V— N

(T-1)(k-1)

K42

(from Eqn. (B.11))

fl
II

= (IN)ik’ (from Eqn. (B.13a))

from which it follows that E NE}? 2 I N- I

Remarks

1. The column vectors of E N are orthonormal, that is, ejiHek = 6,-k, where 6,}, is
the Kronecker delta.

2. Since EN is unitary so too is the NA/I x NM matrix B N (8 I M-

3. The NM x 1V1 matrices e, (8) IM are such that (e,- <8) IM)H(ek (8) 1M) = 6,-kIM.

Next we derive a relationship between the Fourier and flip matrices.
2 H 2
Theorem B.5 EN 2 RN = (EN) . Cl

PROOF. First we show that Egv = ENEN = K. N using the same approach as the
proof of Theorem B.4. For 1 S i,k S N and for any integer m the (i,k) entry of

194

ENEN is given by

r=1
N 1 (MIN (1)(k1)
i— r— r— —
= —w —w fomE . BM
1; N N N N (1' qn( ))
N—l
___1_ wr(i+k—2)
— N
N r=0
2 (RN 1k, (from Eqn. (B.13b))

2
from which it follows that Bi, = KN. Finally, the result EN = (E716) follows

from conjugation and transposition of K. N = E NE N, and by invoking the properties
2
5% 2 RN and (13%,)H = (E716) . I

A number of properties follow directly from Theorem B.5.

Corollary B.1 Let EN, RN, and IN be the N x N Fourier, flip, and identity matri-

ces. Then
i. ENICN ZK'NEN;
.. 2 '
ii. nN=IN or KN: x/IN;

m; Ej‘V =IN or EN = ,4/IN. o

Propterty (i) says that the flip and Fourier matrices commute or, since EN is unitary,
that K, N is invariant under a unitary transformation with respect to E N- Hence K N
is not diagonalizable by E N. Properties (ii) and (iii) give alternative definitions of
the flip and Fourier matrices, respectively. Moreover, since the (possibly fractional)
power of a diagonal matrix can be obtained by raising each diagonal element of that

matrix to the power in question, if follows that the eigenvalues of n N are 21:1 and

those of E N are :l:1 and :tj, each with the appropriate multiplicities.

195

B.4.3 Diagonalization of the Cyclic Forward Shift Matrix

In this section it is shown that the Fourier matrix diagonalizes the cyclic forward shift

matrix. For this purpose, it is convenient to introduce diagonal matrix

9N = diag (1,wN,u’}2V,. .. ,w% 1), (8.14)

which has as its diagonal elements the distinct N M roots of unity.
Theorem B.6 EKIUNEN = RN. [3

PROOF. For 1 S i, k. S N the (i, k) entry of ENQNERT, is given by

N N
(ENQNENLk = Z Z(EN)-zp(9N)pr(Eifi)rk

r=1p=1

N N
1 wu— Duo-1),, w<r—1>1 —<r-1><k—1>

1 w
IWwN PTNmN

r=1p=
1 N
__ngzr- 1)(— 1) wg—1)w—(r—1)(k—1)
jvrzl
1 N ( 1)(‘ k+1)
=3?sz
r=1
N—l
__;L_ ufl(i—k+1)
_ N
Arr=0
= (O’Nh’k’ (from Eqn. (B.13c))

from which it follows that ENQNEH = UN. The desired result follows by multiplying
from the right by E N and multiplying from the left by Eh? I

Theorem 8.6 implies that a' N is unitarily similar to a diagonal matrix whose
diagonal elements are the nonnegative integer powers of the primitive N th root of
unity. Since the eigenvalues of a matrix are preserved under such a transformation

(this is guaranteed by Theorem A.1), it follows that

2 IV—l
/\(0'N) = {1,wN,wN,...,wN },

196

where A( ) denotes the matrix spectrum. The eigenvectors of a N are the linearly
independent columns of E N 2 [e1, e2, . . . ,eN], which are given by Equation (8.12).1

In light of Corollary A.1, we have the following result.

Corollary B.2 BEYOREN 2 ”IN for any k E Z+. (:1

B.4.4 Diagonalization of a Circulant

It will be convenient to define
N

g(t,'r) = Ztk_1 (8) 1', (8.15)
k=1
where t and 1' are arbitrary square matrices. Then the general circulant and block

circulant matrices given by Equation (8.4) and Equation (8.6) can be represented by

(UN, ek)_1r—ck — c1rc (61,02, . .. CN) , (B.16a)

IIMZ

(0’)=N,Ck EON—1 ®Ck =CiI‘C (C1,C2,...,CN), (B.16b)

respectively. W hat 1s meant 1by the notation g(O‘N, ck), for example, is to substitute t
with a N and T with ck in Equation (8.15) and then perform the summation observing

any indices k introduced by the substitution. Note also that

9(9N,7)=i:ciiagN (9(wiv1,r))=:diagN 1?;pr (z 1) , (B-17)

which is diagonal (resp. block diagonal) when 1' is a scalar (resp. matrix).

Theorem B.7 Let C E “KN have generating elements 01,02, . . . ,cN. Then

F -

Elf/GEN =

 

 

 

1In fact, all circulant matrices share the same eigenvectors ek, which is shown in the next section.

197

is a diagonal matrix, where

N
A,- = g(wfilmk) = chwS—lxz—U, i =2 1,. . . ,N
k=1

are its diagonal elements. [I]

PROOF. Consider the representation

C = g(a'N, ck) (from Eqn. B.16a)

= g(ENflNE%,ck) (from Thm. B6)

= ENQ(QN, ck) E7115. (from Thm. A.2, Thm. 13.4)

Thus EffiCEN = g(flN,ck), where the diagonal matrix g(flN,ck) = ' diagN(/\,-)
i: ,...,

follows from Equation (31?). I

Remarks

1. The Fourier matrix E N diagonalizes any circulant matrix.

2. Eff/CE N is a unitary transformation, and hence preserves the eigenvalues of C.
Thus A,- (i = 1, . . . ,N) are the eigenvalues of C.

3. All cirulants share the same linearly independent eigenvectors (the columns of
the Fourier matrix E N), which are given by Equation (312).

The eigenvalues of a matrix C E .7ng with generating elements c1,c2, . . . ,cN

are given by

 

 

N/2 .
c1+ 22% cos(27r(k—]b)(z—1)) + (—1)i_lcfl2t2, N even
, _ k=2
A, — (N+l)/2 . (B.18)
c1 + 2 Z ck cos(2fl(k_](r)(l_fl) , N odd
k=2

a result that is proved in [36]. In this case there are repeated eigenvalues due to
the presence of the cosine term. The eigenvalue /\1 is distinct, but the remaining
eigenvalues /\.,' = AN+2_,- appear in repeated pairs. However, when N is even A NP

is also distinct.

198

 

 

 

Example B.6 Let C = cire (4, -1,0, ——1). Then

'1 1 1 1"4 —1 0 —1' ’1 1 1 1
11—'—1'—14—1011'—1—-'
E?CE4=_ J J _ J J
fl1—11—10—14—1fi1—11—1
1 j —1 —jJ —1 0 —1 41 _1 —j —1 j

’1 1 1 1”1 2 3 2 '

_11—j —1j 1 23' —3 -2j

£11 —1 1 —1 1 —2 3 —2

1 j —1 —j 1 —2j —3 2]“,

 

 

 

 

= diag (2,4,6, 4) .

Hence the eigenvalues ofC are 2, 4, 6, 4. Since C E Y‘él, these can be verified using
Equation (B.18) for N = 4, which yields

A,- =4—2cosg(i— 1)

for-i=1,...,4. [SD

The determinant of a circulant matrix C = circ(c1,c2, . . .,cN) is simply the

product of its eigenvalues and is given by

N'.N
Hzcwi-W-i

i=1k=1

.N
det C = H A,- = (13.19)
i=1

where the eigenvalues A,- are defined in Theorem B.7 or by Equation (B.18) if C is

also symmetric.

B.4.5 Block Diagonalization of a Block Circulant
Theorem B.7 can be generalized to handle block circulants.
Theorem B.8 Let C E .2?ng and denote its M X M generating matrices by

Cl,C2,.. . ,CN. Then

(Eh1g ® IM)C(EN <59 1M) =

 

 

199

is a block diagonal matrix, where
N
Az‘ = g(wjgl,Ck) = ZCk'lL’ES-l)(i_l), i=1,...,N
k-I
are its M x A] diagonal blocks. [:1

PROOF. Consider the representation

N

C = Z afafl ® Ck (from Eqn. B.16b)
k=1
N

= 2(ENQIIVIER?) ® Ck (from Cor. B.2)
k=1
N

= 2(EN a 1M) (mg—ls Ck) (E31, 8) 1M) (from Eqn. A.4)
k=1

= (EN ‘81 IMMQN, Ck) (E716 <59 1M), (from Eqn- 315)

where the block diagonal matrix

0(9N,Ck)= diag (At)
i=1,...,N

follows from Equation (31'?) Since (EN <8) I M) is unitary the desired result follows
by multiplying from the right by (EN <8) I M) and from the left by (BK, ® IM) =
(EN e It)“. .

Remarks

1. The unitary matrix E N (8 I M can reduce any NM x NA! block circulant matrix
with M x M blocks to a block diagonal matrix with A! x M diagonal blocks.

2. (E716 <8) I M)C(E N (8 I M) is a unitary transformation, and hence preserves the
eigenvalues of C, which are the eigenvalues of the N, M x M matrices A,.

3. If v,- is an eigenvector of the ith eigensystem A,, then the corresponding eigen-
vector of C is u,- = 9,- ® vi.

200

Example B.7 Consider the matrix C = circ (A,B,0, B) from Example B.5. It can

be block diagonalized via the transformation

 

 

' 0 —1g 0 0 0 0 0 0 '
:..1..--9-§..9 ..... 9-5--9 ..... 9 ..... 9 ..... 9.-
0 0 5 2 —13 0 O 5 0 0
EH 1 c E 1 = ..9 ..... 9‘120 ..... 99 ..... 9-
(4®2)(4®2) OOgO 0g4—1goo
.9 ..... 9-5-.9 ..... 9.-5.'.t.-1.--.f¥..f..9 ..... 9..
0 0 g 0 0 g 0 0 g 2 —1
_O 0§0 OEO Oi—12J
= diag (A1, A2, A3, A4),
from which the eigenvalues can be obtained from the 2 x 2 matrices A,- fori = 1,. . . ,4.

In particular, /\(A1) = {—1,1}, /\(A3) = {3, 5}, and A(A2) = A(A4) = {1,3}. CED

B.4.6 Some Generalizations

Let C 6 38(6)”, N and denote its generating matrices by C1, C2, . . . ,CN. Then if

( ' )* and ( - )# denote arbitrary matrix operations and for any matrices A 6 CN XN

and B E CI’UXJW’

N
(A* e) B#)c(A e) B) = (A* ®B#) Eats-1 e c,, (As B)
k=1

(”42

[(A*a’;,,‘1) ® (B#Ck)] (A e B)

a...
H
H

(Amy-1A) e (B#CkB), (3.20)

(”42

3‘
1L

where Equation (A.4) has been employed. The importance of this result is that C
can be decomposed into a summation of direct products of two separate equivalence
transformations, one that operates on ”TV—1 and the other on C k- This decomposition
justifies the diagonalizing matrix employed in Theorem B.8 and it also motivates some
generalizations.

In light of Theorem B.6 together with Corollary A.1, it is clear that the choice of

A = E N and ( ' )* = ( - )H accomplishes block diagonaliztion of a matrix C E 35% M, N-

201

Then if one chooses B = 1M, the appropriate diagonalizing matrix to block decouple
C without operating on its generating matrices is E N (8) I M- However, if B and ()#

are kept general, we have the following result.

Theorem B.9 Let C E 53‘ng have M x NI generating matrices C1, C2, . . . , CN.
Then for an arbitrary matrix B E CMXM and operator (-)#

 

 

“1:1 0 1
H # ‘1’?
(EN ®B )C(EN s3):
is a block diagonal matrix, where
11,-: g(wjglfittcke) = ZB#Cka)V_1)(2_1), i=1,...,N
k=1
are its ll! x M diagonal blocks. [:1

This result is useful if there exists an equivalence transformation B#CkB that
simplifies each of the generating matrices. For example, if each Ck is a circulant of
type M then the additional choice of B = E M and ( )# = ( - )H fully diagonalizes a

block circulant matrix C E fi‘fiv’ M with circulant blocks.

Corollary B.3 Let C E fichw have generating matrices C1,C2, . . .,CN 6 ch
and denote the generating elements of each C, by C(12), cg), . . . ,CEC}. Then

-Agi) 0 -
A(2t)
(BK, ® E},)C(EN 69 EM) —_— diag 2
i=1,...,N -
(i)
_ 0 AM,

 

 

is a diagonal matrix, where

N M

(2) 22(1) (l—1)(p—1) (k-1)(i-1)

AI) = Cl “’M “’N
k=11=1

th th

is the 19 diagonal element of the i diagonal block. C]

202

Example B.8 Reconsider the matrix C = circ (A, B, 0, B) from Example B. 5. Since

A, B 6 (fig it can be diagonalized via the transformation

 

 

'-10:0 0:0 0:0 0'
01000000
0 0:1 0:0 0:0 0
EH E“ C E. E = 00030000
(4®2)(4® 2) 00:00:30200,
90000500
0 0:0 0:0 0:1 0
_0m0m0moa
from which is follows that /\(C) = {—1, 1,1,3,3,5, 1,3}. ED

203

APPENDIX C

N oninteger Engine Order Excitation

It is well-known that a perfectly cyclic system (i.e., one without parameter uncertainty,
or mistuning) under engine order excitation will respond in one and only one mode
corresponding to p = nmodN + 1, where n is the order of the excitation. For a
bladed disk assembly in a jet engine the excitation order corresponds to a problematic
harmonic in an expansion of the axial gas pressure field, and is therefore a positive
integer. HoWever, noninteger engine order may occur in other applications and in
what follows we briefly consider the general case of n E R+. Then orthogonality
between the normal modes and system forcing vector generically breaks down and
this gives rise to additional system resonances.

Consider the prototypical cyclic system of Section 2.4 under engine order excita-
tion. In the steady-state it responds according to Equation (2.34), which indicates
that the total forced response is simply a superposition of modal responses. For
the case of n E Z+ only mode p = nmodN + 1 survives, which follows from the
orthogonality condition given by Equation (2.35), that is,1

N

H
pf

e wyfc—an-H—p) (n E R+) (C.1)

91*
ll

f, n+1—p=mN
(TL 6 Z+) (CZ)

r—’\—\
E)

0, otherwise

 

1The result holds even for n S 0, but physically we restrict n > O.

204

where m is an arbitrary integer. (This result follows from Theorem B.3 on page 192.)
However, if the engine order is noninteger such that n E R+/Z+ Equation (C.2) does
not hold and therefore every system mode contributes to the total response, which
is given by Equation (2.34) by replacing the pth model forcing term eyf with the
right hand side Equation (C.1). Correspondingly, for a given engine order there are
N system resonances and these occur whenever the excitation frequency matches a
natural frequency, that is, when no = (12p, where the pth eigenfrequency is defined
by Equation (2.38). Thus in addition to the “principle resonance” corresponding to
p = n mod N + 1, there are (N — 1) / 2 (resp. N / 2) resonances due to the noninteger
excitation if N is odd (resp. even). An example is shown in Figure C.1 for a system
with N = 10 sectors and for various engine orders n E lR+. The additional resonances
can be clearly observed, and they become more pronounced for larger deviations of

the engine order away from the integer value it = 3.

205

l
H

 

O
[\D
.0
A
.0
Ci
CD
CD
H

 

 

H .

0.2 0.4 0.6 0.8
(c) n = 3.01

 

 

 

0.2 0.4 0.6 0.8 1
(b)n=3.001

Principle
Resonance

    

 

 

0:2 04 0.6 08 1

 

 

 

 

 

 

(d) n = 3.1
(I)
2 .P
1.5 > /
J"
“/1
TA
1 N L
L—- Principle
0'5 ’ Resonance

 

 

0.2 0.4 0.6 0.8 1
(e) n = 3.1

Figure C.1. Frequency response curves for the system shown in Figure 2.4 on page 23 for
N = 10, V = 0.5, and f = 0.01: (a) n = 3; (b) n = 3.001; (c) n = 3.01; and (d) n = 3.1; and
(e) corresponding Campbell diagram for n = 3.1. In ((1) and (e) the principle resonance
corresponding to n + 1 = 4.1 is indicated by the black circle.

206

APPENDIX D

The Critical Linear Detuning and Tuning
Order

The critical linear absorber tuning order ft is defined implicitly by Equation (4.36) and
represents the limiting slope of the natural frequencies (By) in the frequency versus
0 curves of Figure 4.5. It is convenient to express ft in terms of the absorber tuning

order fr, which is introduced via Equation (4.39). Then

 

 

. ~ &1u+&0+\/&%p2+51u+58
n(n) = 2712 , (D-l)

 

where
60 = 712(sz + 6)
130 = 73.2012 — 6)

61 = a2(7”z2 +1)(fi2 + g + 1)

 

131 = 2a262(a2 + 1) ((73,2 +1)(6.2 + 6) - gm? — 6))
J
The critical absorber undertuning can be obtained by replacing it with n(1 + B) in
Equation (D.1), setting 71 = n, and then solving for fl = Ber. Then ficr is given

implicitly by

 

2 a1u+co+\/b2,u2+b1u+c8
(1+ 661—) = , (13.2)
01# + 200

 

207

where
a1=a2(%—n2(2+%)) ‘
bl = —262n2(n2 — 6)(n2 + 1) (2 + g)
b2 2 a262(n2 +1)2 > .
C0 = 712(712 — (5)

01 = 20276.2(712 — i)

 

a J

Note that when p. = O, the right hand side of Equation (D2) is unity, which implies
x3” = O (i.e., the no-resonance gap vanishes), and the same is true when a = O (in
which case (11 2 bl = b2 = c1 E 0, but co 75 0).

It is clear from Figure 4.9 that changes in ficr due to parameter variations decrease

for increasing engine order. In the limit as n —> 00 Equation (D2) reduces to

1

1+ 2:—
< 6..) ”a?”

, (n —> 00) (13.3)

which is shown by the dashed line in Figure 4.9d. It depends only on Oz and n and

approximates flcr to within 3% of its limiting large—n value for n 2 10.

208

 

APPENDIX E

Averaging: The Usual Approach

E. 1 Introduction

In the process of averaging it is customary to expand all parameters involving 5 and
to keep only (9(5) terms in the resulting expressions. This amounts to replacing it
with n and also or with the constant or = 1 / m, which is the resonant rotor
speed of an isolated blade without an absorber. In what follows, we carry out the
averaging for the isolated nonlinear system of Chapter 5 in this way and show that the
results are adequate for large linear under- or overtuning of the absorbers but they are
completely unsatisfactory Within the no-resonance zone (precisely where the results
are of the most interest) due to the latter substitution. The aim of this appendix
is to document the rather significant difference in the averaged models obtained via
the usual approach employed below and by means of the slightly modified approach
employed in the nonlinear analysis in Chapter 5.

We begin by defining a detuning scheme (different from that employed in Chap—
ter 5) and then present the averaged equations in both polar and Cartesian forms.
The appendix closes with some representative frequency response curves and a short

discussion.

209

E.2 Detuning Scheme

In order to investigate the nonlinear dynamics near perfect linear tuning and close to

the rotor speeds of interest, we take

73 = n(1+ 5A), (E.1a)

0 = 0,.(1+ 8A), (E.1b)

where /\ serves as the linear order detuning parameter and A plays the role of the
rotor speed. Then the order detuning parameters are related by B = 5A, which is
clear by comparing Equation (E.1a) to Equation (4.44). It should be pointed out
that the detuning scheme adopted here is slightly different than the one employed in
Chapter 5, that is, Equation (5.9). (Equation (B.1) was the original detuning used
by the author before the improved method of averaging was adopted.) This does not,
however, preclude qualitative comparisons between the results of the two approaches.

Next the averaged equations are derived in polar and Cartesian forms.

E3 The Averaged Equations

The nonlinear analysis is carried out exactly as it was done in Section 5.2.4 except
that all appearances of 0 are replaced by the constant or and, as we shall see, this
gives rise to a significantly less accurate model, one that serves as a benchmark for
the modeling approach employed in Chapter 5.

Under the detuning scheme defined in Section B.2, Equation (5.8) on page 127

reduces to1
66%, — 6262 = —52A + 0(52)

-2 2 2 E 2M2
UJ —n U =
22 712—6

(B.2)
+ 0(52)

 

 

1For the detuning employed in Chapter 5 the 0(5) terms are —5A and 902.

210

 

Table E. 1. Shorthand notation.
5:5/271 n —6 A 2(62 —6)A £b=m/n?—6éb f=(n2—6)f
(1' = (n.2 +1)a 5x = 2n2A Ea = nVn2 — 655

 

 

After the appropriate substitutions are made, Equation (5.6) is averaged over one

period T = 27r/n0 and the shorthand notation given in Table E1 is introduced.2

Then if v = (a, g, '1), GT the result is
(11', flé',17’,f2€’)T = gem + 0(53/2), (B.3)

where

G3(v) = we sin 6 — g") — ‘5,»6

G4(v) = —dfl cos(§— 5+) X27 + 371273
are the elements of the vector G. In Equation (E. 4), note that the rotor speed appears
in Gg(v) only (implicitly via the shorthand detuning parameter 5). However, in the
model employed in Chapter 5, the rotor speed appears in each Gp(v) (p = 1,. . . ,4).
It is this absence of the rotor speed that gives rise to less accurate results.

The corresponding Cartesian form of the averaged equations is
w’ = €P(w) + 0(53/2), (B.5)
where w = (A, B, C, D)T and the functions

P1(w) = —aD — AB — ébA
P2(W) = +5zC + AA — EbB + f
P3(w) = —aB + :\D — $0 + §n(D3 + 021))

P4(W) = +C_YA — AC — gal) - 9177(CD2 + C3)

 

2The shorthand notation in Table B.1 applies to this appendix only; some of these symbols are
defined differently elsewhere in the thesis.

211

 

compose the elements of the vector P.
Some representative frequency response results are given next, and we briefly point

out their reduced accuracy.

E.4 Forced Response of the Undamped System

For the purpose of comparison we consider the forced response of the undamped av-
eraged model corresponding to the isolated nonlinear system, that is, Equation (B.3)
with Ea = Eb = 0. Figure E.1 summarizes the results in terms of the speed detuning
parameter A (which is related to the dimensionless rotor speed 0 via Equation (E.1b)),
for several order detuning values fl, and for a hardening absorber path; a representa-
tive set of softening frequency response curves is shown in Figure B.2. In these figures,
the nonlinear frequency response loci are indicated by thick solid (stable) and dashed
(unstable) lines and the corresponding linear frequency response is given by the thin
gray line. The simulation data from Chapter 5 is superimposed in Figure E.2, which
corresponds to the full nonlinear model, i.e., Equation (3.14) together with the gen—
eralized absorber path described in Section 3.4.4, with representative damping levels
a, = 2 x 10-3 and 5;, = 2 x 10—6.

For large under- or overtuning values B the approximate averaged results given
above are, in fact, quite satisfactory. This is clear from the hardening (resp. softening)
frequency response curves shown in Figure E.1a—b (resp. E.2a—b) and Figure E.1e-g
(resp. E.1e-g), which correspond to absorber detuning values outside the no—resonance
zone. In these regions, the averaged model captures the essence of the linearized
resonance ’RL in addition to the first-order nonlinear effects of the absorber, which
are manifested in the additional auxiliary resonance 721:1” and the hardening / softening

bends in the primary resonance REL}

 

3The linearized, nonlinear auxiliary, and nonlinear primary resonances 72L, ”REL, REL are dis-
cussed more fully in Section 5.4.

212

 

 

fi=—01

A = —1.690
,3 = —0.005

A = —0.0845
5 = —0.001

A = —0.0169
(No-Res. Zone)
fi=A=0
(No-Res. Zone)
fi = +0.001

A = +0.0169
£3 = +0.005

A = +0.0845
fi=+01

A = +1.690

 

 

(N
E.

rattan;

 

 

 

 

 

     

-—10 0 10 20 30 A
1%W
l r
0 .
.1
_2
—3 ,
'4AA;;2w~'r -£
10 20 30 A
10
l
0
_1
—2
_3
_4
A

 

20 30

 

 

 

A

 

40 0 w m“ a

 

 
 

 

— Nonlinear (stable)
---- Nonlinear (unstable)
A Linearized

 

 

 

 

 

 

 

l
' I
f
|

 

 

 

 

 

 

L&&Lo~5

 

 

m m m

Or

40

on
E

,
f
I
»’

 

 

 

m a

 

 

 

 

 

 

40 o m m :mA

 

Figure E.1. Blade and absorber frequency response curves for the same conditions in
Figure 5.4 on page 143 (hardening absorber path) based on the less accurate model defined
by Equation (E3)

213

 

 

 

— Nonlinear (stable)
---- Nonlinear (unstable)
_ Linearized

0 0 0 Simulation

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10g lfll log 1m
1 1
0 0
®)fi=—01 3 3
A = —1.690 4 4
~4 —4
(b) B = —0.005
A = —0.0845
(c) a = —0.001
A = —-0.01691 , 4, a»
(No-Res. Zone) 411%. O .7. “.1. A _4
-10 0 13 20 30 —10 6 10 20 30 A
110g IUI lfg |o|
(d) o = A = 0 3 :12 . . . “=1”, -----..
(No-Res. Zone) _3 130' *H1+o~o+
‘4 O O o o . .4
.10 A
1' 1
log lul 1o
0 l 0
_ —l _1
(e) 13 — +0.001 _2 . i __2
A = +0.0169 4 , _3
_4 1W ‘ .4
- A v . Q .‘4
—10 0 10 '20 30 A A
10 lo
lfm 1 ,
0 . 0
_ —1 ‘ .1
(f) o— +0.005 _2 1 _2
A = +0.0845 .3 _3
M ‘4
Q’Okna;: A A
10
1
0
(g) L3 = +0.1 :21
A = +1.690 _3
-4 “
A - . A

 

 

 

 

—10 0 10 20 30

Figure E.2. Blade and absorber frequency response curves for the same conditions in
Figure 5.5 on page 144 (softening absorber path) based on the less accurate model defined
by Equation (E3)

214

 

In contrast, for order detunings ficr S E < 0 inside the no—resonance-zone the

results are poor at best or completely erroneous. This is shown in Figure E.1c-d (reSp.

E.2c-d) for a hardening (resp. softening) absorber path. For example, in Figure E.2c
the linear theory predicts no resonances over the full range of rotor speeds, and this
is verified by the simulation data, yet the nonlinear theory gives rise to a primary
resonance. As another example, the nonlinear auxiliary resonance in Figure E.2d is
expected, as it also appears in the improved model employed in Chapter 5 (compare
with Figure 5.5d and Figure 5.6d), but the nonlinear theory does not adequately
capture the linear branch of absorber motions.

Since the no—resonance zone is where the results are of the most interest, and since
that is precisely where they are least accurate (or completely erroneous), the averaged

model described above is not at all satisfactory.

215

l

 

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