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MODEL REDUCTION OF NONLINEAR STRUCTURAL
SYSTEMS USING NONLINEAR NORMAL MODES AND
COMPONENT MODE SYNTHESIS

presented by

Polarit Apiwattanalunggarn

has been accepted towards fulfillment
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6/01 c:/ClRC/DateDue.065-p. 15

MODEL REDUCTION OF NONLINEAR
STRUCTURAL SYSTEMS USING NONLINEAR
NORMAL MODES AND COMPONENT MODE
SYNTHESIS

By

Polarit Apiwattanalunggarn

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

2003

ABSTRACT

MODEL REDUCTION OF NONLINEAR
STRUCTURAL SYSTEMS USING NONLINEAR
NORMAL MODES AND COMPONENT MODE
SYNTHESIS

By

Polarit Apiwattanalunggarn

This work addresses the general problem of model size reduction for describing the
nonlinear vibration of structural elements and systems. The aim is to provide
computational tools that allow one to accurately capture nonlinear dynamic behavior
using a minimal number of degrees of freedom. In typical applications the finite element
(FE) method is used to generate structural dynamic models, and model size reduction is
carried out using linear modal analysis with truncation. However, in some cases one must
retain many modes in order to accurately capture essential nonlinear coupling between
the linear modes. In this work we utilize nonlinear normal modes (NNMs) defined in
terms of invariant manifolds for the purposes of model size reduction, since it directly
addresses modal coupling. This approach, which makes use of master and slave modes,
along with the concept of dynamic invariance, allows one to generate accurate reduced
order models (ROM) with only a few DOF, while capturing the effects of all modeled
linear modes without directly simulating them. There are three main contributions of the

present effort:

(1) Two new numerical approaches for solving the invariant manifold equations are
introduced. Both approaches employ master modal displacement and velocity
coordinates and are based on weighted-residual techniques. When compared with
previous methods that utilize amplitude and phase variables, the new methods are found

to be superior in terms of computational time but inferior in terms of accuracy.

(2) A specific application is considered: the finite amplitude vibrations of a rotating
beam, which is a crude model for a rotorcraft blade. This system is known to possess
essential nonlinear coupling between axial and transverse displacements, thereby leading
to slow modal convergence. The proposed method systematically captures this coupling
and provides an accurate single degree of freedom ROM. These results demonstrate the
utility of NNM-based ROM, since they combine the versatility of the finite element

method with the accurate NNM model reduction technique.

(3) A model reduction technique suitable for structures that can be partitioned into
substructures is developed. This allows one to build ROMs using NNMs at the
substructure level and to assemble these using a component mode synthesis (CMS)
technique. It is found that the proposed nonlinear CMS technique generally provides an

accurate model only when the couplings between substructures are weak.

To my father and mother

ACKNOWLEDGMENTS

I would like to express my gratitude and appreciation to many people who have contributed

to this work. Without them, it could not be finished.

I would first like to thank my advisor, Professor Steve Shaw, for his generous support,
patience and guidance throughout my research work. I would also like to thank my co-
advisor, Professor Christophe Pierre, from the University of Michigan for his helpful insight
on Component Mode Synthesis. I would also like to thank Professors Alejandro Diaz,

Ronald Averill, Milan Miklavcic, and Brian Feeny for serving in my dissertation committee.

I would like to acknowledge the contributions of professors who taught me the knowledge in
engineering and mathematics. Professor Steve Shaw deserves thanks for teaching me solid
foundations in system dynamics, vibrations and control and most importantly training me
to use my physical insight to cross check with mathematical results. Professors Brian Feeny,
Alan Haddow, Ronald Rosenberg, Clark Radcliffe, Hassan Khalil, N ing Xi, and Fathi Salem
also deserve thanks for their lectures in system dynamics, vibrations and control. I would
also like to thank Professors Ronald Averill, Patrick Kwon, Hungyu Tsai, and Darren Ma-
son for their lectures in finite element method and mechanics. I wish to thank Professor
Abraham Engeda for allowing me to visit his class ME802 Advanced Classical Thermody-
namics. I also wish to thank Professors Milan Miklavcic, Jerry Schuur, John McCarthy,

Patricia Lamm, and Michael Hazier for expanding my knowledge in mathematics.

I would like to thank the colleagues from the University of Michigan, Dr. Eric Pesheck,

a former Ph.D. student, and Mr. Dongying Jiang, a Ph.D. student, for their suggestions,

supplying some results, and allowing me to use their computational programs to generate

some results in this work.

Thanks also goes to the colleagues in the System Dynamics and Vibration Laboratory,
Dr. Choong-Min Jung, a former Ph.D. student, Brian Olson, Tyler Nester, Peter Schmitz,

Jeffrey Quinby, and Jeff Rhoads, for their suggestions, comments, and friendship.

I wish to thank the Royal Thai Government and Kasetsart University for giving me the

opportunity and support to pursue higher education at l\x’lichigan State University.

I would also like to thank a friend of mine from Thailand, Mr. l\-’Ianoch Srinangyam, who
I met during my freshman year at Chulalongkorn University and travelled with me on the
first trip to the U.S., for being a good friend and giving helpful suggestions during my

study at Michigan State University.

I wish to express my thanks to the special person, Dr. Kunlakarn Lekskul, who has walked
in to my life for seven years. It has been the joyful time to get to know her during all these
years. I could not possibly go through many difficulties without her encouragement and

support.

Finally, I would like to express my special thanks and love to my mother who always values

the importance of education. Her farsightedness is the basis for my success today. I also

thank my sister and brothers for supplying all my needs from Thailand.

vi

TABLE OF CONTENTS

LIST OF TABLES ix
LIST OF FIGURES x
1 Introduction 1
1.1 Motivation .................................... 1
1.2 Nonlinear Normal Modes ............................ 2
1.3 Beam (Blade) Dynamics ............................ 7
1.4 Substructure Synthesis ............................. 9
1.4.1 Component Mode Synthesis ...................... 9
1.5 Dissertation Organization ........................... 11

2 Comparison of Methods for Constructing Invariant Manifolds for Non-
linear Normal Modes 13
2.1 Introduction ................................... 13
2.2 Formulations of N NM Invariant Manifold ................... 15

2.2.1 Invariant Manifold Equations in Terms of Modal Position and Velocity 16
2.2.2 Invariant Manifold Equations in Terms of Modal Amplitude and Phase 22

2.3 Solution of the Invariant Manifold Equations ................ 26
2.3.1 Solution by Asymptotic Series Expansion ............... 27
2.3.2 Solution by Galerkin and Collocation Methods ............ 31
2.3.3 Comparison of Computational Efforts ................. 47

2.4 Examples .................................... 53
2.4.1 A Two-DOF Nonlinear Spring-Mass System ............. 54
2.4.2 A Finite-Element-Based Model of a Rotating Beam ......... 58

2.5 Conclusions ................................... 62

2.6 Tables ...................................... 63

2.7 Figures ...................................... 64

2.8 Appendix 2A .................................. 80

2.9 Appendix 28 .................................. 81

3 Finite-Element-Based Nonlinear Modal Reduction of a Rotating Beam

with Large-Amplitude Motion 85

3.1 Introduction ................................... 85

3.2 Rotating Blade Formulation .......................... 88
vii

 

3.3 Modal Reduction ................................ 94

3.3.1 Galerkin-based Invariant Manifold Approach ............. 94

3.3.2 Collocation-based Invariant Manifold Approach ........... 99

3.4 Results ...................................... 100
3.4.1 Linear System Convergence ...................... 100

3.4.2 Nonlinear Model Development ..................... 101

3.4.3 System Response ............................ 103

3.4.4 Computational Considerations ..................... 106

3.5 Conclusions ................................... 107
3.6 Tables ...................................... 108
3.7 Figures ...................................... 109
3.8 Appendix 3 ................................... 121

4 Component Mode Synthesis Using Nonlinear Normal Modes 128
4.1 Introduction ................................... 128
4.2 Formulation of the NNM Invariant Manifolds ................. 130
4.2.1 Invariant-Manifold Governing Equations ............... 130

4.2.2 Invariant Manifold Solution ...................... 133

4.3 Fixed-Interface Nonlinear Component Mode Synthesis ............ 136
4.3.1 System Representation in Physical Coordinates ........... 137

4.3.2 System Representation in Linear Modal Coordinates ........ 138

4.3.3 Synthesis with Nonlinear Modal Reduction .............. 143

4.4 A Five-DOF Nonlinear Spring-Mass System ................. 147
4.5 A Class of Systems ............................... 151
4.5.1 Model Development ........................... 152

4.5.2 Example ................................. 154

4.6 Conclusions ................................... 165
4.7 Tables ...................................... 166
4.8 Figures ...................................... 168
4.9 Appendix 4 ................................... 198

5 Conclusions and Erture Works 205
BIBLIOGRAPHY 211

viii

2.1

2.2

3.1

3.2

4.1

4.2
4.3

4.4

LIST OF TABLES

Comparison of the total number of unknown coefficients and the number of
evaluation points needed for the integrands in order to compute the integrals
for the global domain approaches ........................
Comparison of the total number of unknown coefficients and the number of
evaluation points needed for the integrands in order to compute the integrals

for the local domain approaches .........................

Dimensionless linear natural frequencies. (.01 and tag were computed using
results from [1]. (123 and (124 were computed using results from [2] .......
Comparison of the linear natural frequencies obtained by power series and
finite element methods. The analytical results for an and tag were computed
using results from [1]. The analytical results for rug and M] were computed

using results from [2] ...............................

Fixed—interface linear natural frequencies of substructure [3 using the second—
trial parameters. ................................
Fixed-interface linear natural frequencies of substructure 7 ..........
Comparison of the linear natural frequencies of the three—DOF linear-CMS
nonlinear model and the three—DOF nonlinear-CMS model ..........
Comparison of the first-sixteen linear-natural frequencies of the forty-one-
DOF linear-CMS nonlinear model and the forty-one-DOF original model. .

63

63

108

108

166
166

167

167

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

LIST OF FIGURES

A two-DOF nonlinear spring-mass system. ..................
The contribution of the second linear mode position 712 = P2(a1, m) to the
first nonlinear mode manifold obtained by the global a — (b approach.

The contribution of the second linear mode position 02 : X2071, in) to the
first nonlinear mode. manifold obtained by the global u — v approach with
Ub =1.5 and Vb =16. .............................
The contribution of the second linear mode position 772 =
X2(n1(a1,¢1),171(a1, Q51» to the first nonlinear mode manifold obtained by
the global u - v approach with Ub = 1.5 and Vb = 1.5. ...........
The contribution of the second linear mode position 772 = X 2071,51) to the
first nonlinear mode manifold obtained by the global u — v approach with
Ub = 1.7 and Vb = 1.17 ..............................
The contribution of the second linear mode position 172 =
X2(7)1(a1, $1),7'71(a1,gb1)) to the first nonlinear mode manifold obtained by
the global u — v approach with Ub = 1.7 and Vb = 1.17 ............
The contribution of the second linear mode position 712 = X2(7)1,1'71) to the
first nonlinear mode manifold obtained by the global u — v approach with
Ub = 2.6 and Vb = 1.79 ..............................
The contribution of the second linear mode position 772 =
X2(7)1(a1, r151),r)1(a1, 951)) to the first nonlinear mode manifold obtained by
the global u — v approach with Ub = 2.6 and Vb = 1.79 ............
The time responses of the first nonlinear mode displacement for the ROM
of the global u — 12 approach with 111(0) 2 1.3, 7'}1(0) = 0.0, the ROM of the
global (L — (15 approach with (11(0) 2 1.3, ¢1(0) = 0.0, and the original model
with initial conditions from the shooting algorithm. .............
The time responses of the second linear mode displacement n2 on the first
nonlinear mode manifold for the ROM of the global u — 21 approach, the
ROM of the global a — d) approach, and the original model ..........
The time responses of the displacement $1 from Figure 2.1 for the ROM of
the global u — v approach, the ROM of the global a — d) approach, and the
original model. .................................
The contribution of the second linear mode position 02 = P2(al,el) to the

first nonlinear mode manifold obtained by the local a — go approach. . . . .

64

64

65

65

66

66

67

67

68

68

69

69

2.13

2.14

2.15

2.16

2.17

2.18

2.19

2.20

2.21

2.22

2.23

2.24

2.25

2.26

2.27

2.28

The contribution of the second linear mode position 772 = X2(r}1,7')1) to the
first nonlinear mode manifold obtained by the local u — v approach .....
The time responses of the first nonlinear mode displacement for the ROM
of the local u — 22 approach with 771(0) = 1.5,7'}1(0) = 0.0, the ROM of the
local 0. — ¢ approach with a1(0) = 1.5, (1)1(0) : 0.0, and the original model
with initial conditions from the shooting algorithm. .............
The time responses of the second linear mode displacement 712 on the first
nonlinear mode manifold for the ROM of the local u — '11 approach, the ROM
of the local (I — (15 approach, and the original model ..............
The time responses of the displacement $1 from Figure 2.1 for the ROM of
the local u — v approach, the ROM of the local 0. — (i) approach, and the
original model. .................................
Finite element representation of a rotating beam with Q = constant. . . . .
The contribution of the second linear flapping mode 772 = P2(a1,¢1) to the
first nonlinear mode manifold obtained by the local a —- d) approach .....
The contribution of the fifth linear flapping mode 02 = P5(a1,¢1) to the
first nonlinear mode manifold obtained by the local (1. — qb approach .....
The contribution of the first linear axial mode 7710 = P10(a1,¢)1) to the first
nonlinear mode manifold obtained by the local (1 — ab approach. ......
The contribution of the sixth linear axial mode 015 = P15(a1,¢1) to the first
nonlinear mode manifold obtained by the local (2 — 45 approach. ......
The contribution of the second linear flapping mode 1);; = X2(n1,r')1) to the
first nonlinear mode manifold obtained by the local u — 11 approach .....
The contribution of the fifth linear flapping mode '02 2 X5(n1,7°)1) to the
first nonlinear mode manifold obtained by the local u — v approach. . . . .
The contribution of the first linear axial mode 7710 = X10(n1,1'71) to the first
nonlinear mode manifold obtained by the local u — v approach ........
The contribution of the sixth linear axial mode 7715 = X15071, 7'71) to the first
nonlinear mode manifold obtained by the local u — v approach ........
The time responses of the first nonlinear flapping mode displacement m for
the ROM of the local u — v approach with 171(0) = 2.375, {71(0) 2 0.0, the
ROM of the local 0. — (i) approach with (11(0) = 2.375, ¢1(0) = 0.0, and the
original model with initial conditions from the shooting algorithm ......
The time responses of the second linear flapping mode displacement in on
the first nonlinear mode manifold for the ROM of the local u - 12 approach,
the ROM of the local a — (23 approach, and the original model. .......
The time responses of the fifth linear flapping mode displacement 775 on the
first nonlinear mode manifold for the ROM of the local u — v approach, the
ROM of the local (1 — g5 approach, and the original model. .........

xi

70

70

71

71

72

72

73

73

74

74

76

77

2.29

2.30

2.31

2.32

3.1
3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

The time responses of the first linear axial mode displacement 7710 on the
first nonlinear mode manifold for the ROM of the local u — 12 approach, the
ROM of the local a — d) approach, and the original model. ......... 78
The time responses of the sixth linear axial mode displacement 7715 on the
first nonlinear mode manifold for the ROM of the local u — 12 approach, the
ROM of the local a — (35 approach, and the original model. ......... 78
The time dependence of the beam—tip flap deflection, w(L, t), for the ROM
of the local u — 21 approach, the ROM of the local a — qt) approach, and the
original model. ................................. 79
The time dependence of the beam—tip axial deflection, u(L, t), for the ROM
of the local u — v approach, the ROM of the local a — d) approach, and the
original model. ................................. 79

Finite element representation of a rotating beam with Q = constant. . . . . 109
Comparison of the static axial blade deflection, us, obtained from an ana-
lytical solution [3] and by the finite element method. ............ 110
Convergence of the first four linear natural frequencies (flapping modes) of
the finite element model ............................. 111
The first nonlinear flapping natural frequency as a function of initial mode
amplitude a(0) with (b = 0 obtaining using three methods: the FE—based
ROM, the Rayleigh-Ritz based ROM, and the reference model (182-element
and 45-kept—modes). .............................. 112
The projection of the motion occurring on the first nonlinear flapping mode
manifold onto the linear eigenspace of the first linear flapping mode, in am-
plitude and phase coordinates .......................... 112
The time responses of the first nonlinear flapping mode displacement 171 for
the F E—based ROM, the Rayleigh-Ritz based ROM, and the reference model. 113
The contribution of the second linear flapping mode 772 = P2(a, (15) to the first
nonlinear mode manifold, solved by the collocation method, as a function of
the first nonlinear mode amplitude and phase. ................ 113
The time response of the second linear flapping mode deflection 772 =
P2(a(t), ¢(t)) on the first nonlinear mode manifold for the FE—based ROM,
the Rayleigh-Ritz based ROM, and the reference model. .......... 114
The contribution of the tenth linear flapping mode 1710 = P10(a, (ii) to the first
nonlinear mode manifold, solved by the collocation method, as a function of
the first nonlinear mode amplitude and phase. ................ 114
The time response of the tenth linear flapping mode deflection 7710 =
P10(a.(t), ¢(t)) on the first nonlinear mode manifold for the F E-based ROM,
the Rayleigh-Ritz based ROM, and the reference model. .......... 115
The contribution of the first linear axial mode 1722 = P2201, Q5) to the first
nonlinear mode manifold, solved by the collocation method, as a. function of

the first nonlinear mode amplitude and phase. ................ 115

xii

3.12

3.13

3.14

3.15

3.16

3.17

3.18

4.1
4.2
4.3

4.4

4.5

The time response of the first linear axial mode deflection 1722 =
P22(a(t), ¢(t)) on the first nonlinear mode manifold for the FE—based ROM,
the Rayleigh-Ritz based ROM, and the reference model. ..........
The contribution of the twelfth linear axial mode n33 = P33(a, (b) to the first
nonlinear mode manifold, solved by the collocation method, as a function of
the first nonlinear mode amplitude and phase. ................
The time response of the twelfth linear axial mode deflection 7733 =
P33(a(t), ¢(t)) on the first nonlinear mode manifold for the F E—based ROM,
the Rayleigh-Ritz based ROM, and the reference model. ..........
The time dependence of the beam-tip transverse deflection, w(L, t), for the
FE—based ROM, the Rayleigh-Ritz based ROM, and the reference model. .
Transverse deflection w(:v, t) for a quarter-period of motion on the first non-
linear mode manifold. The motion starts at the maximum deflection (the
bottom curve) and moves as shown to the zero deflection at a quarter-period,
after which it moves upward in a symmetric manner about w = 0 and then
repeats. .....................................
Angle deflection 0(1v, t) for a quarter-period of motion on the first nonlinear
mode manifold. The motion starts at the maximum deflection (the top
curve) and moves as shown to the zero deflection at a quarter-period, after
which it moves downward in a symmetric manner about 0 = 0 and then
repeats. .....................................
Axial deflection u.(;1:, t) for a quarter-period of motion on the first nonlinear
mode manifold. The dashed line denotes the static deflection, u3(:r). The
bottom curve corresponds to the initial condition. The top curve from Figure
3.16 and the bottom curve from Figure 3.17 correspond to the top u(:r) curve
here, all at one quarter-period. Note that the axial motion occurs at twice the
frequency of the transverse and angle motions, due to symmetry. As time
progresses beyond the quarter-period, the axial motion moves downward
again, and oscillates between the top and bottom curves shown. ......

A five-DOF nonlinear spring-mass system. ..................
System substructures. .............................
The contribution of the second fixed-interface linear mode amplitude 779% =
Xzfimiv’fi, 7.7{Vfl) to the first fixed—interface nonlinear mode manifold of sub—
structure 6. ...................................
The contribution of the second fixed-interface linear mode amplitude "$7.7 =
X3079”, 779,”) to the first fixed-interface nonlinear mode manifold of sub-
structure 7. ...................................
The time response of the displacement $1 of mass ml from Figure 4.1, which
corresponds to 513?, :13? , and :517 from Figure 4.2, for the nonlinear-CMS model,
the linear—CMS nonlinear model, the linear-CMS linear model, and the orig-

inal model .....................................

xiii

116

116

117

117

118

119

120

168
168

169

169

170

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

4.19

4.20

The time responses of the displacement $2 from Figure 4.1, which corre-
sponds to :13? from Figure 4.2, for the nonlinear—CMS model, the linear-CMS
nonlinear model, the linear-CMS linear model, and the original model. . . 170
The time responses of the displacement $3 from Figure 4.1, which corre-
sponds to I? from Figure 4.2, for the nonlinear-CMS model, the linear-CMS
nonlinear model, the linear-CMS linear model, and the original model. . . . 171
The time responses of the displacement :54 from Figure 4.1, which corre-
sponds to it); from Figure 4.2, for the nonlinear-CMS model, the linear-CMS
nonlinear model, the linear-CMS linear model, and the original model. . . . 171
The time responses of the displacement 11:5 from Figure 4.1, which corre

sponds to 173 from Figure 4.2, for the nonlinear—CMS model, the linear-CMS

nonlinear model, the linear-CMS linear model, and the original model. . . . 172
A class of nonlinear spring-mass system. ................... 172
System substructures. ............................. 173
The first four fixed-interface LNMs of substructure [3 using the first-trial
parameters. ................................... 173
The coefficients of the pqr cubic nonlinear terms expressed in fixed-interface

linear modal coordinates and associated with the first mode (master mode)

of substructure B using the first-trial parameters. .............. 174

The contribution of the second fixed-interface linear mode amplitude 77? ”[3 =

X2fi(n]V,fi, niv’fi) to the first fixed-interface nonlinear mode manifold of sub-
structure 5 using the first-trial parameters ................... 174

5:

to the first fixed-interface nonlinear mode manifold of sub-

The contribution of the third fixed-interface linear mode amplitude 1797’
3 (771 v 771 )

structure 6 using the first-trial parameters ................... 1

The contribution of the fourth fixed-interface linear mode amplitude 1751,13 =

X f (nfv ’fl , nil/fl ) to the first fixed-interface nonlinear mode manifold of sub—

"‘1

CI!

structure 6 using the first-trial parameters ................... 175
The time responses of the first fixed-interface mode displacement nil/’3 for the
ROM and the original model of substructure 6 using the first-trial parameters. 176
The time responses of the second fixed-interface linear mode displacement
név’fi = Xgmiv’fi, nil/fl) for the ROM and the original model of substructure
B using the first-trial parameters. ....................... 176
The time responses of the third fixed-interface linear mode displacement
név’fi = Xgmiv’fl, niv’fi) for the ROM and the original model of substructure
6 using the first-trial parameters. ....................... 177
The time responses of the fourth fixed-interface linear mode displacement
nil/fl = X f (77]V ’5 , 7-)]Vfl ) for the ROM and the original model of substructure

L3 using the first-trial parameters. ....................... 177

xiv

4.21

4.22

4.23

4.24

4.25

4.26

4.27

4.28

4.29

4.30

4.31

4.32

The time responses of the first fixed-interface mode displacement 779% for
the linearized one-mode model, the one-mode model, and the ROM of sub-
structure 6 using the first-trial parameters ...................
The first four fixed-interface LNMs of substructure 3 using the second-trial
parameters. ...................................
The coefficients of the pqr cubic nonlinear terms expressed in fixed-interface
linear modal coordinates and associated with the first mode (master mode)
of substructures ,6 using the second-trial parameters. ............

The contribution of the second fixed-interface linear mode amplitude 779513 =

Xgmiv’fi, niv’fi) to the first fixed-interface nonlinear mode manifold of sub-

structure 6 using the second-trial parameters. ................
The contribution of the third fixed-interface linear mode amplitude név’fi =

X g (7)]V ’3 , niv’fi ) to the first fixed-interface nonlinear mode manifold of sub-

structure 6 using the second-trial parameters. ................

The contribution of the fourth fixed-interface linear mode amplitude nil/(3 =

X f (nil/fl , rift/6 ) to the first fixed—interface nonlinear mode manifold of sub-
structure 6 using the second-trial parameters. ................
The time responses of the first fixed-interface mode displacement 779MB for
the linearized one-mode model, the one-mode model, the ROM, and the
original model of substructure 3 using the second-trial parameters.

The time responses of the second fixed-interface linear mode displacement
név’fi = Xgmiv’fi, niv’fi) on the first fixed-interface nonlinear mode manifold
for the ROM and the original model of substructure ,8 using the second—trial
parameters. ...................................

The time responses of the third fixed-interface linear mode displacement

N, N, .N,
773 fi=X§0Il fin] 5)

for the ROM and the original model of substructure (3 using the second-trial

on the first fixed-interface nonlinear mode manifold

parameters. ...................................
The time responses of the fourth fixed-interface linear mode displacement
nf’fl = X f (77?“3 , 7.7]Vfi ) on the first fixed-interface nonlinear mode manifold
for the ROM and the original model of substructure B using the second-trial
parameters. ...................................
The time responses of the displacement 2:? from Figure 4.11 with the inter-
face fixed for the linearized one—mode model, the one-mode model, the ROM,
and the original model of substructure 6 using the second-trial parameters.
The time responses of the displacement mg] from Figure 4.11 with the inter—
face fixed for the linearized one—mode model, the one-mode model, the ROM,

and the original model of substructure 6 using the second—trial parameters.

XV

178

178

179

179

180

180

181

181

182

182

183

183

4.33

4.34
4.35

4.36

4.37

4.38

4.39

4.40

4.41

4.42

4.43

4.44

i3
i
fixed-interface linear mode manifold (linear eigenspace) and on the first fixed-

Comparison of deflections 2: (t) for a quarter-period of motion on the first
interface nonlinear mode manifold of substructure B using the second-trial
parameters. The motion starts at the maximum deflection (the bottom
curve) and moves as shown to the zero deflection at a quarter-period. Note
that linear mode shape is normalized such that r? is the same as 1’13 of the
nonlinear mode shape at each time .......................
The first four fixed-interface LNMs of substructure 7 .............
The coefficients of the pqr cubic nonlinear terms expressed in fixed-interface
linear modal coordinates and associated with the first mode (master mode)
of substructure 7 .................................
The contribution of the second fixed-interface linear mode amplitude 17?,” =
X g (nil/’7, 179,”) to the first fixed-interface nonlinear mode manifold of sub-
structure 7. ...................................
The contribution of the third fixed~interface linear mode amplitude 77:]; {’7 =
X g (179,”, 17¢”) to the first fixed-interface nonlinear mode manifold of sub—
structure 7. ...................................
The contribution of the fourth fixed-interface linear mode amplitude 041‘,” =
XZ(77[V’7,1'7[V’7) to the first fixed-interface nonlinear mode manifold of sub-
structure 7. ...................................
The time responses of the first fixed-interface mode displacement nix,” for
the linearized one-mode model, the one-mode model, the ROM, and the
original model of substructure 7. .......................
The time responses of the second fixed-interface linear mode displacement
179,” = X 3 (779m, 7'79”) on the first fixed-interface nonlinear mode manifold
for the ROM and the original model of substructure 7. ...........
The time responses of the third fixed-interface linear mode displacement
n?” = X g (7)]V ’7, nil/’7) on the first fixed-interface nonlinear mode manifold
for the ROM and the original model of substructure 7. ...........
The time responses of the fourth fixed-interface linear mode displacement
nil/’7 = X 207?”, fly”) on the first fixed-interface nonlinear mode manifold
for the ROM and the original model of substructure 7. ...........
The time responses of the displacement £133 from Figure 4.11 with the inter-
face fixed ( as? (t) = 0 ) for the linearized one-mode model, the one-mode
model, the ROM, and the original model of substructure 7 ..........
The time responses of the displacement $31 from Figure 4.11 with the in-
terface fixed for the linearized one-mode model, the one-mode model, the
ROM, and the original model of substructure 7. ...............

xvi

184
184

185

185

186

186

187

187

188

188

189

189

4.45

4.46

4.47

4.48

4.49

4.50

4.51

4.52

4.53

4.55

Comparison of deflections 2:270) for a quarter-period of motion on the first
fixed-interface linear mode manifold (linear eigenspace) and on the first fixed-
interface nonlinear mode manifold of substructure 7. The motion starts at
the maximum deflection (the bottom curve) and moves as shown to the zero
deflection at a quarter-period. .........................
The time responses of the first synthesized fixed-interface mode displacement
779”" for the three-DOF linear-CMS linear model, the three-DOF linear-CMS
nonlinear model, the three-DOF nonlinear-CMS model, and the forty-one-
DOF linear-CMS nonlinear model of substructure B. ............
The time responses of the second synthesized fixed—interface linear mode
displacement 779% for the three-DOF nonlinear-CMS model and the forty-
one—DOF linear-CMS nonlinear model of substructure fl. ..........
The time responses of the third synthesized fixed-interface linear mode dis-
placement 17f;v ’fl for the three-DOF nonlinear-CMS model and the forty-one-
DOF linear-CMS nonlinear model of substructure fl. ............
The time responses of the fourth synthesized fixed-interface linear mode
displacement niv’fi for the three-DOF nonlinear-CMS model and the forty-
one-DOF linear-CMS nonlinear model of substructure fl. ..........
The time responses of the first synthesized fixed-interface mode displacement
nil/37 for the three-DOF linear-CMS linear model, the three-DOF linear-CMS
nonlinear model, the three-DOF nonlinear-CMS model, and the forty-one—
DOF linear-CMS nonlinear model of substructure 7 ..............
The time responses of the second synthesized fixed-interface linear mode
displacement 1);] ’7 for the three-DOF nonlinear-CMS model and the forty-
one-DOF linear-CMS nonlinear model of substructure 7. ..........
The time responses of the third synthesized fixed-interface linear mode dis-
placement "97,7 for the three—DOF nonlinear-CMS model and the forty-one-
DOF linear-CMS nonlinear model of substructure 7 ..............
The time responses of the fourth synthesized fixed-interface linear mode
displacement nil/’7 for the three-DOF nonlinear-CMS model and the forty—
one-DOF linear-CMS nonlinear model of substructure 7. ..........
The time response of the displacement $1 of mass m1 from Figure 4.10,
which corresponds to r?,x?, and 2517 from Figure 4.11, for the three-DOF
linear-CMS linear model, the three—DOF linear-CMS nonlinear model, the
three-DOF nonlinear-CMS model, and the forty-one—DOF original model. .
The time response of the displacement 51:2 of mass m2 from Figure 4.10,
which corresponds to as? from Figure 4.11, for the three-DOF linear-CMS
linear model, the three-DOF linear-CMS nonlinear model, the three-DOF
nonlinear-CMS model, and the forty-one-DOF original model. .......

xvii

190

190

191

191

192

192

193

193

194

194

4.56

4.57

4.58

4.59

4.60

The time response of the displacement 1:21 of mass mm from Figure 4.10,
which corresponds to 22% from Figure 4.11, for the three-DOF linear-CMS
linear model, the three-DOF linear-CMS nonlinear model, the three-DOF
nonlinear-CMS model, and the forty-one—DOF original model. .......
The time response of the displacement 1:22 of mass mm from Figure 4.10,
which corresponds to 2:; from Figure 4.11, for the three-DOF linear-CMS
linear model, the three-DOF linear-CMS nonlinear model, the three-DOF
nonlinear-CMS model, and the forty-one—DOF original model. .......
The time response of the displacement $41 of mass m41 from Figure 4.10,
which corresponds to 2:31 from Figure 4.11, for the three-DOF linear-CMS
linear model, the three—DOF linear-CMS nonlinear model, the three-DOF
nonlinear-CMS model, and the forty-one—DOF original model. .......
The time responses of the first synthesized fixed-interface mode displacement
179w for the three-DOF nonlinear-CMS model and the forty-one—DOF linear-
CMS nonlinear model of substructure 6 with ”955(0) 2 1.3, "{qu) =
1.3, nC(0) = 0.04,Nfl(0) = 0, fifWO) = 0,1')C(0) = 0. ..........
The time responses of the first synthesized fixed-interface mode displacement
nil/’7 for the three-DOF nonlinear-CMS model and the forty-one-DOF linear-
CMS nonlinear model of substructure 7 with nil/(3(0) = 1.3, n[v’7(0) =
1.3, 770(0) = 0.04,[V’5(0) = 0, 77]“(0) = 0. r;C(0) = 0. ..........

xviii

195

196

196

197

197

CHAPTER 1

Introduction

1.1 Motivation

h‘lodern engineers are sometimes required to accurately model the motions of complex
structural systems, i.e., rotorcraft, turbo machines (turbines), vehicles, aircraft, bridges,
off-shore platforms, robotic arms, etc., so that they can function properly and safely under
certain operating conditions. Nowadays such structural systems are designed to be lighter,
have more complex geometry, and operate at high speeds in order to enhance performance.
These conditions can cause structural systems to experience responses in the nonlinear
regime. To mathematically describe such motions, one must formulate nonlinear models,
either discrete or continuous in nature. If they are continuous, in practice engineers typ-
cially discretize them by using either direct modal discretization (Rayleigh-Ritz method)
or the finite element (FE) method. The FE method is typically prefered in practice since
it is a versatile tool for modeling very complex structures. However, it has a disadvantage
in that it often requires a very large number of degrees of freedom (DOF) to accurately
model the structure ([4]). The dynamic analysis (e.g., determination of natural frequen-
cies, simulations, etc.) of nonlinear structures is typically more easily performed in modal
coordinates than in physical coordinates, since it requires a smaller number of DOF. How-
ever, many modal coordinates are still needed for some nonlinear structures, typically those
with some type of essential coupling between the linear modes. Therefore, this can result

in considerable computational effort when ones try to analyze their dynamics ([5]).

The main objective of this research is to develop a class of techniques for generating ac-
curate reduced order models for nonlinear structural systems. Our starting point is gener-
ally a nonlinear model, from either modal or FE analysis, whose dynamics are expressed
by equations of motion expressed in terms of the linear modal coordinates. First, this
work demonstrates a general framework on how to obtain nonlinear reduced order mod-
els (ROMS) using nonlinear normal modes (NNMS) that are defined in terms of invariant
manifolds. Next, a methodology for obtaining nonlinear ROMS from FE based descriptions
of structural elements is demonstrated through an example, the nonlinear rotating beam,
which is a crude model for a helicopter rotor blade. Lastly, this work demonstrates how
one can synthesize nonlinear ROMS using NNMs of relatively simple substructures that are

assembled to form a complex structural system.

In this first chapter we offer brief overviews and literature surveys of the main themes
considered in the thesis, Specifically: nonlinear normal modes, rotating beams, and
component mode synthesis. These are considered sequentially, and are followed by a

description of the outline of the remainder of the thesis.

1.2 Nonlinear Normal Modes

Linear modal analysis is a fundamental tool used in linear vibration theory. It allows one to
decompose a general motion into a linear combination of the fundamental motions (modal
motions) that take place on the eigenspaces in the system phase space; this is the essence of
superposition. These modal responses are invariant, since, if one starts with a purely modal
response, the system stays in that mode for all time. Similarly, if the system is started with
energy in a subset of modes, only those modes will be active during the ensuing response.
For general responses, the individual modal responses remain uncoupled from one another

during the motion.

Of course, these facts do not generally hold for nonlinear systems. Certainly superposition
does not extend to nonlinear systems. But, the concept of fundamental invariant responses
does carry over, even if one cannot construct general responses by combining them, whether
in a linear or nonlinear manner. Rosenberg and Atkinson ([6] and [7]) provided the pio-
neering ideas which tried to extend the idea of fundamental motions to nonlinear systems.
Rosenberg and coworkers carried out several studies along these lines ([6], [8], [7], [9], [10],
[11]) and summarized these results in a classical review paper [12], in which Rosenberg
offered a break-through definition of nonlinear normal modes (N NMS). In that work, the
NNMS of conservative systems with 71 degrees of freedom were defined as vibration in uni-
son, i.e., vibrations such that all degrees of freedom reach their extrema at the same time
and pass through zero at the same time. By this definition, one can express all generalized
displacements as functions of a chosen generalized displacement. When these systems pos-
sess certain types of symmetries, the constraint relations are linear in the configurations
space (just as they always are in the linear case) and are referred to as “similar” NNMS. For
more general, but still conservative, systems the constraint relations are nonlinear and the
nonlinear modes are called “non-similar” N N Ms, which can be depicted by curved lines in
the configuration Space. This definition was picked up and used by subsequent researchers
who used it to construct NNMS and study their stability and bifurcations that occur due to
changes in system parameters and system energy levels. These were typically conservative
systems with two degrees of freedom (DOF) system ([13], [14], [15]). This definition, to-
gether with the conservation of energy, were employed by Vakakis to construct non-similar
NNMS of a two DOF-conservative system in [16]. It was also used to study the steady
state motions of two DOF systems subjected to periodic forcing; see [17] for Similar N NMS
and [18] for nonsimilar NNMS. Vakakis also discovered that the important phenomenon of
mode localization can occur for NNMS, even for perfectly tuned subsystems [16]. This is
in contrast to linear systems, wherein localized modes exist only when the subsystems are
Slightly mistuned. A good account of NNMS, their stability and bifurcations in unforced
nonlinear systems, their existence in forced nonlinear systems, and their application to

engineering systems is given in the monograph by Vakakis [19].

The N NM concept has been generalized to a wide class of systems, which includes discrete
systems with dissipation and gyroscopic terms, as well as quite general continuous systems
by Shaw and Pierre in ([20] and [21]). Therein, a definition of NNMS is given in terms of
invariant manifolds, which are a natural way to define and construct fundamental motions
of nonlinear systems. The procedures used to obtain the NNMS by this approach are
closely related to center manifold theory, which is used for bifurcation analysis ([22]),
and inertial manifold theory, which is used to study the long time behavior of dissipative
partial diflereiitial equations (PDES) ([23], [24]). Using this definition, the nonlinear system
equations are restricted to a two-dimensional invariant manifold that describes the NN M of
interest. The behavior on the N N M manifold is governed by a single second order differential
equation of motion, which corresponds to the equation of motion for a nonlinear single mode
reduced order model. In Shaw and Pierre ([20], [21]) approximate solutions of the invariant
manifold equation were obtained by asymptotic expansions using polynomials expressed in

terms of a generalized position-velocity pair of state variables.

King and Vakakis ([25]) developed an energy—based N N M approach based on [21] to investi—
gate N N Ms of one dimensional, conservative, continuous systems. N ayfeh and N ayfeh ([26])
computed NNMS of continuous systems based on a complex amplitude / phase formulation
and the method of multiple scales. Nayfeh ([27]) compared the various methods for con-
structing N NMS of continuous systems as developed by Shaw and Pierre, King and Vakakis,
Nayfeh and Nayfeh, and a new approach that employed normal form theory. They con-
cluded that all expansion methods yielded the same results, but claimed that the method

of multiple scales with complex variables was the simplest to implement.

All of these approaches are in some manner equivalent, but differ in terms of formulation,
solution, and range of applicability. However. the invariant manifold definition is the most
general, since it covers the widest range of systems and responses. It can also be generalized
to the case of multiple modes, as required for the case of internal resonances, or if one

fecmires a model that is valid over a wide frequency range. By defining the NNMS in terms

of two-dimensional invariant manifolds, the individual NNMS can be constructed, but they
can not interact with each other once a motion is initiated on any one of them. Therefore,
a motion involving multiple modes cannot be captured by this definition. In addition,
the concept of mode superposition cannot be applied to construct non-linear multi-mode
models using individual NNMS, since the essential interaction between the NNMS will be
missing. Boivin et al. ([28]) generalized the individual N NM concept by defining a motion
involving M N NMS as a motion that takes place on a 2M -dimensional invariant manifold
in the system’s phase space. By this definition, the non-linear ODES are restricted to the
M-NNMS invariant manifold of interest. The behavior on the manifold is governed by
M second order, coupled equations of motion (a nonlinear M -degree-of-freedom system).
Boivin et al. also described how to detect the case of internal resonances from the multi-
mode invariant manifold formulation [29]. N NMS were also constructed for discrete systems
with internal resonances, based on the complex formulation, by Nayfeh et al. [30]. King
et al. ([31]) also extended the energy-based NN M approach to cover the case of internal

resonances. These latter two studies were for the case of M = 2 N N Ms.

Slater ([32]) developed a numerical method for determining individual NNMS based on
numerical searching techniques for periodic solutions of conservative non-linear systems.
Using this approach, individual solutions are found, but the entire family of motions on
the NNM manifold cannot be obtained, therefore one cannot develop ROMS using this ap-
proach. A similar shooting technique has been used in this work and by previous resarchers
in the author’s research group ([3], [33], [34], [35] and [36]). In this thesis it is employed
for comparing ROMS with simulations of the original system, which has the full number of

DOF.

Other works, related to the use of invariant manifolds for the generation of ROMS, include
the use of Karhunen-Loeve (K-L) decomposition to develop accurate low-order models, by
using data obtained from transient simulations of large-scale systems. See [37], [38], [39],

and [40], for example.

In references [20], [21], [28], [41], and [29], the N NM invariant manifolds were approximated
by a polynomial expansion (asymptotic series) of position-velocity pairs of chosen (master)
modes, which provides a solution that is locally valid. The approximate invariant manifolds
obtained by such an asymptotic series will diverge from the actual invariant manifold in
some amplitude regime. Typically, the domain of validity of the approximation is not
known a priori. Hence, the reduced equations of motion will generate inaccurate time
responses when the amplitudes of the modes are “too large”. Pesheck et al. ([33]) improved
the approximation of the invariant manifolds by expressing the invariant manifolds as an
expansion of basis functions defined over a specified domain and numerically solving the
invariant manifold equations. The expansion of basis functions is introduced into the PDEs
governing the NNM invariant manifold, and, using a Galerkin projection, the nonlinear
equations for the expansion coefficients were obtained and then solved numerically. By
this approach, the domain of approximation can be selected by the user, and the error
of approximation can be minimized over the chosen domain by selection of the number
and type of basis functions. Since the computational cost associated with the Galerkin
projection can be quite expensive, the collocation method has recently been adopted in [35]
to minimize computational efforts associated with this method. In this approach, instead
of projecting each manifold governing equation onto each basis function, each manifold
governing equation is projected onto a set of Dirac delta functions in the master coordinates,
thereby providing a solution that minimizes an error that is measured in a point-wise

manner.

Research on the construction and use of NNMS continues; here we outline recent work by
others in the MSU/UM NNM research group, in particular by Mr. Dongying Jiang, a
Ph.D. student at UM. Jiang et al. ([36]) have applied the work of Pesheck et al. ([33])
to construct NNMS of piecewise linear systems, and are working on systems with friction
elements. They have also extended the work of Pesheck et al. ( [33]) to numerically construct
multi-NNM models which can capture internal resonances among participating modes, and
are valid over a large range of amplitudes ([42]). They have also numerically constructed

NNMS of nonlinear systems under periodic excitation ([43]). In this case the manifolds are

time-periodic in nature, and responses on them represent the steady-state responses of the
full system. Current work is aimed at distilling ROMS from detailed FE models for rotor
blades.

A detailed summary of the invariant manifold approach to NNMS, and the computational

issues associated with its solution, are presented in Chapter 2 of this thesis.

1.3 Beam (Blade) Dynamics

The dynamic analyses of helicopter blades, turbopropeller blades, wind-turbine blades and
robotic arms has provided motivation for investigations of the vibration of rotating beams.
To predict the dynamic characteristics of rotating flexible structures, the kinematics must
be carefully modeled, which leads to nonlinear coupling effects between degrees of freedom
(DOF) in different directions. These coupling effects can cause slow modal convergence,
thereby often requiring large system models for accurate dynamic representation. Simula-
tion of such large—scale models is a time consuming process, which slows parametric studies

and design cycles.

Much work has been done using finite elements (FE) to model the nonlinear, large amplitude
vibrations of rotating beams, including [44], [45], [46], [47], [48], and [49]. These models are
typically complex in nature due to their geometry, degrees of freedom (flap, lead-lag, axial,
and torsion), and nonlinear coupling effects. Furthermore, because of the nature of the
finite element approach, many elements are required in order to obtain an accurate model.
A common approach is to use linearization of the finite element model about the nonlinear
static equilibrium position and solve the eigenvalue problem of the resulting linearized
model to obtain the linear natural frequencies of the system ([44] and [49]). Bauchau and
Hong ([45]) also utilized finite elements in time to obtain nonlinear responses and stability

results of the rotating beam undergoing large deflections. However, the computational time

associated with obtaining the equilibrium solution was expensive, because all of the spatial
degrees of freedom are coupled at all time steps. Perturbation modes ([50] and [51]) were
applied to the finite element model of a helicopter rotor blade in order to obtain a reduced
order model ([46]). Bauchau and Bottasso ([52]) applied the perturbation modes to the
Space-time finite element model of a beam subjected to a sinusoidal load in order to obtain
a reduced order model. Crespo da Silva ([53]) utilized a truncated set of eigenfunctions or
eigenvectors obtained from the linearized system of PDES or the linearized finite element
model about the nonlinear static equilibrium position in order to obtain a reduced order
model of a beam in planar motion. Crespo da Silva ([54]) also extended his work to handle

multi-beam structures in planar motion.

In most nonlinear structures, there is no simple expansion of basis vectors which decouples
the DOF (i.e., modes) in the frequency range of interest from those outside that range. For
the rotating-beam problem, this is evident in the nonlinear coupling between transverse and
axial motions. Therefore, some (potentially important) nonlinear effects may be ignored
in the truncation process, unless one is careful. Generally, many linear modes must be
retained in the nonlinear model in order to minimize these effects. NNMS is a natural

approach for handling this issue.

Over the past decade, systematic procedures have been developed to obtain ROMS via
NNMS that are based on invariant manifolds in the state space of nonlinear systems,
as described above. These procedures initially used asymptotic series to approximate
the geometry of the invariant manifold and have been used to study the nonlinear
rotating Euler-Bernoulli Beam ([3]). More recent work has employed a numerically-based
Galerkin approach to obtain the geometry of the NNM invariant manifolds out to large
amplitudes ([33]). These procedures can be applied to more general nonlinearities over
wider amplitude ranges, and have been recently applied to study the vibrations of a
rotating Euler-Bernoulli beam ([34]). These approaches have provided accurate models
for the fundamental nonlinear flapping mode, by systematically capturing the essential

dynamic coupling that exists between the linear modes of the system. Chapter 3 of this

thesis considers this problem in detail, by investigating the N NMS of a rotating beam that

is modeled using a nonlinear finite element formulation.

1.4 Substructure Synthesis

Many complex structures are composed of several relatively simple substructures that
are assembled together. This occurs in trusses, bladed disk assemblies in turbine rotors,
aerospace and ground vehicles, and other applications. In such cases it is convenient to
develop a dynamic model for the overall structure by taking advantage of the dynamic
properties of the substructures. Methods for doing this for linear structural models are
well developed and have been used extensively, especially in the aerospace industry ([55],
[56], [57], [58], etc). These techniques construct ROMS of the overall structure by making
use of modal-based ROM descriptions of the substructures and combining these using
a technique known as Component Mode Synthesis (CMS). In this section we offer an
overview of CMS for linear systems and describe the two main CMS approaches, fixed-

and free- interface CMS. A thorough review of substructuring and CMS can be found in [59].

1.4.1 Component Mode Synthesis

CMS was developed to synthesize models that are described in terms of substructures,
and to take advantage of model size reduction carried out at the substructure level ([60],
[61], [62]). In CMS, the dynamics of each substructure is described by a set of dynamic
(normal) modes and a set of static (constraint or residual attachment) modes that are used
to describe the interfaces between the substructures. A set of component normal modes
is selected from each substructure and are chosen and truncated in such a way that the
modes lie-in the frequency range of interest. A set of static (so-called constraint) modes
is a key component for the low frequency response of the structure ([63]) and are used to

couple the substructures together. There are two general types of CMS methods; they are

known as the fixed-interface and the free-interface approaches, as briefly described below.

1.4.1.1 Fixed—Interface Linear CMS

The fixed-interface CMS technique, developed by [62], is widely used, since the procedures
are straightforward. Moreover, it produces very accurate models with very few component
modes ([64]). The dynamics of each substructure is described by its modal description,
which is composed of substructure normal modes (component modes) and constraint
modes. The component normal modes are the normal modes of the substructure derived
for the case when the interface coordinates between the substructures are held fixed. A
constraint mode (static mode) of the substructure is the deflection obtained by imposing a
unit displacement on one of the interface coordinates and holding the remaining interface
coordinates fixed. To obtain all of the constraint modes, the process is applied in turn
to each of the interface coordinates. By applying displacement and force compatibility
conditions at the interface coordinates, one can obtain the reduced synthesized system,
which is described by the component normal-mode coordinates and the generalized

constrained coordinates. This approach iS known as the Craig-Bampton method.

1.4.1.2 Free-Interface Linear CMS

Ree-interface CMS methods are more attractive than fixed-interface CMS methods when
the component modes are obtained from modal testing or when an experimental verification
of the component modes is required ([65]). The free-interface CMS technique developed by
Craig and Chang ([66], [67] and [63]) is the most accurate among the free-interface CMS
techniques. It is a modified version of Rubin’s method [68] and MacNeal’s method [69].
It is superior to the CMS of Craig-Bampton in terms of accuracy, but is more difficult to

implement ([64] ).

10

In this synthesis technique, the dynamics of each substructure is described by a set of
substructure component normal modes and residual attachment modes. The component
normal modes are the normal modes of the substructures for the case when the interface
coordinates between the substructures are free. The residual attachment modes of the
substructure are a Special type of static modes that are used to couple the substructures
together, and they also account (at least partially) for the static deflections of the truncated
normal modes of the substructures ([66], [67] and [63]). By applying the displacement
and force compatibility conditions at the interface coordinates, and neglecting the inertial
effects associated with the generalized residual attachment coordinates, one can obtain the
reduced synthesized system. It is described in terms of only the component normal mode
coordinates since, in this approach, the residual attachment modes can be condensed out
of the equations of motion ([66], [67] and [63]). This approach is known as the Craig-Chang

method.

Chapter 4 of this dissertation describes a CMS methodology developed for nonlinear
systems, wherein the substructure ROMS are developed using NNMS, and these are
assembled using a newly developed technique that is a extension of the fixed-interface

CMS method.

1.5 Dissertation Organization

The Dissertation is organized as follows.

In Chapter 2, the formulation and solution of NN M invariant manifold equations are de-
scribed. Both previous and original work is described. Asymptotic series expansions and
weighted-residual type methods, i.e., Galerkin and collocation, are the methods employed
to solve for the manifolds. In this chapter, two new methods, similar to those described in

[33], are developed and implemented to solve for the single-mode manifold solutions. They

11

are both formulated in terms of modal position and velocity. The first alternative is to solve
for the manifold solution using globally defined basis functions using the Galerkin method.
The second is to obtain the manifold solution over several small patches using locally de-
fined basis functions, and solving these smaller problems using a collocation method. A
rough comparison of the relative computational efforts of these approaches is provided.
Also, a two-DOF nonlinear spring-mass system and a FE model of a rotating beam are
used for demonstrating these approaches and comparing their accuracies, by direct time

Simulations of the corresponding ROMS and the original system models (with full DOF).

In Chapter 3, the generation of nonlinear ROMS from FE based descriptions is demon-
strated through the application to a nonlinear rotating beam, which is a highly idealized
model for a helicopter rotor blade. First, the nonlinear FE model of a rotating beam is gen-
erated and converted into a truncated (but still large—scale) modal form that is convenient
for the NNM analysis. The invariant manifold equations are formulated, and a numerical
collocation method is used to obtain the solution of the NN M invariant manifold for the
fundamental flapping mode of the beam. This invariant manifold is used to construct a
nonlinear single DOF ROM, which is subsequently used for a Simulation study. Note that

the results of Chapter 3 have been recently published [35].

In Chapter 4, the development of N N Ms, as needed for the individual substructures is first
reviewed. The associated invariant manifold equations, parameterized by modal position
and velocity, are formulated, and the numerical collocation method is reviewed, since it
allows one to obtain the solution of the NNM invariant manifold. Then, the procedures
for the fixed-interface nonlinear CMS are described. A five—DOF spring-mass system and a
forty-one-DOF spring-mass system are used to demonstrate the effectiveness of the method,

via direct time—simulation comparisons of different ROMS.

Chapter 5 offers a summary of the contributions of this research to the field of nonlinear

structural dynamics, along with a discussion of potential areas for further work in this area.

12

CHAPTER 2

Comparison of Methods for
Constructing Invariant Manifolds for

Nonlinear Normal Modes

2.1 Introduction

In linear vibration theory one can decompose a general free vibration response as a linear
combination of the fundamental motions (modal motions) that take place on the eigenspaces
of the system. These motions are invariant, that is, if one initiates a purely modal response,
the system remains in that mode for all time. The works of Rosenberg and Atkinson ([6]
and [7]) were the pioneering works which extended the idea of fundamental motions to
nonlinear systems. In those works, nonlinear spring-mass systems with two degrees of
freedom (DOF) were studied. Due to special symmetries possessed by the systems, they
were able to construct linear modal-constraint relations, the attendant nonlinear natural
frequencies, and the stability of each nonlinear mode in terms of its amplitude. Rosenberg
summarized his works ( [6], [8], [7], [9], [10], [11]) on normal mode vibrations of nonlinear
systems in [12], in which a fundamental definition of nonlinear normal modes (NNMS)
for multi-DOF systems was given. In that work the concepts of similar and non-similar
modes were introduced. Caughey et al. ([14]) studied similar normal modes and their

bifurcations in terms of system parameters for two-DOF nonlinear conservative system,

13

 

based on Atkinson’s work ([7]). Rand et al. ([15]) studied the general class of two-DOF
nonlinear conservative systems. In that study, they were able to determine the number and
stability of periodic motions of which the non-similar modes were special cases and also
studied the bifurcation of the periodic motions in terms of system parameters. Vakakis
([16]) studied the non-similar modes of two-DOF nonlinear conservative systems, where

the construction of the non-similar modes exploited the conservation of energy.

Shaw and Pierre ([70] and [20]) generalized the definition of NNMS by using concepts
from invariant manifold theory, which encompassed a wide class of quite general nonlinear
systems. The technique was constructive and allowed for traveling wave (that is, complex)
and damped NNMS. Boivin and co-workers et al. ([28], [29], [71]) extended the construction
of the Single—mode NNM manifold developed by Shaw and Pierre to the multi—mode case,
which allowed for the dynamic interactions among a subset of nonlinear modes of interest.
A summary of these works by Shaw, Pierre, and coworkers ([72], [73], [41], [28], [29], and
[71]) can be found in [74]. In these papers, the solution of the invariant manifold equatiosn

were sought in terms of locally valid asymptotic power series expansions.

Pesheck et al. ([33]) developed a numerical method for solving the invariant manifold
equations, such that the solution, while not analytically available, was valid over a relatively
large amplitude range. An added feature of this approach is that the nonlinear terms
are not limited to being smooth functions, which allows for extensions to more general
classes of problems ([36]). Even though the Galerkin method gives accurate solutions, the
computational cost of acquiring solutions is quite expensive, especially when one attempts
to generate multi-mode models ([75]). This motivated the application of the collocation
method, which significantly reduces computational costs, but does not sacrifice much in the
way of accuracy ( [76], [77], and [78]). This method has been applied in [35] to study the
dynamics of a finite-element model of a rotating beam. In the forthcoming work of Jiang
et al. ([42]), the Galerkin method developed by Pesheck et al. ([33]) has been generalized
to construct accurate multi-mode manifolds that are able to capture internal resonances,

and this method is applied to a model of a rotating beam.

14

In this chapter, we develop two alternative methods for the numerical solution of the
single-mode invariant manifold equations, and offer a comparison of the various methods
that have been used in this and previous studies. The new methods are Galerkin-based,
in the same spirit of that developed in [33], except that the equations are formulated in
terms of modal position and velocity, instead of modal amplitude and phase. The first new
method uses global basis functions and obtains the unknown coefficients for the manifold
solution using the Galerkin method; this method is described in section 2.3.2.2. The second
new method uses a local patchwork of basis functions, and the unknown coefficients for the
manifold solution are obtained using the collocation method; this method is described in
section 2.3.2.4. In terms of computational time, both new methods have advantages over

the approaches used in [33] and [35]; this is summarized in section 2.3.3.

This chapter is outlined as follows. The formulations for the NNM invariant manifolds
are first introduced. The power series expansions methods and the weighted-residual
type methods, i.e., Galerkin and collocation methods, are described in detail, for both
amplitude—phase and displacement-velocity formulations. The computational effort
associated with each approach is then discussed, and the accuracy of each approach is
considered through direct comparisons in general terms and for calculations carried out

for two example problems. Some conclusions are drawn at the end of the chapter.

2.2 Formulations of NNM Invariant Manifold

We begin with a general discrete representation of the vibrations of a nonlinear structural
system obtained either by a finite element model followed by linear modal expansion, or
by a Rayleigh-Ritz approach. We assume that the system at linear order is undamped.
(This assumption greatly simplifies the problem, but can be relaxed, in principle.) In this

case, the equations of motion for a Q-DOF system are uncoupled at linear order and can

15

be expressed in the form:

Ifi+An=f(n,ii) (2.1)

where I is the identity matrix, A the diagonal matrix of squared linear natural frequencies,
f (17, 1'7) a vector of nonlinear forces, 77 the modal position vector, and 7'] the modal velocity

vector. The component form of equation (2.1) is given by

777i+wi2 7h = fiMjflij) (2-2)

for i,j=1,2,3,...,Q

where w,- is the linear natural frequency of mode 2' and Q is the number of retained linear

modes.

2.2.1 Invariant Manifold Equations in Terms of Modal Position

and Velocity

The fundamental concepts for nonlinear normal modes of discrete, conservative, nonlinear
systems were laid out by Rosenberg ([12]). Shaw and Pierre ([70], [20], [21]) used
the theory of invariant manifolds for dynamical systems to generalize the concept of a
nonlinear normal mode to a wide class of systems, including continuous systems and
systems with dissipation and gyroscopic terms. Boivin et al. ([28], [29], [71]) extended

the single-nonlinear-mode construction developed by Shaw and Pierre to the multi-mode

16

case, which is able to capture the dynamic interactions among a set of nonlinear modes of
interest. A summary on the initial series of works by Shaw, Pierre, and coworkers ([72],
[73], [41], [28], [29], and [71]) can be found in [74]. Below we provide a short account of
the geometric definition of NNM invariant manifolds and methods of constructing single-

and multi-mode versions of N N M models.

2.2.1.1 Singe-Mode Manifold Formulation

For present purposes one can consider an invariant manifold to be a low dimensional surface
living in the system state space, such that motions initiated on the surface will remain on
it for all time. This concept is the key to obtaining a reduced order NNM model, that is,
a lower-dimensional dynamical system, Specifically by restricting the equations of motion
to this surface. In an N-degree-of-freedom, undamped, nongyroscopic, linear vibratory
system, individual modal motions are synchronous responses that take place on a two-
dimensional plane (linear eigenspace) in the 2N -dimensional system state space. In this
case the response of the system is a time-harmonic standing wave in which all degrees
of freedom reach their extrema and pass through zero simultaneously. Such motions are
governed by two first-order linear differential equations or, equivalently, by one second-order
linear differential equation, which in this case is a Simple undamped oscillator. For linear
systems with gyroscopic and/or general dissipative terms this concept is also applicable,
however the motions taking place on the invariant planes are generally nonsynchronous,
and represent traveling wave responses of the system. This concept can be extended to
nonlinear systems as well, however, the motions generally take place on non-planar surfaces,
and the motions are governed by two first-order nonlinear differential equations, or by one
second—order nonlinear differential equation. For smooth nonlinear systems, the manifolds

are curved surfaces that are tangent to the manifolds (eigenplanes) of the linearized systems.

In [74], a definition for NNMS based on invariant manifolds was offered, as follows:

A normal mode for a nonlinear system is a motion that takes place on a two-

17

dimensional invariant manifold in the system’s phase space. This manifold
passes through the stable equilibrium point of interest and, at that point, is tan-
gent to a two-dimensional eigenspace of the system linearized about that equi-
librium. On this manifold, the system dynamics are governed by an equation

of motion involving a pair of state variables; that is, it behaves like a single-

degr‘ee-of-freedom system.

Based on the above definition, the construction of the nonlinear-mode manifold is straight-

forward. In order to search for a particular individual NNM, it is assumed that the NNM

manifold is parameterized by a single modal position-velocity pair corresponding to the

mode of interest, referred to as the master mode. This is accomplished by using the fact

that for a NN M response all of the remaining modal positions and velocities are slaved

(constrained) to this master mode. For the kth nonlinear mode, we take uk 2 71k, and

vk = 71k as the master states. The remaining slave states are expressed as

772' = Xi('“ks'vk) = Xif’lkflik)

7h = Witt-Wk.) = 33(77ka'7ik)

for i = 1,2,3,...,Q, i 74 k.

Equations (2.3) and (2.4) constitute a set of constraint equations that are to be determined.

The constraint functions in equations (2.3) and (2.4) are obtained by an invariant manifold

procedure that generates equations that can be solved for the unknown constraint relations.

The process begins by taking a time derivative of the constraint equations, yielding

18

 

 

. 8X, , 0X, ,

- = ' . — i. 2.5
"I auk uh + aUk 1A ( )
.. (913 . BY .
r), — 0a]. uk + —ka vk (2.6)

for i:1,2,3,...,Q, 2'75 k.

The time dependence in these equations is eliminated by using the following relations:
ti), = vk, v'k = 7)), = —w,%, 11k+fk(nj,7'7j), and 77',- = ‘wi2 n,+f,-(r)j,7'7j). Then, the constraints
(equations (2.3) and (2.4)) are substituted in the resulting expression everywhere in place of
the Slave state variables, resulting in a set of partial differential equations for the functions
(X,(uk, vk), Y,(uk, vk)). This set of 2Q - 2, time-independent, partial differential equations

govern the geometry of the kth manifold, and are given by

8X,- 8X, , 2
Buk ,le + 81—ik (—wk uk

+fk(X',Yj, ltk,'Uk)) (2.7)
81" BY'

w? X.- + Miner... vi.) = 57212;. + 5—7 (we? ”is

+fk(Xj,YJ-,uk,vk)) (2.8)

for i,j:1,2,3,...,Q, i,j;£ k.

These equations are not solvable in closed form (except in very Special cases) and meth-

ods for obtaining approximate solutions for X .1- and Y, are described in detail in this chapter.

19

2.2.1.2 Multi-Mode Manifold Formulation

Single-mode motions are not general motions, since they take place on two dimensional
manifolds in the system state space. In structural dynamics, we are also interested in
more general motions in which more than a single mode participates. For linear systems,
multi—mode motions take place in the Space spanned by a set of two-dimensional planes
that are the linear eigenspaces of the modes of interest. Furthermore, using superposition,
multi-mode motions of linear systems can be obtained through a linear combination of
responses of the individual modes of interest. For nonlinear systems, multi-mode motions
cannot be obtained through the superposition of individual NNM responses, since the
coupling between the NNMS cannot be accounted for in this manner. However, the idea
of a nonlinear single-mode manifold can be generalized by defining a nonlinear multi-mode
manifold to be a multi- dimensional surface that is tangent to the multi-dimensional linear

eigenspace of the corresponding linear modes of interest.

Based on these ideas, in [74] the following definition of a nonlinear-multi mode was offered:

A nonlinear ll/I-mode invariant motion of a system is a response that takes
place on a 2M -dimensional invariant manifold in the system’s phase space;
the manifold passes through the stable equilibrium point of interest, and at that
point it is tangent to a 2M -dimensi0nal eigenspace of the system linearized about
that equilibrium (representing M linear modes). On this manifold, the system
dynamics are governed by ll”! pairs of state variables; that is, it behaves like an

M’ -degree-0f— freedom system.

The procedure for constructing the multi-mode manifold is Similar to the procedure given
in section 2.2.1.1, except that the manifolds are now parameterized by 2M variables. Note
that the Single-mode manifold is a special case of the multi-mode manifold with M = 1.
Specifically, we take uk 2 71k: and vk 2 77),, for h 6 SM. where SM is a set of indices

that describes the modes of interest (master modes). For example, if one is interested in

20

constructing a model that captures the interactions between the second, third and fifth
nonlinear modes, then SM = {2,3,5}. The sets {uk} and {vk}, k 6 SM, are denoted by

u M and vM respectively. The remaining slave states are expressed as

m = Xi(UM, vM) (2-9)

le'umwm (2.10)

H

7h

for i=1,2,3,...,Q, re SM.

The procedure described in section 2.2.1.1 is again applied here by taking a time derivative
of the constraint equations, eliminating the time dependence by using the default rela-
tions, the equations of motion of the master modes, and the equations of motion of the
slave modes, and substituting the constraints every where in place of the Slave state vari-
ables, yielding the following set of 2Q — 2M partial differential equations for the constraint

functions:

 

Yi = Z [351(ka + % (_w% “k
keSM
+fk(Xj»}/ja"-Ika ”kl” (2.11)
“W? X1 + fr(XjalG~‘llkvl"k~) I Z [3:ka + 37):: (“9f “k
keSM ,
+fk(Xj1Yj»“k~1/’l.~.))] (2.12)

for i,j=1,2,3,...,Q, i,j ¢ SA].

AS in section 2.2.1.1, closed-form solutions do not generally exist. The method for

21

obtaining approximate solutions for X,- and Y,- in terms of power series expansions is

reviewed in section 2.3.1.2.

2.2.2 Invariant Manifold Equations in Terms of Modal Ampli-

tude and Phase

Pesheck et al. ([33]) developed an alternative form of the single-mode invariant manifold
equations that are expressed in terms of modal amplitude and phase. The procedure was
similar to the formulation in modal position and velocity, and yields results that are, of
course, equivalent. Recently Jiang et al. ([42]) have extended this formulation to the

multi-mode case. These formulations are described in the following sections.

2.2.2.1 Singe-Mode Manifold Formulation

The coordinate transformations used here are typical of those used in the method of averag-
ing ([79]), and are similar to those used in the method of variation of parameters, since they
employ a time-varying amplitude and phase. They relate the master modal displacement
and velocity (nk, 77k) to the master modal amplitude and phase (ak, (bk) via the following

invertible transformation:

77k : ak cos((.bk) (2.13)

ifk = —ak wk 8111(Ok). (2.14)

Hence, the constraints for the slave modes, equations (2.3) and (2.4), can be expressed in

term of ak and 3,, as

22

771' = PAM-.951.) = XiUIt-(a-k,$k)fl7'i.-(ak,¢k)) (2-15)

7h = Qi(aka¢kl = K(72k(ak»¢k)ifik(ak,¢kll (2.16)

for i: 1,2,3,...,Q, 1'72 k.

Using equations (2.13) and (2.14), the equation of motion governing the master mode can

be expressed as two coupled 1st order ODES in ak and risk, as follows,

— .P', -, , . -
(ii.- 2 f1.( 1le :1: <31.) 81110151.), (2.17)
k

.- .P', ',a.,ct). coscb.
@k : wk_f1(JQ] i. A) (A), (2.18)

“k. wk

 

 

for j=1,2,3,...,Q, jaé k.

In order to determine the slave constraints, the procedure is identical to that for the (uk, vk)
formulation, adapted to these coordinates. Using steps similar to those in equations (2.5)

and (2.6), we obtain

 

 

, 3P,- , 0P,- ,-
4 ,- = — .. — . 2.19
7)! aak (1k + 00k Qbh ( )
_ 8Qi . 3Qz‘ '
r), — Oak "k + 00k¢k (2.20)

for i=1,2,3,...,Q, 1:75 k.

23

The time dependence in these equations is eliminated by using equations (2.17), (2.18), and
r); 2: wot-2 771+ fi(7lj, 17]). Then, the constraints (equations (2.15) and (2.16)) are substituted
in the resulting expression everywhere in place of the slave state variables, resulting in a set
of partial differential equations for the functions (Pi(ak, (pk), Q,(ak, pk». This set of 2Q -
2, time-independent, partial differential equations govern the geometry of the kth manifold,

and are given by

 

 

3Pi(-fk(Pinj~ak~C/bkl Sinf‘r’lkl)

 

 

 

 

Qi = 8
(1k wk
8P, fk(Pj9Qj‘a'k3¢k) COS<QDk>
+86% (wk - ak wk ) (221)
, (9 ‘ —fk(P*Qaa“¢) 5111(0)
_w’12Pi+fi(Pjanaakv@k) = a—Q—I( J J k A A )
ak wk
' .P'. ',,..‘..‘I.,
argue _ f1.( 3 Q3111. €151) (08051)) (2.22)
00h. Gk Wk

for i,j:1,2,3....,Q, i,j;£ k.

These equations are more complicated than the invariant-manifold governing equations
represented in modal position and velocity, and have potential singularities at zero modal
amplitude. However, as mentioned in [33], the constraint equations of the slave state
variables are periodic in ct], with period 2rr, and therefore a Fourier basis can be used for
shape functions in the (bk direction. The methods for obtaining approximate solutions for

P,- and Q,- are described in section 4.2.2.

2.2.2.2 Multi-Mode Manifold Formulation

The multi-mode manifold formulation in terms of modal amplitudes and phase angles can

24

be developed in the same manner as was done for the case of modal displacements and
velocities. To avoid redundancy, only the important steps are given here. The trans-
formations relating the master modal position and velocity (77k, in.) to the master modal
amplitude and phase (ak, (bk) are given in equations (2.13) and (2.14); these are employed
with k E S M, where SM has the same definition as before. The sets {ak} and {m} are

denoted by a M and d) M respectively. The remaining slaved states are expressed as

7h = PilaMs (PM) (223)

7h = Qi(aMa (PM) (224)

for i=1,2,3,...,Q, i¢ SA].

Equations (2.17) and (2.18) are still valid with j if S M. Next, we take a time derivative of
the constraint equations, eliminate the time dependence by using the first—order equations
of motion for the master and slave modes, and substitute the constraints every where in
place of the slave state variables. This yields the following 2Q — 2M equations for the

mnlti—mode invariant manifolds:

25

Qt
_ Z [911 (-fi.~(Pj»Qj.ak.ak) sums“)

 

 

 

 

kESAI (90k wk

+11 (Wk _ f1.( JQJ a}. <91.) C05(C’A))] (225)
30k ak Wk

—w,-2 P7; +fi(Pj,Qjaak,¢k)

: Z [991(-fk(Pinjiak~¢L-) Sin(¢kl)
k E S A! 80 k wk

+6121 (wk _ fi( JQJ 0]. Oi.) 009090)] (2.26)
dcfik ak wk.

for i,j=1,2,3,...,Q, i,j¢ SM.

As before, closed-form solutions do not generally exist. A method for obtaining accurate
approximate solutions for the P,- and Q,- is described and implemented on example systems

in [42].

2.3 Solution of the Invariant Manifold Equations

In general, solutions of the invariant-manifold equations cannot be determined in closed
form, since such a solution represents a family of solutions to the original equations of
motion. However, in some special cases involving symmetries, closed-form solutions, which
typically represent flat manifolds, can be determined ([14]). More generally, though, the
manifolds are not flat, and other approaches are required (such as those described in sections
2.3.1 and 2.3.2). In the early works by Shaw and Pierre ([70], [20], [21]), approximate

solutions for the NNM manifolds of smooth systems were obtained in the. form of asymptotic

26

series expansions. Shaw, Pierre, and co—workers ([72], [73], [41], [28], [29], and [71]) used
this approach to solve a variety of problems, as did others, notably [26], [27], [30], [18], [16],
and [19]. However, the asymptotic series approximation of the manifold is only locally valid;
therefore, if one tries to Simulate a NNM model at large amplitudes, the time response of
the NNM model will deviate from the time response of the original model. The primary

difference is in the frequency of oscillation, which results in a phase drift.

As mentioned in section 2.1, the work by Pesheck et al. ([33]) employs a Galerkin method
to solve the invariant manifold equations, which allows one to obtain accurate approximate
individual NNM manifolds up to large amplitude ranges. This approach is also not limited
to smooth systems, so that the nonlinear terms can be more general functions. However, the
computational cost of acquiring such solutions can be quite expensive. Therefore, efficient
means of computing coefficients in the Galerkin expansions were sought and, in particular,
the collocation method has been found to provide accurate solutions with much greater

computational efficiency.

The series expansion method is reviewed briefly in section 2.3.1, for both individual
and multi-mode NNM models. The Galerkin and collocation methods for individual
NNM models are reviewed in section 2.3.2. However, the Galerkin method used to
solve multi-mode NNM manifolds, developed by Jiang et al. ([42]) is not reviewed here.

Interested readers are referred to [42].

2.3.1 Solution by Asymptotic Series Expansion

A good summary of the series expansion method for individual and multi-mode NNM
manifolds was given in [74]. A brief version of the method is given here. To use the series
expansion method, the nonlinear terms in equation (2.2) are assumed to be second and

third degree polynomials in the modal displacements. This assumption can be generalized

27

such that velocity-dependent and mixed nonlinear terms are included, but the solutions
will be more complicated ([20]). Therefore, the form of the forces f,- in equation (2.2) is

given by

Q Q Q Q Q
f1 = —ZZa1pq ups—2:23am 11111., 11. (2.27)

p=1q=p p=lq=PT=q

2.3.1.1 Solutions for Individual NNM Manifolds

To construct the single—mode manifold, approximate local solutions of X,- and Y,- can be

found using a power series expansions in terms of uk and vk as

X, = afiuk + 03,171}; + aging, + allaukvk

+a§7ivg + aging + afiuzvk + ogiukvg + agivg + . .. (2.28)
Y1 = b11111: + birl’k + biaui + bi,1’uk'l’k

+b§ge§ + 115,112 + 11%,.11711, + 115,111,112, + 1153,11}: + . .. (2.29)

The a’s and b’s are the unknown coefficients to be determined, and there are 9 x (262 — 2)
of them. These coefficients can be determined by substituting equations (2.28) and (2.29)
into equations (2.7) and (2.8) and collecting like powers in uk and 11),. This yields a set
of linear equations for the unknown coefficients, which can be solved successively. The
first-order (linear) coefficients are solved for first, and they are all found to be zero. This
is because equation (2.2) is given in terms of the linear modal coordinates, and therefore
the linear parts are already uncoupled, and must remain so since the linear model satisfies
invariance for its modes. Next, the 3 x (262 — 2) second—order coefficients are obtained, and
they are required in the solution of the 4 x (2Q — 2) third-order coefficients (as is typical

in expansion solutions). General formulas for the as and b’s can be found in [74].

28

 

Once all of the expansion coefficients are obtained, the X’s and Y’s, which describe the
slaved modes, are known functions of the master states (11k, vk). For i = k, these known
functions are used to express fk in equation (2.2) in terms of only uk and 11),, rendering a

single-DOF oscillator as the reduced-order-model for the kth NNM.

The coeflicients of the polynomials in the expansions become Singular when there exists
low order resonances between modes. For the quadratic and cubic terms, 1:1, 2:1, and/or
321 resonances ([79]) between the linear natural frequencies of the master and slave modes
will give rise to these Singularities. In these cases there exists nonlinear coupling such
that energy is exchanged among the resonant modes, and therefore these modes cannot
be dynamically separated from one another, as is required for the construction of an
individual NNM model. We should note that in this case the response of an individual
N NM is not stable, and any perturbation will cause the response to leave the neighborhood
of the NNM manifold, as the energy flows from that mode to others (and in most cases,
back to that mode again at a later time). In these situations, the multi-mode formulation
described in section 2.2.1.2 is required, with the necessary inclusion of the resonant modes.

The series solution of these multi-mode manifolds is described in the next section.

2.3.1.2 Solution of Multi-NNM Manifolds

Here there are M pairs of master displacements and velocities. The assumed forms of the

slave constraints X ,- and Y,- in terms of 11k and U}, are given by

29

k k k.l k,l k,l
Xi(uM,vM) = 2 (anti), + a2,,ei1k)+ Z Z ((23,,l-Itkll11‘l' a4’i’kaUl + 05,11‘kvl)

kESAy ICES)” [65]”
+ Z Z Z (algj‘qukujuq + aéj‘qukujvq + agj’qukvlvq

ICES)” lESM (1681”
+ ugli’qvkvlvq) + . .. (2.30)

1(uM. vM) — Z ( 1,114 + 2,114.) + Z Z ( 3,111,11111 + 4,1141% + 5,2-vkv1)
kESM ICES!” [63M
k 1, k1, '.l.

+ Z: Z Z (b6:1‘ qukuluq + biz-qukulvq + bggi'qukvlvq

kESAJ 165)” (1651”
+ bgjg‘qvktqvq) + . .. (2.31)

The as and b’s are the unknown coefficients to be determined, of which there are (2M +
3M2 + 4M3) x (26.2 -— 2M ). These coefficients can be determined by substituting equations
(2.30) and (2.31) into equations (2.7) and (2.8) and collecting like powers in the master
displacements and velocities. This yields a set of (2M + 3M 2 + 4M3) x (2Q - 2M) linear
equations for the (2M +3M2 +4M3) x (262- 2M ) unknown coefficients, which can be solved
at successive orders. Since the multi—mode invariant—manifold equations are formulated in
terms of the linear modal coordinates, the linear terms in the above expansions are again

zero. The formulas for the remaining a’s and b’s can be found in [74].

The coefficients a’s and b’s in equations (2.28) and (2.29) of the individual NN M manifold
formulation become Singular when w,- : wk and/or to,- = 2112;, and/or w,- = 3wk. This is
a flag indicating that there exists 1:1, 2:1, and/or 3:1 internal resonance(s) among slave
modes 2' 75 k and master mode is: ([79]). In this case there exists energy exchange among the
resonant modes and the motions necessarily take place on higher-dimensional (dimension
greater than two) manifolds. Once all of the participating modes in the internal resonance
are included in the set S M, the singularities in the coefficients a’s and b’s will be removed (at
least up to polynomial degree 3) in uk and 11),. Then we obtain the multi-mode invariant

manifold that captures the internal resonance, and allows energy exchange among the

30

master modes.

Sin'iilarly, there are possibilities that the coefficient a’s and b’s in equations (2.30) and (2.31)
of the multi-mode manifold formulation become singular when 112, 2 wk and/or to, = 2wk
and/or w,- = 3a)];C and/or 112,- : [wk ital] and/or to,- = [2112), ital] and/or to,- 2 [wk in); :Ewml.
This is another flag indicating that there exists 1:1, 2:1, and/or 3:1 internal resonances
among a Slave mode i d S M and a master mode k, l, m E S M- The singularities in the
coefficients a’s and b’s can be removed at least up to polynomial degree 3 (third order) in

uk and 11],, by redefining the new set SM such that it includes those extra modes.

Once all of the expansion coefficients are obtained, the X ’s and Y’s, which describe the
slaved modes, are known functions of the master states (11 k1 vk), where k E S M- For i = k,
these known functions are used to express fk in equation (2.2) in terms of only the uk
and vk, rendering a M -DOF oscillator. which is the reduced—order—model that includes the

desired NNMS.

2.3.2 Solution by Galerkin and Collocation Methods

The Galerkin and Collocation methods are techniques that employ a. variational formulation
to solve for discrete approximate solutions of continuous problems. Both methods belong
to the general class of techniques known as weighted—residual methods. A good account of
the theoretical aspects of these methods, and examples of how to apply them to engineering
systems, can be found in [77]. Many useful guidelines and rules of thumb on how to use the

methods effectively can be found in [78]. A brief summary of the methods is given here.

Consider the operator equation of the form

31

A(u(a:)) = f(;1:) (2.32)

in an open domain Q with apprOpriate homogeneous boundary conditions, where :1: E Q is
the independent variable, A is an operator, u(r) is the dependent variable to be determined
such that it satisfies the operator equation and the boundary conditions, and f (:13) is a
known function that represents a driving term. In the weighted-residual method, u(:r) is

approximated by uN(a:) in the expanded form

N
1e) 2111(1) = Z ea) (2.33)

i=1

where {dz-(12)} are elements of a complete set of basis functions that are differentiable up to
the order of differential operator A and satisfy all homogeneous boundary conditions, and
the c, are constants to be determined. Substituting equation (2.33) in (2.32), we obtain

the residual, R N

RNl$1 C1) Al'urvtv» — f(~’r), (1334)

which is generally not zero, since uN(r) is an approximation of u(.r). Note that uN(a:) lies
in a finite-dimensional space since uN(;1:) is represented by a finite linear combination of
elements of the basis. However, u(a?) lies in infinite—dimensional space, since to represent

u(.r) all elements of the basis are generally required.

If {tb,(:r)} is a complete set of basis functions (possibly different from {gb,-(.r)}) in the space

32

that A(u(;r)) and f(:r) live in, then by projecting RN(a:,c,-) onto 111,,(1) using the usual

L2(Q) inner product, and equating the resulting terms to zero, we obtain

<RN($161);¢’A~.($)> = fflRniiricz'W’kffldx = 0 (235)

If the set {'1l',(:r)} is an orthonormal basis, 7‘}, = < R N(:r, oi), u’1k(:1:) > is the representation
of RN(:r, Ci) in that basis. We then have N algebraic equations with N unknowns, the {C1}.
Solving for the {C1} that satisfy equation (2.35) would make RN(a:,c,-) small over $1. In
the limit N —1 00, R N(r, ci) must go to zero. This method is called the weighted-residual

method.

If one chooses different basis functions for expansion and projection, 1141A. # pk, the approach
is known as the Petrov-Galerkin method. If one chooses the same basis functions wk 2 (pk,
it is known as the Bubnov-Galerkin method, or, more commonly in the West, the Galerkin

method.

If one chooses "l’k : 6(r —:rk), where 6 (:13) is the Dirac delta function and it], are N selected
points (called collocation points), this becomes the collocation method. The collocation
method will force the residual R N(a:, c2) to be zero at points 313;, in (I. This greatly speeds

up the calculations, due to the simplification of the integrals.

In the following sections, the Galerkin method will be applied to equations (2.21) and
(2.22) in a domain ak E [0,a0] and dk E [0,27r], and also to equations (2.7) and (2.8) in
a domain uk 6 [—Ub, U5] and vk E [—Vb, Vb]. The collocation method will be applied to

equations (2.21) and (2.22) in a local domain defined later, and also to equations (2.7) and

33

(2.8) in another local domain defined later.

2.3.2.1 Galerkin Method with Global-Basis Functions in Modal Amplitude

and Phase Angle

Pesheck et al. ([33]) applied the Galerkin method to solve equations (2.21) and (2.22) for
P2: and Q,- in the domain (1;, E [0, no] and 1,5,, 6 [0, 2r] using an expansion of basis functions.

A brief summary of the approach in [33] is reviewed here.

The unknown position and velocity constraints (P,, (2,) are expanded as a double series in

the master modal amplitude ark and phase (bk as

Na N92
, l.-
P1<111e1f> = [Zamnmeeea (2.36)

l=1m=1
Na N11 1
Q1(a~i.-1¢1~.) = ZEDi’mUz,1—n(ak1¢k) (2-37)

l=1rn=1

for i=1,2.3,...,Q,i71- k,

where sz‘m and Dg’m are coefficients to be determined, T1,",(abqbk) and Ul,m(ak~ m) are
known basis functions. The basis functions T1,",(ak. m.) and Ul,111(aka¢k) are products of
selected basis functions in the (1k and (bk directions, and they are defined over the domain
61k E [0, a0] and 1.9;, E [0, 27r]. Na and Ng are the number of shape functions used in ak and
Qk respectively. Since. (pk has period 2rr, the basis functions in the (1);, direction are chosen
to be the Fourier basis, {1, cos(ncf)), sin(no)}. The basis functions in the (1;, direction

are Chosen to be polynomial functions, L1(ak). Therefore the form of T17m(ak,q)k) and

Ul,m(ak1¢k) are given by

34

71111011119511) = LAG/1) 608((771-1)¢k) (2.38)

Ul,m(akv¢>k) = L1(ak)sin(m¢k). (2.39)

It can be shown that for a conservative non-gyroscopic system to posses synchronous mo—
tions only the cosine functions are needed for basis functions in 111,, for Tim» and only sine
functions are needed in 115k for Ul,m- The polynomial functions L,(ak) are chosen to be a
set of polynomials defined over the domain ak E [0. a0] with zero value and zero slope at
ak = 0, which will satisfy the conditions that the invariant manifold pass through the ori-
gin (ak, pk) = (0,0) (the stable-equilibrium point) and be tangent to the two-dimensional
eigenspace at that point . Also, for convenience, they are chosen to satisfy the orthogonality

condition

“0 (1.", 1 for 'l :j
/ —2- Li(ak) Lj(ak) dak = (2.40)
0 (10 O for i 75 j,

which can be achieved by starting with a set of polynomial functions
“k 2 0k 3 “k

{(—) ,(--) .(——
a0 a0 00

of the L1(ak) can be found in [33]. Equations (2.36) and (2.37) are substituted into the

)4, }, and then applying the Gram—Schmidt process ([80]). The form
invariant-manifold governing equations, equations (2.21) and (2.22), and each of these is

projected onto each basis function using the Lg-inner product over the entire domain.

This leads to

35

 

or, _
0 ___ / (1_2 )Up.q ‘ak EDI, n'liU 1,111+ +20%, 111 l. n( fk a}; SiIl(¢1))
a

 

 

k ‘f’k 031,111 l,m 8(1k(wk
0T .
+2: Cl 171 lm( l. 1.1.111. - M) dak (lgbk (2.41)
[771 wk
1 1 8U1 —f a sin (b )
0 = / <—2> T11 .. aizC’ ",—T1m 111 “2080111 U; (k)
“A ilk (’0 l, m l. m k k
0U . 1
+ 2: D1 m [m( 1. , wk — ————f]‘ (02%)) dak 111,11. (2.42)

l m. wk

for i: 1,2,3...-,Q, 1% k;

The integrations in equations (2.41) and (2.42) are computed numerically. Therefore, equa-
tions (2.41) and (2.42) result in a set of 2(Q — 1)N1,N11, nonlinear algebraic equations in the
C ’s and D’s. For simplicity in referring to the method subsequently, this method is named

“the global a —— ct approach”.

Note that for the special case of a conservative non-gyroscopic system with only cubic
nonlinearities in the modal positions, all harmonic terms in the expansion of basis functions
for each state are needed. Therefore, the number of nontrivial coefficients remains 2(Q —
1)NaN¢. This fact is observed by performing numerical experiments. In contrast, for the
global u — 11 approach, this special case results in a reduction in the number of coefficients,

as described in section 2.3.2.2.

Once all of the expansion coefficients are obtained, the PS and Q’s, which describe the
slaved modes, are known functions of the master states ((11,31). These known functions

are used to express f1. in equations (2.17) and (2.18) in terms of only a1. and (111., rendering

36

a single-DOF oscillator as the reduced-order-model for the km NN M.

2.3.2.2 Galerkin Method with Global-Basis Functions in Modal Position and

Velocity

In this section, we apply the Galerkin method to solve equations (2.7) and (2.8) for X,-
and Y,- in the domain uk 6 [—U1,,U1,] and v1, 6 [—V1,, V1,] using an expansion of polynomial
basis functions. For nonlinear structures, the form of nonlinear terms is often a polynomial
function, and solving the invariant manifold equations in modal position and velocity would
have advantages over those in modal amplitude and phase angle. This is so Since the terms
that are necessarily zero from symmetry , i.e., the trivial terms, are easily identified and can
be removed from the expansion of basis functions. Moreover, the integration of polynomial
functions can be computed exactly using Gaussian integration, while the integration of
sine and cosine functions in equations (2.41) and (2.42) cannot be computed explicitly.
Therefore, the integrands in equations (2.41) and (2.42) have to be evaluated at many

points, which in turn increases the computational effort of the procedure.

The unknown position and velocity constraints (X1, Y1) are expanded as a double series in

the master modal position 111,. and velocity vk as

Np,u Npgv l
X1('tik,vk) = Z Z Czi’lel/I1,m(uk,111.) (2.43)

[:1 m=1
Nan Nprv

1’.
3301111011) = Z ZDiijlIlflnhl'kivk) (244)

1:1 "1:1

for i = 1,2,3, ...,Q, i 729 k,

l,m l,-m . .
where C,- and DI. are to-be-determmed coeffic1ents, and the L1,",(1.1.k.vk) are known

37

basis functions. The basis functions L1,",(uk,vk) are chosen to be two—dimensional

polynomial-basis functions in uk and 11k such that L1g.,,,(0,0) = 0, (31m (0 0): 0,
11k

and #(0,0) = 0 , which insure that the invariant manifold passes through the ori-
Uk

gin (11.1., v11) = (0, 0) (the stable equilibrium point) and is tangent to the two-dimensional

eigenspace at that point. The construction of LM1,m(u1.,vk) is obtained by performing the

tensor product of two sets of polynomial functions given by

 

1_11 11.2111. 3 11.1. 1_1
L = 1 —— — ,... 2.45
(I’kb (L7; 2 “A: 3 ”k m—l
[I : Q 0.. 0.. O I

v . u .
The terms 1, (vi), and (F2) are eliminated from the set {Lh11’m(uk,vk)}, since they do

not satisfy the boundary conditions at the origin. The Gram-Schmidt process is applied

to the reduced set {LML,,,(111.,v1.)}, such that the following orthogonality conditions are

satisfied:
Vb Ub 1 1 1 for [771 2 pg
/ / (U)(V) L’ll1m(u1.,11)L'llpq(u1.,1k)duk (111: (2.47)
Vb Ub b b 0 for lrn. sé pq.

Samples of the {LA/1,",(uk, 111)} can be found in Appendix 2A. Equations (2.43) and (2.44)
are substituted into the invariant-manifold governing equations, equations (2.7) and (2.8),
and each of these is projected onto each basis function using the Lg—inner product over the

entire domain. This leads to:

38

1 1 lm
0 = f —, — LAI, — 0’ LM
11k;11k( lb) (V1,) pq Z i l,m

(l-l-m.)yé2,3
1.171 aLi’l/Mn l.m CLAIM" 2
't' 2 Ci ’Uk W 'i- 2 Ci W(—wk ltk + fk) dllk dUk
(l+m)¢2,3 (l+m)7é2.3
_ Yrq
: Fz‘ (2.48)
_ 1 1 , 2 1,171
0 _ (-U—) (V) LMW .1,- 2 C, LMW — f,-
aka/'1. b b (l+m)7é2,3
l.m 814])!th [.m aLjubm 2
‘i‘ Z _ Di l'k T + 2 Di w(—wk 11k + fk) d111, d‘l’k
(l+m)¢2,3 (l+-m);£2,3
s Ff“ (2.49)

for i = 1,2,3,...,Q, i 79 k;

p :1""’1N1)‘u;

7

q :1,...,1‘\lp.11;

and (1) + (1) 7f 2, 3 .

The integration in equations (2.48) and (2.49) is computed numerically. Therefore, equa-
tions (2.48) and (2.49) represent a set of 2(Q — 1)(1’Vp,u1’Vp,v — 3) nonlinear-algebraic equa-
tions in the C ’s and D’s. For simplicity in referring to the method subsequently, this

method is named “the global u — 11 approach”.
For a conservative non-gyroscopic system to posses a synchronous modal response, some of

the C ’s and D’s must be zero. The basis functions corresponding to such C ’s are the ones

that are generated from

39

“Exit we). (“433(3), (“—kW—h, (fire).

It,

31:3 36:23 “$233 $323 E5433 35533

I) . 11 ’U . U. . U . ll . ‘U . ll, . ”U . ”(1. U.
—‘— 5, <—" —* 5 i 2 —")5 —"—>3 —">5 —‘— 4 ——‘—>5 i— 5 —")5, ...}, (2.50)

Vb Ub Vb ’ Ub Vb ’ Ub Vb ’ Ub Vb ’ Ub Vb
. ‘ , . I . 'Uk 1 NA. 3 'l’k 5 ‘3‘ .
where the elements in the set are linear in (—) , (—) , (———) , The basis functions
Vb Vb Vb

corresponding to such D’s are the ones that are generated from

on,

{(302, (E93, (”—w, (“—k>5,.

Ub Ub Ub Ub
952 a a2 3.23:2 a3a2 £5432 32.532

1&4 ”_k 21:4 2521;“ 113114 E434 “4524 251
Vb ’(Ub Vb ’Ub Vb ’Ub) Vb ’(Ub) Vb ’Ub) vb) "'"} (' )

. . , v . v. v .
where the elements in the set are linear 1n (~15)0 —A 2 a ($7504
b

Vb ’Vb

the number of nonlinear algebraic equations to be solved for the remaining 0’5 and D’s is

(Q _ 1)(J’Vp‘u]\’rpyv "' 3).

, Therefore, in this case

For a conservative non-gyroscopic system with cubic nonlinearities, the additional C ”s and
D’s that correspond to l +m 2 an even number, would all be zero. Therefore, the number of
nonlinear-algebraic equations in the C ”s and D’s to be solved is about 15(6) - 1)(i’Vp,uNp,.,/. —

3).

Once all of the expansion coefficients are obtained, the X ‘s and Y’s, which describe the
slaved modes, are known functions of the master states (uk, vk). For i = k, these known
functions are used to express fk in equation (2.2) in terms of only uk and ck, rendering a

single—DOF oscillator as the reduced-order—model for the km N NM.

40

2.3.2.3 Collocation Method with Local-Basis Functions in Modal Amplitude

and Phase Angle (Annular Strip Representation)

In [33], Pesheck et al. mentioned that as the number of retained linear modes Q increases,
the computational time associated with solving equations (2.41) and (2.42) increases drasti-
cally. Therefore, they developed a new approach to solve the invariant manifold equations.
In this approach, the desired domain in the (IA. direction is divided into K small sections,
producing annular subdomains given by (35}: E [0, 27r] and ak E [Cl/(J, (Maj + Aak]. For each
subdomain, as before the shape functions in the qbk direction are chosen to be the Fourier
basis. However, the shape functions in the ak direction are chosen to be piecewise linear
segments, which are accurate, providing the subdomains are sufficiently small. Therefore

Pi(ak, (bk) and Q,(ak, (bk) can be expressed over the jth interval as:

. 1,- ,.
Paar-9 99k) = Z Ci‘mfl-mUl/c» at)
1,171
NO 0) (1k -
l,m ‘ _ “,j
= c, __
z [ . < >
m=l
, a . — (1.,-
+C.2""(1 — i—Ei) cos((m — n.3,.) (2.52)
1' Auk
. l,
Qliak’gok) : ZDimUlmlmke‘rl’k)
Lm
Né 1 m 0k — Gk], 2 m 0k "' akj
= 2 [DIN (——'——) + Dz? (1 — —) sin(mc;')) (2.53)
77121 Aak Auk

for i:1,2,3,...,Q,i7$k.

The Galerkin method is then applied over each subdomain. Therefore, equations (2.41)

1 . Y
and (2.42) without the scaling factor (7) are valid for each subdomam as well. Note that.

“'0

41

there are K sets of 0’5 and D’s, one for each Aak interval. These individual solutions
are assembled to construct the invariant manifold. Note that this approach was applied to

construct the accurate ROM of a nonlinear rotating beam in [34].

The computational effort associated with evaluating and solving equations (2.41) and (2.42)
can be quite high. To reduce it, Apiwattanalunggarn et al. ([35]) implemented the collo-
cation method to construct an accurate ROM for a nonlinear finite element model of the
rotating beam. For this implementation of the collocation method, the nonlinear-algebraic

equations (similar to equations (2.41) and (2.42)) are given by

- _, . l,m
() = / ()(a;C -— ”1.31)st — (Dhql) “'01; 2 Di Ul,m
(‘ki‘f’k

1,771

 

+2 Cl, m 0T! III (—fk a}; Si“(ok))

 

(In aak< wk
+ Z Cl md _l_0m( kwk — M) dak dqbk (2.54)
(In wk
- l. .
0 = / , 001k - ak,p~¢k - 65m?) w? 6% ZCi‘mTIm - 0k fi
ah ¢k l,m
[JuaUl m (“f/,- ak 5111(6)”
+2 0 a . . >
I In ”k wk
. 0U . co.‘ '.
+ 2 D9," 0;:1 (aka) wk— iii—M) . dak dcfik (2.55)

w .
[.In I"

for i=1,2,3,...,Q,i# 1:;

13:12.
()2 =1 "NC“

where (ak.p~¢k,q1) E [(Lk‘j,(lk‘j + Auk] x[0,27r] are collocation points associated with

42

Qi(ak,¢k) and (ak,p~¢k.q2) E lak,jaak,j + Auk] ><[0,27r] are collocation points associated
with [Dz-(alt, an). The collocation method greatly simplifies the integrals. Equations (2.54)
and (2.55) are a set of 4(Q —- 1)N¢ nonlinear-algebraic equations in the C ’s and D’s. Recall,
that one must solve K such sets of equations to obtain the manifold over the entire domain.
For simplicity in referring to the method subsequently, this method is named “the local

a — qb approach”.

For a conservative non-gyroscopic system, only half of the harmonic terms in the expan—
sion of the basis functions for each state are needed. Therefore the number of nontrivial
coefficients is 2(Q — 1)N¢,. This fact is known from the symmetry of such solutions, and is

also confirmed by numerical calculations.

Once all of the expansion coefficients are obtained, the P’s and Q’s, which describe the
slaved modes, are known functions of the master states (ak, (bk). These known functions
are used to express fk in equations (2.17) and (2.18) in terms of only ak and (bk, rendering

a single-DOF oscillator as the reduced-order—model for the kth NNM.

2.3.2.4 Collocation Method with Local-Basis Functions in Modal Position and

Velocity (Patch Representation)

The approach developed in section 2.3.2.2 has the same disadvantage as that described in
section 2.3.2.1 when the number of retained linear modes Q increases. Hence, in this section
we develop an approach similar to the approach given in section 2.3.2.3. However, we take
the slightly different approach from that of section 2.3.2.3, wherein we use rectangular
subdomains defined in the modal displacement and velocity. The solution is expanded
over each such subdomain using two-dimensional polynomial basis functions. Since these
subdomains are small, we can use low degree polynomial functions to describe the invariant
manifold over the subdomain, which significantly reduces the computational time of solving

for the invariant manifold.

43

Here we describe the development of the method for a local domain described by u): E
[—ub, ub] and v}: E [-'ub, vb]. Once this procedure has been developed, it can be applied to
each patch, and the final result is obtained by collecting the results for all patches such that
the entire domain is covered. The relations between the coordinates in the global domain

(uk, ck) and those in the local domain (’uz, vi) are given by

 

(I). = (1” +112, (2.56)

L’k : (11} + vi, (257)
2U

where (1“ : —Ub + ( N:)(eu — 1) + (1),, (2.58)
u

r 2"?) '—

and (iv = —l’b + (WW2), — 1) + vb, (2.09)

L!

where eu and 6,, are the patch indices in the uk and ck directions respectively, and N5 and
N5 are the number of patches used in the uk and ck directions, respectively, and (in and

~ - - , e .,e e
(11, represent the shift from the or1g1n of (uk, wk) to (uk, bk). Here cu runs from 1 to Nu

and ev runs from 1 to NS.
Substituting equations (2.56) and (2.57) into equations (2.7) and (2.8), the partial differen-

tial equations governing the geometry of the kth manifold in the local coordinates (iii, vi)

are given by:

44

—w,-2 Xi + fiinanadu + afidv + 12f.)

for

6X- 3 -
’(d. + vi.) + ’(—w£ (d. + at)

 

57; av;

+ fk(XJ-. Yjadu + ui..dv + vi.» (2.60)
ar- 01/-

ggéw. + vi.) + 51%(_°°’12c (.1, + at)

+ fk(XJ-, Yj,du + 11):, dv + 1%)) (2.61)

2'.I=1,2,3.....Q. zyjsék.

The solution of equations (2.60) and (2.61) on the local domain is obtained by expanding

(X i: Y,) in terms of basis functions as:

1Vpau Npav
x,(uf;,u,f) = Z ZCf'mLML7,,(ui,vg) (2.62)
[:1 m=1
NW NW
Yi(ui.,v£) = Z Z Dg’mLA/Il’mhtz,vi), (2.63)
[:1 m=1
where Lfl>ll,,.,,(u‘):;,'v;) = Tl_1(ui./ub)me_1(v,‘;/vb), (2.64)

where T1_1 (:L‘) and Tm_1 (.1) are standard Chebyshev polynomials defined over :17 E [—1, +1],

and the C ’s and D’s are the to-be-determined expansion coefficients. Equations (2.62) and

(2.63) are substituted into the local manifold-governing equations, equations (2.60) and

(2.61). Normally, each of the resulting equations is projected onto the basis functions, but

here we employ a collocation method, which is computationally more efficient, yet retains

very good accuracy. This is carried out by projection of the equations onto Dirac delta

functions in the local master coordinates over the local domain, as follows:

45

for

, 1.171
/ e 6(112 — 1124,. 1'; - 'vi‘q) — 2 D, LMI,rIz

'“Z’Uk
l,m e BLAIIJH
"i”:Cz- (d1) +Uk)—_8—(;Z—-
l,m ‘
Fri/J”!

l

l,m

BLAIM,

Lm 2
+ Z Ci w(—wk(du ‘i‘ HZ) + fk) duck (iv/i:

1,171

(2.65)

, _. . 2 [.771 ,
/ e 5(112 _ {1.24), v; — vixq) “’2' 2C1 Ljubm _ f2.

6 .,,.
11k,1k
8 0141111.,"

1m
+2 DY d»,+1..’.—,—.—
l (i 1‘) 011;;
lm

RX 1%]

Z

i=1,2,3,...,Q,i7€k;

q :19 ..., AGLU,

1,171

BLI’ULm

Lm . 2 , € . e ,e
’i‘ E: Di W(—wk(du + 11k) + fk) duk dl'k

,m

(2.66)

where (ui‘pwg‘q) E [—ub,ub] x[—vb,vb] are the collocation points, which are zeroes of
TNp‘u(uf;, / 11b) and er,L,(Ui)~/vb)~ respectively ([78], [81]). Equations (2.65) and (2.66) con-
stitute a set of 2(Q — 1)} nuNpm nonlinear equations in the C ’s and D’s. Note that there
are N5 x N5 sets of Cs and D’s. However, if the system is conservative, non-gyroscopic,
and the nonlinear terms are functions of solely the modal positions, then only :14 x N5 x N5
sets of C’s and D‘s need to be obtained, and the remaining coefficients can be generated
using symmetries in the solution. For simplicity in referring to the method subsequently,

this method is named “the local 21 -— 11 approach”.

Once all of the expansion coefficients are obtained, the X ’s and Y’s, which describe the

slaved modes, are known functions of the master states (1123,, iii). For 2' = k, these known

46

functions are used to express fk in equation (2.2) in terms of only 11% and vi, rendering a

single-DOF oscillator as the reduced-order-model for the km NN M.

2.3.3 Comparison of Computational Efforts

To compare the computational efforts of the approaches described above, there are three
main aspects that we must consider: the total number of unknown coefficients, the
number of evaluation points needed for the integrands in order to compute the integrals,
and the methods used to solve the nonlinear-algebraic equations. First, the reduction
in the number of unknown coefficients C’s and D’s for a specific class of systems of all
four approaches (the global a — 66, the global u — v, the local (1 — <25, and the local 11 — v
approaches) is described. Next, a. comparison of the number of unknown coefficients 0’8
and D’s and a comparison of the number of evaluation points needed for the integrands of
the global approaches (the global a — (b and the global u — v approaches) are described.
Then, a comparison of the number of unknown coefficients 0’3 and D’s of the local
approaches (the local (1 — (b and the local 11 — v approaches) are described. This section is
closed with a comparison of the methods used to solve the nonlinear-algebraic equations
in this study. Summaries for the total number of unknown coefficients and the number of
evaluation points for the global-domain and local-domain approaches are shown in Tables

2.1 and 2.2, respectively.

2.3.3.1 Coefficient Reduction for a Specific Class of Systems

For the most general class of systems, that is, those with general (i.e., non-Caughey)
damping, gyroscopic effects, etc., one must obtain the entire set of C and D coefficients.
However, in some commonly encountered cases the number of coefficients can be reduced
simply due to symmetries in the equations of motion and the resulting responses. These

symmetries may be in terms of temporal or spatial dependence. Specifically, consider the

47

forms of equations (2.41), (2.48), (2.54), and (2.65). If the nonlinear terms are functions
of the modal positions only, it can be seen that the D’s can be expressed as functions of
the C’s. By using this result in equations (2.42), (2.49), (2.55), and (2.66), the number
of equations and the number of unknowns in all four approaches can be reduced by half,
thereby reducing the computational time accordingly. We now turn to the details of the

four approaches.

2.3.3.2 Comparison of Coefficients and Evaluation Points for the Global-

Domain Approaches

Here a comparison of the number of unknown coefficients C ’s and D’s of the global a — (b
and the global u — v approaches is described first. This comparison is based on the systems
having nonlinear forces of polynomial types. The numbers of unknown coefficients 0’3
and D’s of the global a — (b and the global u — v approaches for this class of systems
have been described in sections 2.3.2.1 and 2.3.2.2. Second, a comparison on the reuse
of previously-computed integrals, which is based on how the slave states are represented
by the expansion of basis functions, is described. Then comparisons of the number of
evaluation points needed for the integrands in order to compute the integrals of the global

(L — 45 and the global u -— v approaches are described.

For the global-domain approaches, solving the invariant—manifold equations in terms of
modal position and velocity has three advantages over solving them in modal amplitude

and phase angle.

First, for systems having specific polynomial nonlinear terms, either quadratic or cubic,
the trivial basis functions of the global u — v approach (mentioned in section 2.3.2.2) can
be easily determined, and therefore the number of coefficients can be significantly reduced.
However for the global a. — (I) approach (mentioned in section 2.3.2.1), we can not take such

an advantage from this specific class of systems.

48

Second, the basis functions of the slave positions and slave velocities, expressed in terms
of the master position and velocity, are the same, therefore equations (2.48) and (2.49)
share the same integrals. However, this is not true for equations (2.41) and (2.42) for the
global a — 65 approach, since the basis functions of the slave positions and slave velocities

expressed in terms of the master amplitude and phase angle are different.

Third, as mentioned in [82], the number of evaluation points in the (bk direction of the
global a — (I) approach is taken to be at least 10N¢ to provide good results, and typically
M, 2 12. The number of evaluation points in the 1);, direction of the global u — 11 approach,
for which gaussian integration ([80]) is used, is (ngxv — 2). Assuming that the number of
terms used for the modal displacement is comparable to that used for the modal amplitude,
i.e. Npru % Na, therefore the number of evaluation points in the uk and the 0.), directions,
for which gaussian integration are used, are comparable, i.e., (ngm — 2) z (3N0 + 3).
Hence, one can make a comparison by considering the number of terms needed for the
modal velocity versus those needed for the modal phase. If NW, is assumed to be 12, which
is the minimum number required for N¢, it can be seen that the number of evaluation
points in the 63;, direction is much higher than in ii), direction, i.e., (10M), = 10 x 12) >>

(314,,“ — 2 = 3 x 12 —- 2).

2.3.3.3 Comparison of Coefficients for the Local-Domain Approaches

Here, a comparison on the reuse of already-computed integrals, which is based on how the
slave states are represented by the expansion of basis functions, is described first. Second,
a comparison of computational times, which is based on how many terms in the expansion
of basis functions are needed to sufficiently describe the slave states, is described. Note
that the number of terms in the expansion of basis functions is determined from experience
gained from numerical experiments. Note that a comparison of the number of evaluation
points needed for the integrands in order to compute the integrals for the local a.——<z> and the

local 'u — v approaches is not considered since both approaches use the collocation method.

49

First, the basis functions for the slave positions and slave velocities, expressed in terms of
master position and velocity, are the same, in the local 21 — I) approach, therefore equations
(2.65) and (2.66) share the same integrals. However, this is not true for equations (2.54)

and (2.55) for the local 11 - (f) approach.

Second, the numbers of basis functions in the a), and uk directions for the local a — 45
and the local 11 — v approaches, respectively, are comparable to each other since the
manifold is described in the “k and U), directions using local basis functions and typically
Na = 2 and typically NW) 2 3. Many harmonic terms, typically N¢ Z 12 (as previously
mentioned), are needed to describe the manifold globally in the 45k direction. In contrast,
only a few Chebyshev-polynomial basis terms, typically Np,” : 2, are needed to describe
the manifold locally in the 2);, direction. Therefore, the dimension of each subproblem of
the local 11 — v approach is much smaller than the local a — 65 approach, which implies
that the computational time for each subproblem of the former method is much less than
that for the latter method. Even though for each grid (piece) along the uk direction
many pieces along the 1);, direction are needed, the total computational time for each grid
(piece) along the u), direction is just the sum of the computational times for the pieces
along the 1);, direction. However the computational time for each grid (piece) along the
a), direction does not grow proportionally with the number of harmonic terms N¢, i.e.,
it grows nonlinearly with Ngb' Hence, there is a tendency that the local 21. — 12 approach
uses less computational time than the local 0. — (25 approach. Note that this is only an
educated estimate. In section 2.4.2.1, the computational times for the local a — q!) and the
local 11 — v approaches will be measured and reported. Note that for both local-domain
approaches applied to conservative non-gyroscopic systems, the conservative condition and

the associated symmetry in the manifold solution are fully employed.

5O

2.3.3.4 Comparison of Methods Used to Solve the Nonlinear-Algebraic Equa-

tions

In this study, three methods have been implemented to solve the system of nonlinear-
algebraic equations for the unknown coefficients. They are Powell’s Hybrid method ([83]),
the Newton-Raphson method ([80]), and the Secant method ([80]). All three methods have
been used in this study, and by previous researchers in this line of research. Note that this
is a cooperative research program between the Department of Mechanical Engineering at
Michigan State University and the Department of Mechanical Engineering at the University
of Michigan. Powell’s Hybrid method, as implemented via the NAG (Numerical Algorithms
Group) routines, is available at the University of Michigan, and was used in the studies
by Pesheck et al. ([33], [34], [71], and [82]), and in the study by Apiwattanalunggarn
et al. ([35]). However the NAG routines are not available at Michigan State University,
and therefore the Newton-Raphson method was implemented via the Numerical-Recipes
routines([84]), and the Secant method was developed in-house and used in other parts of

this study.

Powell’s Hybrid method is a method used to find a minimum solution of an objective
function, which in our case it is a summation of nonlinear-algebraic equations squared,
whose global minimum point is the best available solution of the system of nonlinear-
algebraic equations. The method requires no information on derivatives of the objective
function. However it maintains a set of independent directions, where the number of
directions is equal to the number of dimensions of the solution, and it performs successive
line searches along the set in a cyclical manner. The set is also updated at each iteration.
As mentioned in [84] and [83], for a problem with N unknowns, the quadratic-objective
function would take N iterations of the basic procedure, which results in N 2 + 0(N) exact
line searches, i.e., the minima is obtained along each line search. This estimate can be
used as a guide for a general function since locally it behaves like a quadratic function at

an arbitrary point. As mentioned in [82], it takes on average about 1.25N iterations to

51

reach a solution to equations (2.41) and (2.42) (with the D’s expressed as functions of the
C’s), for either the global domain approach or the local domain approach (for one single
strip) using this approach. Also, as mentioned in [82], the computational effort of the local
(1 — 45 approach using Powell’s Hybrid method is only £2926 of the global a - (I) approach
using Powell’s Hybrid method. This is because for large :mplitude motions many terms of
polynomial basis (Na) are needed to describe the manifold globally along the ak direction
for the global (L — (p approach. This causes the number of iterations to reach a solution
to equations (2.41) and (2.42) to increase as Na increases, and the number of evaluation
points needed for the integrands increases as well. However this is not the case for the
local a — (I) approach since Na is fixed at 2, the number of evaluation points needed for the
integrands is fixed, and the total computational effort for the whole domain is just the sum

of the computational efforts for each annular strip.

The Newton-Raphson method uses gradient information (the J acobian matrix) of the sys-
tem of nonlinear-algebraic equations, evaluated at a current step, to determine the direction
of the subsequent step, which advances the process towards the solution. The gradient in—
formation of the Newton-Raphson method is determined in closed form from the system
of nonlinear-algebraic equations. For a single nonlinear equation, the order of convergence

([80]) of the method is determined from lim M = ' w = c, where 7‘ is the

n—+oo [enlp 71—400 [Tn — TIP

th iteration, rn+1 is the approximate

exact solution, Tn is the approximate solution at the 71
solution at the (n + 1)th iteration, c is a constant called the asymptotic error (c 75 0 ), and
p is a constant called the order of convergence, where p 2 1. For a system of nonlinear
equations, similar definitions can be formed using norms instead of absolute values ([85]).
For the Newton-Raphson method, the order of convergence is 2 (quadratic convergence)
([80], [85]). There is no such estimate for the number of iterations needed in order to reach
the solution from an initial guess, since in practice the solution is not known beforehand, so

the error at each iteration is not known. Also, the order of convergence and the asymptotic

error constant hold only when the iterates are close to the solution.

The Secant method is similar to the Newton-Raphson method. However, the gradient

52

information is determined approximately using finite difference methods. The order of
convergence of the Secant method is 1.6 (between linear and quadratic convergence) ([80],
[85]). Just as for the Newton-Raphson method, there is no estimate for the number of

iterations needed in order to reach the solution from an initial guess.

Among these three methods, Powell’s Hybrid method has an advantage over the Newton-
Raphson and the Secant methods. Since both N ewton-Raphson and Secant methods require
the Jacobian matrix, and the cost of computing the J acobian grows rapidly as the dimension
of the problem increases. The Newton-Raphson method would have an advantage over the
Secant method since it generally requires fewer steps to reach the solution, due to the
order of convergence ([80]). However, when the system of nonlinear-algebraic equations is
complicated, the Secant method is efficient for computing the Jacobian matrix, since each
entry of the J acobian cannot be determined in closed form, or is complicated and therefore

it takes significant time to compute.

In the next section, direct comparisons of the accuracy of the four approaches, both
global-demain and local—domain, each using modal displacement/velocity and modal

amplitude / phase, are made by comparing time simulations from two example systems.

2.4 Examples

A two-DOF nonlinear spring-mass system and a finite element (FE) model of a rotating
beam are used as examples to demonstrate the accuracy of the four approaches used to
solve the invariant manifold equations. The first example is used as a “proof of concept,”
while the second demonstrates the utility of the approach for systems with several DOF.
For simplicity in identifying the various approaches, they are labeled as follows: global
(L — (6, global u — 1), local (1 — (f), and local 11 — v, as described in sections 2.3.2.1, 2.3.2.2,

2.3.2.3, and 2.3.2.4, respectively. For the two—DOF nonlinear spring-mass system, all four

53

methods are considered. For the rotating beam, only the local (I — a) and local 11 — v

approaches are considered.

2.4.1 A Two-DOF Nonlinear Spring-Mass System

The two-DOF nonlinear spring-mass system considered here is the same as the system
studied in [33]. The system has nonlinear springs, described by linear spring stiffnesses In
and 163 and nonlinear spring parameters kg and k4; it is schematically depicted in Figure
2.1. The system parameters are m1 = n12 2: 1, and k1 = 1,162 = 2,k3 = 5, and k4 = 1. A
comparison of a — gb and u — v global methods is considered first, followed by a comparison

of local methods in a -— (f) and u — v.

2.4.1.1 Comparison of Global Methods

The accuracy of the global methods is studied through a comparison of motions taking
place on the NNM manifold of the first mode of the system. This manifold can be depicted
as a pair of constraint surfaces which depend on either the first modal amplitude-phase
or the first modal position—velocity. The two constraint surfaces restrict the second mode
displacement and velocity, respectively, and describe motions that take place on the 2-

dimensional NNM manifold in the 4-dimensional state space.

The C coefficients describing the manifold solution in equation (2.36), with the D’s con-
densed out, were solved using Powell’s Hybrid method implemented in the C language on a
work-station computer (this was carried out by Mr. Dongying J iang, a graduate student at
the University of Michigan, and a collaborator on this project). A sample of such surfaces
is shown in Figure 2.2, which corresponds to the contribution from the second linear mode

displacement to the NNM manifold.

The boundary of all constraint surfaces in terms of modal amplitude is given by a0 : 2.6.

54

These surfaces are obtained by using Na = 5 polynomials and N¢ = 12 harmonics, which
are the same as used in [33]. Note that Na 2 5 and Nd) = 12 result in 60 coefficients
for each state when the conservative (symmetry) condition is used. Here the range of
validity of the surfaces is based on comparing simulations of the original model and the
N NM ROM. If the motion is initiated beyond a certain amplitude of the master mode, the
time responses of the original model begin to have high frequency components. We define
this amplitude to be the boundary of the valid domain. As mentioned in [33], the surfaces
are valid up to a = 1.5 with (10 = 2.22. Here with (10 = 2.6, the surfaces are valid up to
a = 1.7. Therefore, we suspect that an instability of the first NNM manifold may exist at
near this amplitude limit. This suspicion is based on the fact that instabilities of NNMS

were observed in 2-DOF systems with symmetry by Rosenberg et al. ( [6], [15], and [19]).

The C coefficients describing the manifold solution in equation (2.43), with the D’s con-
densed out, are solved using the secant method implemented with the symbolic processor

TA!

Mathematica on a personal computer. To yield a comparable number of C’s for this

case, the number of polynomials along uk and 1),, would need to be NW, 2 11 and Np,” = 11.

TM , and are therefore computa-

Here the manifold-solver codes are written in Mathematica
tionally slower. Thus, for acceptable computational time we used Npm = 8 and NW, = 8.
Note that Np,“ = 8 and Np,” = 8 result in 15 coefficients for each state when the con-
servative and odd-polynomial conditions are imposed. The C coefficients are solved with
three different sets of boundaries, given by: (Ub, Vb) = (15,15), (Ub, Vb) = (1.7,1.7 X an),
and (Ub, Vb) = (26,26 x wl), where w] = 0.69 rad/sec is the linear natural frequency of
the first mode. The contributions from the second linear mode position to the first NN M
manifold for the three different sets of boundaries are depicted in Figures 2.3, 2.5, and 2.7,
respectively. Using the transformations given in equations (2.13) and (2.14), the previous
contributions can be expressed as functions of modal amplitude and phase, as depicted in
Figures 2.4, 2.6, and 2.8. It can be seen from Figure 2.5 that the surface bends sharply
at around 11 = 1.3. Also, from Figure 2.8, which is solved for the larger amplitude, the

solution surface shape looks completely different from the lower amplitude surfaces. These

two observations are a sign of some type of instability, either dynamic or numerical, in the

55

first N N M manifold or its solution. Therefore, in order to compare the accuracy of the
global a. — d) and global u. — v approaches, the surface in Figure 2.4 is used to generate the

time responses.

Figure 2.9 depicts the time responses of the first nonlinear mode displacement 771 obtained
from the ROM of the global u —- 22 approach, the ROM of the global a — 62 approach, and
the original model. The dotted line is the time response of the ROM defined at the end
of section 2.3.2.2 with initial modal displacement 771(0) 2 1.3 and velocity 771(0) = 0.0.
The dashed line is the time response of the ROM defined at the end of section 2.3.2.1 with
initial modal amplitude (11(0) = 1.3 and phase angle 631(0) 2 0.0. The solid line is the
time response of the original model (2 DOFS) initiated quite precisely on the first-NNM
manifold (as determined by a shooting algorithm that finds periodic responses). As these
responses all appear to be quite close, an examination of the slaved modes is used to offer

a more refined look at the methods.

Figure 2.10 depicts the time responses of the second linear mode displacement obtained
from the ROM of the global u - v approach, the ROM of the global a — 4) method, and
the original model. The dotted line is obtained by applying the master—slave constraint
(depicted in Figure 2.3) to the numerical solution of the ROM of the global u — v approach
(the dotted line in Figure 2.9). The dashed line is obtained by applying the master-slave
constraint (depicted in Figure 2.2) to the numerical solution of the ROM of the global
(L — (b approach (the dashed line in Figure 2.9). The solid line is obtained from the 2-DOF

original model initiated on the NNM manifold.

Figure 2.11 depicts the time responses of the physical displacement 2:1 from Figure 2.1 for
the ROM of the global u — 22 approach, the ROM of the global a - qb approach, and the

original model. The line types definitions are the same as in the previous figure.

Horn Figures 2.9, 2.10, and 2.11, it can be seen that the time responses from the global

a — 63 approach are closer to the original model than the global u — v approach. However,

56

this is not surprising, since the number of basis functions used for the global (L — ab approach

was twice the number of basis functions used for the global u — v approach.

2.4.1.2 Comparison of Local Methods

Similarly, the accuracy of the two local methods is studied through a comparison of motions

taking place on the first NNM manifold.

For the local 0. — (.75 method the C coefficients describing the manifold solution in equation
(2.52), with the D’s condensed out, were solved using Powell’s Hybrid method implemented
with the C language on a computer work-station (again, by Mr. D. Jiang). A sample of
such a surface is shown in Figure 2.12, which depicts the contribution from the second
linear mode displacement to the first NNM manifold. This N N M manifold is obtained over
an amplitude range of a E [0, 2.6], which is generated using 52 piecewise linear manifold
segments in a of width Aa = 0.05, and N4, = 12 harmonics in ¢- Note that Na = 2 and
N9), = 12 result in 24 coefficients for each state within each strip when the conservative
symmetry condition is used. The total number of coefficients for the local a — 65 approach

is (Q — 1) x 2N¢ x Nsm‘p = 1248. Here the surfaces are valid up to a = 1.7.

For the local 11 - v method the C coefficients describing the manifold solution in equation
(2.62), with the D’s condensed out, are solved using the secant method implemented with
the C language on a personal computer. The contribution from the second linear mode
displacement to the first NNM manifold is depicted in Figure 2.13. The boundaries of all
surfaces along the modal position and velocity are Ub = 2.6 and Vb = 1.79, respectively.
All surfaces are obtained by using Chebyshev polynomials along 11; and v]: with Np," = 3
and NW) = 2, respectively, and the number of pieces along 11;, and vk to be N5 = 100 and
N5 = 100, respectively. The total number of coefficients for the local 11 — v approach is

I
(Q _ 1) X 2N1),uNp,v X ENSIVS = 30,000 .

57

Figure 2.14 depicts the time responses of the first nonlinear mode displacement m obtained
from the ROM of the local 11 — v approach, the ROM of the local a - (15 approach, and
the original model. The dotted line is the time response of the ROM defined at the end
of section 2.3.2.4 with initial modal displacement 711(0) 2 1.5 and velocity 1')1(0) = 0.0.
The dashed line is the time response of the ROM defined at the end of section 2.3.2.3 with
initial modal amplitude (11(0) = 1.5 and phase angle (61(0) = 0.0. The solid line is the
time response of the original model (2 DOFS) initiated quite precisely on the first-NNM
manifold (again, found using a shooting algorithm). Again, since these responses all appear

to be quite close, an examination of the slaved modes is in order.

Figure 2.15 depicts the time responses of the second linear mode displacement obtained
from the ROM of the local 21 — v approach, the ROM of the local a — 45 approach, and
the original model, using the previously defined line types. Figure 2.16 depicts the time
responses of the physical displacement 2:1 from Figure 2.1 for the ROM of the local 11 — v

approach, the ROM of the local (I — (15 approach, and the original model.

From Figures 2.14, 2.15, and 2.16, it can be seen that the time responses from the local
a — 45 approach are generally closer to the original model than those from the local 11 — v
approach. Hence, for this example, and considering the number of coefficients used for
each method, we conclude that the local a — qb approach provides more accurate manifold

solutions than the local 11 — v approach.

2.4.2 A Finite-Element-Based Model of a Rotating Beam

The finite element (FE) model of a rotating beam used here is the same as that described
in [35], and analyzed in detail in the next chapter. The system consists of a uniform,
unloaded Euler-Bernoulli beam attached to a hub that is rotating at a constant rate. The

beam is restricted to vibrate in a plane that is parallel to the axis of rotation and rotating

58

with the hub, so that only transverse vibrations in one direction and axial vibrations are
allowed. The corresponding transverse vibrations are commonly referred to as “flapping”
motions, while transverse vibrations out of this plane are known as “lead-lag” motions. In
this model, torsion, lead-lag, and other motions are neglected. The system is schematically
depicted in Figure 2.17. The rotating beam parameters are L = 9 m, m = 10 kg/m,
E1 = 3.99 x 105 N -m2, EA = 2.23 x 108 N, o = 30 rad/s, and h = 0.5 m.

The objective here is to compare the accuracy of the methods for generating NNM
manifolds (this model is discussed in more detail in the next chapter). The beam is
modeled using 182 elements, which are used to construct linear modes and associated
nonlinear coupling terms. This model is truncated to Q = 21 DOF, expressed in linear
modal coordinates, and this is used as a starting point for developing the N N M ROM. This
model has 9 transverse modes and 12 axial modes, selected according to the information
described in the next chapter. For this system only the local methods in a — d) and u —- v
are considered. As mentioned in section 2.3.3.4 the computational effort for the local
a — d) approach with Powell’s Hybrid method implemented is only %,6—§% of the global
a -— 65 approach with Powell’s Hybrid method implemented (as estimated by Pesheck
in his dissertation [82]). Based on this estimate, it is reasonable to assume that the
local approach is generally much faster than the global approach. Therefore the global

approaches are not considered in this example.

2.4.2.1 Comparison of Local Methods

The accuracy of the local methods is studied by direct comparison of motions taking place

on the computed first N NM manifold.

In collaboration with Mr. D. Jiang, the C coefficients describing the manifold solutions
in equation (2.52) (with the D’s condensed out) are obtained using Newton’s method,

implemented via the Numerical-Recipes routines ([84]) with the C language on a personal

59

computer. Note that for a single NN M there are 40 slave functions for this problem, that
is, 40 constraint surfaces. Samples of these surfaces obtained by the local a — 66 approach
are shown in Figures 2.18, 2.19, 2.20, and 2.21. These correspond to the contributions from
the second and fifth linear flapping mode displacements and the first and sixth linear axial
mode displacements to the first NNM manifold, respectively. The first NN M manifold is
obtained over an amplitude range of a 6 [0.6.0], which is generated using 120 piecewise
linear manifold segments in a with width AG. 2 0.05, and M), = 16 harmonics in 4). Note
that Na = 2 and Ne = 16 result in 16 coefficients for each state within one strip, when the
conservative symmetry condition is used. Thus, there are a total of 120 problems, each with
16 unknowns, that must be solved, and it took about 12 hours to solve for the manifold
solution using a 2.0 GHz personal computer. We will consider simulations of this model

after describing the solution obtained using the local 11 — v method.

The C coefficients describing the manifold solution in equation (2.62) (again, with the
D’s condensed out) were next solved using the secant method implemented with the C
language on a 2.0 GHz personal computer. To yield a comparable number of the C ’s over
the whole domain, the number of segments along 11;, and vk are taken to be approximately
N5 = 40 and N5 = 40 with NW, = 3 and Np, = 2 to cover 11 E [-6.0,6.0] and v E
[—6.0 x w1,6.0 x 621], where 6.21 = 34.03 rad/sec is the linear natural frequency of the
first flapping mode. It takes about 40 minutes for the manifold solution with N5 = 40 and
N5 = 40 to converge. However, the residuals of the manifold-governing equations are large.
To obtain the appropriate boundary Uh. we fix Ub to be 6.0 and then increase the number
of patches, for instance, N5 = 100 and N5 = 100. Then we simply observe the residuals
for pieces along the 1)), direction for a few of the first pieces along the uk direction. From
these observations, we can determine the range in the vk direction that gives acceptable
residuals. This allows one to compute the appropriate Vb and the appropriate Ub. We then
choose the same N5 and NS for these reduced values of Ub and Vb, i.e., we choose N5 = 100
and N5 = 100. Therefore, we take N5 = 100 and N5 = 100 to cover the smaller domain,
11 E [—4.0,4.0] and v E [-4.0 x 621,40 x 1.11]. This procedure yields small residuals over

the entire domain. In this manner it takes about 4 hours to obtain the manifold solution.

60

Samples of constraint surfaces obtained by the local 11 — v approach are shown in Figures
2.22, 2.23, 2.24, and 2.25, which correspond to the contributions from the second and fifth
linear flapping mode displacements and the first and sixth linear axial mode displacements

to the first NNM manifold, respectively.

Figure 2.26 depicts the time responses of the first nonlinear mode displacement 771 obtained
from the ROM of the local 11 — v approach, the ROM of the local (1 — cf) approach, and
the original model with the full 21 DOF. The dotted line is the time response of the ROM
of the local 11 —- v approach with initial modal displacement 771(0) = 2.375 and velocity
171(0) = 0.0. The dashed line is the time response of the ROM of the local (1 — 63 approach
with initial modal amplitude (11(0) = 2.375 and phase angle 961(0) 2 0.0. The solid line
is the time response of the original model (21 DOF) initiated on first NNM manifold as
obtained from a shooting algorithm. Figures 2.27, 2.28, 2.29, and 2.30 depict the time
responses of the second and fifth linear flapping mode displacements and the first and sixth
linear axial mode displacements obtained from the ROM of the local 11 — v approach, the
ROM of the local (1 — (25 approach, and the original model, respectively. Figures 2.31 and
2.32 depict the time responses of the beam-tip flapping deflection and the time responses

of the beam—tip axial deflection, respectively.

From the above figures, it is obvious that the time responses from the local (1 — 65 approach
are closer to the original model than the local u—v approach. We conclude that the local (1-
¢ approach yields more accurate manifold solutions than the local 11 —— v approach. However
the local 11 — v approach yields the manifold solutions with significantly less computational
time. Therefore, the local 11. — v approach can be used as a preliminary study or for a study

in which very high accuracy is not required.

The drawbacks of using 11 —— v coordinates might be eliminated by first applying the
invertible van der Pol transformation ([22]) to the master u -— v coordinates. This would
results in a new rectangular coordinate system that rotates with the linear response,

which would uncouple the governing equations of the master states up to linear order. In

61

other words, we would parameterize the invariant manifold using a dynamically motivated
rotating-coordinate system. Once the invariant-manifold governing equations are reformu-
lated, then the patch approach can be applied again with low-degree polynomials. Note

that this idea has yet to be tested.

2.5 Conclusions

In this chapter, we have proposed two new methods for solving for individual NN M in—
variant manifolds. For both of these methods the manifold is parameterized by the master
displacement and velocity and the manifold is obtained using Galerkin approaches. For
the first method, the manifold is solved using global polynomial basis functions over the
domain of interest. For the second method, the domain of interest is subdivided into small
pieces and the manifold is solved using the collocation method over these sub domains,
and then pieced together. These methods are compared, in terms of computational times
and accuracy, to similar methods that employ amplitude and phase master coordinates, as
developed by Pesheck et al. ([33]). In terms of computational time, the proposed methods
have advantages over the. methods of Pesheck et al.. In terms of accuracy, the methods
of Pesheck et al. are superior to the methods proposed herein. In general, a combination
of these approaches, which uses rotating rectangular coordinates, as described above, may

offer good efficiency and accuracy.

62

2.6 Tables

 

 

 

 

 

 

 

 

Approaches No. of Coefficients No. of Eval. Points
Consv. Syst. Consv.-Cubic-NL Syst. per integrand
Section 2.3.2.1 (Q — 1) (Q — 1) (3N0 + 3)
Global (1 — Q X2NaN¢ XZNQNQ, x10N¢
Section 2.3.2.2 (Q — 1) (Q - 1) (57%,, — 2)
Global 11 — 6 x (Np,,,Np,., — 3) x §(NP,.,NP,, - 3) x (336,, — 2)

 

 

Table 2.1. Comparison of the total number of unknown coefficients and the number of
evaluation points needed for the integrands in order to compute the integrals for the global

domain approaches.

 

Approaches

No. of Coefficients
for Consv. Syst.

No. of Eval. Points
per Integrand

 

 

Section 2.3.2.3 (Q —— 1) 1
Local (1 — ¢ x2N¢, x Ngmp
Section 2.3.2.4 (Q - 1) 1

Local 11 — v

 

 

 

 

x2NEuM x iNfiNg

 

Table 2.2. Comparison of the total number of unknown coefficients and the number of
evaluation points needed for the integrands in order to compute the integrals for the local
domain approaches.

63

2.7 Figures

 

 

 

 

X1 X2
—> —>
; k3’ k4
/,
,/ m, m,
2;

 

 

 

 

 

 

 

s\\ \\\\\ x
\ \

Figure 2.1. A two-DOF nonlinear spring-mass system.

‘5 c
”. '0

Q o 0 0. a o
\ o c

3 1’... 9:44'0,
’l’

 

 

Figure 2.2. The contribution of the second linear mode position 712 = P2(a1, Q1) to the first
nonlinear mode manifold obtained by the global a — cf) approach.

64

  

  

:

~
s~§~
-

.
s
¢

:V

J a.
a...
.-

      

      

-
-
-
Q
s

  

Figure 2.3. The contribution of the second linear mode position in = X2(n1,I'71) to the

1.5 and

first nonlinear mode manifold obtained by the global u — v approach with Ub

V6

1.5.

 

3.952.954? s

The contribution of the second linear mode position 772

X2(171(a1, $1), 7')1(a1, (91)) to the first nonlinear mode manifold obtained by the global u - v

approach with Ub = 1.5 and Vb = 1.5.

Figure 2.4.

65

       

       

~ .
~ 4 w .
¢~¢-~-
. t .3

       

  

ow .
.041.“ v.6 .
. c .1

~“QQJQQO‘.
. :3 cf \

h ~0§§\\

.4

 

   

        

        
 

   

y l
n. ..:...... . . 0
. . - . . ~
~ ~ ~ . . ~
..........:.s~
. 3

 
     

 

-~4

    

33ft:

Figure 2.5. The contribution of the second linear mode position 112 = X2(n1,1'71) to the

1.7 and

first nonlinear mode manifold obtained by the global u — v approach with U),

V, =1.17.

 

agiééeééax n 3

linear mode position 1n =

second

The contribution of the
X2(n1(a1,¢1),1)1(a1,¢1)) to the first nonlinear mode manifold obtained by the global u— v

approach with Ub = 1.7 and Vb

Figure 2.6.

1.17.

66

    

        

     

o ~ ~
~ ~
.-...~...
.1... I
1...:
.1 I -~
. Jo-~- -Q
42...
. 1
.3...
.~
0
O
Q

 

O
O r
- QOQa. A
99 .fi
.0

   

O

A

 

           

    

         
 

X2(171,7'71) t0 the

Figure 2.7. The contribution of the second linear mode position 712

2.6 and

first nonlinear mode manifold obtained by the global u — v approach with Ub

Vb

1.79.

 

O _

3.96.295?” a

mode position 02

linear

second

The contribution of the

Figure 2.8.

1.79.

2.6 and Vb =

X2(n1(a1, 1151), 1')1(a1, ¢1)) to the first nonlinear mode manifold obtained by the global u — v

approach with Ub

67

 

 

2 GloLbal u—v. Nth. 1 DOF
[ — — Global a—(p. NNM. 1 DOF
-~— Original Model, 2 DOF

 

 

 

 

 

 

 

O
N)-
A
"0,,
co
.1
O
.1
N

Figure 2.9. The time responses of the first nonlinear mode displacement for the ROM of
the global u — v approach with 771(0) = 1.3171(0) = 0.0, the ROM of the global a — 12
approach with 111(0) = 1.3, 651(0) 2 0.0, and the original model with initial conditions from
the shooting algorithm.

 

 

0.2 ~ ' Gldbal u—v, Nth, 1 DOF _
— - Global a—¢. NNM, 1 DOF
—~ Original Model. 2 DOF

 

 

0.15"

 

0.1 r

0.05[
[

:N0

-0.05 *

 

—O.1 r

—0.15* 1

 

 

 

Figure 2.10. The time responses of the second linear mode displacel'nent 772 on the first
nonlinear mode manifold for the ROM of the global u — v approach, the ROM of the global
a - (25 approach, and the original model.

68

 

 

Gldbal u-v, NfiM, 1 DOF
— - Global a—Q, NNM, 1 DOF

 

 

 

 

 

 

 

1 )- ———— Original Model, 2 DOF H
bx
0.5— \
x" 0i
—0.5 r
..1 i- "
O 2 4 6 8 10 12

Figure 2.11. The time responses of the displacement 51:1 from Figure 2.1 for the ROM of
the global u — v approach, the ROM of the global a — Q approach, and the original model.

 

 

Figure 2.12. The contribution of the second linear mode position 772 = P2(a1,Q1) to the
first nonlinear mode manifold obtained by the local a — Q approach.

69

                               

      

  

xx
.x
xexx

xxxxxx
xx xx xx
xx xxx xx .
xx x xx x n
xxx
xx

xxx\\\xxx\ x H
xx xxx xxx xx x
xx xx xx xx x
xx xx xx xxxxxx xx
xx xx xx x xxxxxxx

\ x x
x x x
x. 8.. xxx

xxxx
\ xx
x x
x xx xxxxxx
xxxxxxxxxx
x
xxxxxxxxxxxxxx
xx xx xxx x
xx xxxxxxx
xxxxx
xx

   

x
x
x
xxxxxx
xxxx
xxxxxx
x x x
x x x x
xx xxxx .
xxxxxxxx xx xx xx
xxxxxxx.
x x x x.x
xxxxxxxxxx
xx x
x x x x
x x x x
xxxxxx..x
xxxxx

x x
xxxx xxx
x xxxxxxxx
x x x x
xxxxxxx x

x
\ ‘
x x
x‘zx“¢‘
x‘ 8‘ ‘
K ‘\
x

       

x
x“

x

‘x

x‘.

lilim.
4202 .
. . 00
00 __

3.5;" 9

xx
x
x x
xxx xxx
xxx xxx
xxx xx
x xx xx
x x x x
x x x xx
xxx xxx xx
xxxxxx
x x
xx xx
xx x

\\

xx xx xx xx
xxxxxxxx xxxxxxxxxx
xxxxxx xxxxx x xx

xx xx .xx» xxx
xx x. x x x.

xx xx xx
xxxxxxx
xx xx xx xx
xx xx
xxxxxxxxx
xx x x

x
x
xx
xx

          

       

.x .x i

x x x

xxxxxxxxx
xxxxx

xxx xxx
xx

       

 

Figure 2.13. The contribution of the second linear mode position n2 = X2(n1,1'71) to the

first nonlinear mode manifold obtained by the local 11 — v approach.

 

 

 

Local u—v. NNM, 1 DOF
Local a—Q. NNM, 1 DOF

— Original Model. 2 DOF

l

 

 

 

 

2..

1.5“

12

10

t

Figure 2.14. The time responses of the first nonlinear mode displacement for the ROM

the ROM of the local 11 — Q

1.5, Q1(0) = 0.0, and the original model with initial conditions from

7

0.0

)

O

= 1.5,7)1(

0)

(

of the local 11 — v approach with 171

approach with a1(0)

the shooting algorithm.

70

 

 

0.3% I ~ Lgal u—v, NNrM, 1 DOF l.
— — Local a—Q, NNM, 1 DOF
——— Original Model. 2 DOF

 

 

 

 

 

 

[

 

Figure 2.15. The time responses of the second linear mode displacement W on the first
nonlinear mode manifold for the ROM of the local 11 — v approach, the ROM of the local
a — Q approach, and the original model.

 

 

Local u-v, NfiM, 1 DOF
_- ~ Local a-o, NNM, 1 DOF
1 _ —— Original Model. 2 DOF

 

 

 

 

 

 

 

Figure 2.16. The time responses of the displacement 3:1 from Figure 2.1 for the ROM of
the local 11 — v approach, the ROM of the local (1 — Q approach, and the original model.

71

We(xe: 1)

 

 

®e(xe,t)
Ue(xe.t) e=1. 2. 3...,, M
l l l l l e1 I | | l l l J
x9 xe+1 \E, A, I, L, m

 

 

Figure 2.17. Finite element representation of a rotating beam with Q = constant.

P2(a1,¢1) (2"d Flapping Mode)

 

Figure 2.18. The contribution of the second linear flapping mode ”()2 = P2(a1,Q1) to the
first nonlinear mode manifold obtained by the local a — Q approach.

72

   
  

) (5th Flapping Mode)
.3:
8

P5(a|1’¢1
.0
03

Figure 2.19. The contribution of the fifth linear flapping mode 712 = P5(a1,Q1) to the first
nonlinear mode manifold obtained by the local (1 — Q approach.

   
 

O
'_.

O

51 .
P1o(a1.¢1) (1 Ax1a| Mode)

Figure 2.20. The contribution of the first linear axial mode 1710 = P10(a1,Q1) to the first
nonlinear mode manifold obtained by the local a — Q approach.

73

$8.2 53. 52 333

m—

 

n.

la] mode 7715 = P15(a1,Q1) t0 the first

lnear ax

Figure 2.21. The contribution of the sixth l

nonlinear mode manifold obtained by the local (1 — Q approach.

. x xxx
. x
1 xxxxxxxxx
x xxx
xxx
xx x
..xxxx
Xxx

  

     

 

      

   

                           

   

 

     

  

             

  

    
  

      

       

.o xxx xxx x xx .
x .x xx .xx. xx x ..
xxxxxxexxxxxxxxxxxxxxxxxx. .xxx.
waxes......s.x........................ ii.
xxxxxxxxxxxxxxxxxxxx xxx. xx x xxx. xx x x xxx.
x x xx xxxx xx x x x x
x. xx xx. x xx ...x
.11.. 1.2....
xx x xx x x xx x x
xxxxxxxxxx x xx xxnxxxxxxxxx “xx
4......,......,.....N........a..n.....a. .
\ x 9 x x s x x x Q
x x x x x x . x x
x xxxx xxxx xx xxxx xx x xxxx x x
xxxxxxxx xxxxxxx xxxx xxxx x xxx xxxxx
xx ..xx x. xxxx xxcxxxxxxxxxxxxx
x xx x x x x xx x x
xxxxxxx x x
x x

         

                     

Figure 2.22. The contribution of the second linear flapping mode 712 = X2(n1,r']1) to the

first nonlinear mode manifold obtained by the local 11 — v approach.

74

X5(T}1,fil) t0 the first

x
xxxxxxxx
xxxx xx

x

x
\\\\
x xx
xxx xxxxx xx
x x x x x
x\xxxx\\\\ \\
.xxxxx xxxx xx x
x «xxx \i\\~\\ \ \\
xx xxxxx x xx xxx

x
x

\
xx
xxx\\

x
xx xx
xx

      

x
x
x
xxx xx \ K
x xxxxxxxx xx x

      

x xx \ \ \
. x x x x x
xx xx xx .xxxx xxxxx xxx xx xxxx \
. . x . x x x x x x x x
xx...xx xxxxxx
x . x x . x x x x x x x
xx xx xx x x x xxxxxxxxxxxxxxxx xx
...x... .x xxx xx .xxxxxxx
. . ..xxxxx xx xxxxxx
xxxxxxx x x

x xx x x \
x xx xxxx .

                     

   

    

  

       

x xx . xx. xxxxx xxxx xx
xxxxx x . x xxxx. x xxxx
xxxx xx.xxxx xx xx x x x
xx .xxxxx. xxxx xxxxxxxxxxxxxxxxxnxxxxx .
x.x xxxxx xxx xxxxx. xxxxxxxxxxxxxxxxxxx x x x
xxxx. xxx. xxxx xxxxxxx xxxx xxxx x x xx
x xxxxxx xxxxxx.
x x x x x . x x x
x xxxxxxxxxx xxxxxxxxxx
x x xxxxxxxxxxx

     

         
 

xxx xx xxx xxxxxx x

      

ion of the fifth linear flapping mode 712

nonlinear mode manifold obtained by the local u — 11 approach.

 

Figure 2.23. The contribut

ial mode 7710 = X10(n1,i)1) to the first

nonlinear mode manifold obtained by the local u -— 11 approach.

inear ax
75

ion of the first 1'

but

1

Figure 2.24. The contr

\\
\ \ \ \ \ \
\ x \ x x
x“ \x\\\\\\\\\x\ \\x\
xx\\xxxx\x xx
‘ ‘ x ‘x\‘ \‘ x‘ x“x~\
x x \\\‘x

\‘x‘x‘
x \ \\
\\\\\\

 

Figure 2.25. The contribution of the sixth linear axial mode m5 = X 150113771) to the first
nonlinear mode manifold obtained by the local u — 11 approach.

 

- —- Local 8—4). NNM. 1 DOF

Y
Local u—v. NNM. 1 DOF
3' — Orlginal Model, 21 DOF

 

.1
r

 

 

 

 

Mode 1 (1St Flapping Mode, Master Mode) Deflection
O

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
t

Figure 2.26. The time responses of the first nonlinear flapping mode displacement n1 for
the ROM of the local u — v approach with n1(()) 2 2.375,7‘]1(0) = 0.0, the ROM of the
local (1 — ¢ approach with (11(0) = 2.375, (151(0) = 0.0, and the original model with initial
conditions from the shooting algorithm.

76

 

 

 

 

 

   

 

 

 

~ Local u-v, NNM, 1 DOF
0.15% ~ Local a—o, NNM, 1 DOF -
: —— Original Model. 21 DOF
0
§ 0.1 l‘ \
"“5
o 0.05 R,
a) \
é o~ / .
a K \
.s /
Q-O.O5 ~
3 /
LI.
‘2 -O 1 r '
N
—0.1 5 ~ j
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 2.27. The time responses of the second linear flapping mode displacement in on the
first nonlinear mode manifold for the ROM of the local u —— v approach, the ROM of the
local a — Q5 approach, and the original model.

—3

 

 

 

 

 

6 x 10
Local u—v. NNM. 1 DOF
Local a-o. NNM. 1 DOF
4 _ Original Model. 21 DOF
2 h i

".1va..

 

5th Flapping Mode Deflection

 

 

 

 

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 2.28. The time responses of the fifth linear flapping mode displacement 775 on the
first nonlinear mode manifold for the ROM of the local u — v approach, the ROM of the
local a — qb approach, and the original model.

77

 

 

Local u-v. NiuM, 1 06F
0.01 i — - Local a-¢, NNM, 1 DOF F,
~-— Original Model, 21 DOF

O'- /\
/ fl
/
—o.02L ’
—o.03
—o.04~
4.057
—o.06~ \ ‘

—0.07 4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

t

 

 

 

T

I

1St Axial Mode Deflection

 

 

 

 

Figure 2.29. The time responses of the first linear axial mode displacement 7710 on the first
nonlinear mode manifold for the ROM of the local u — 11 approach, the ROM of the local
or — 96 approach, and the original model.

 

 

 

 

x 10"4
9 . Local u-v. NNM. 1 DOF
—- - Local a-c. NNM, 1 DOF
3. , .. Original Model. 21 DOF 7
.8 7
3 6
8 5
§
4
2
E 3
5 2
(D
1
0

 

 

 

 

l
..A

0.25 0.3 0.35

o
o
o
01
.0.
A
o
d
m
o
m

Figure 2.30. The time responses of the sixth linear axial mode displacement 7715 on the
first nonlinear mode manifold for the ROM of the local u — 11 approach, the ROM of the
local a — qb approach, and the original model.

78

 

 

0.8 fl 1 T I I I f
, Local u—v, NNM. 1 DOF
— — Local a-o, NNM, 1 DOF

 

E

 

 

 

 

 

 

0.6 r - — Original Model, 21 DOF
g 0.4 ~
C
8 0.2 ~
:3
‘55
o 0 r \
8'
L? -O.2 — -
.9 l /
—O.6 r ‘
’0'80 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 2.31. The time dependence of the beam-tip flap deflection, w(L, t), for the ROM of
the local u — v approach, the ROM of the local a -— o approach, and the original model.

 

 

Local u~—v. NNM. 1 06’:
a n Local a—o, NNM. 1 DOF
4- —-— Original Model. 21 DOF

 

 

 

O

—2

-4

-6—

—8

Tip-Axial Deflection (m)

—10

~12

 

 

 

-14

 

Figure 2.32. The time dependence of the beam-tip axial deflection, u(L, t), for the ROM
of the local u — v approach, the ROM of the local (1 — ab approach, and the original model.

79

2.8 Appendix 2A

The samples of {L1’lIL.,n(uk. 1.115)} for NW, 2 3 and NIH) = 3 in section 2.3.2.2 are as follows:

Lilli,1(“ka“k)
Lrl11.2(ur~~ vk)
“11.30%: 'l’k)
L1112.1(‘llk,llk)
L1\[2,2(“k~1’kl
LAI‘ZBULk, PA.)

LAI3.1(“kavk)

LM3.2(UA~»'U1)

LA'13,3(UA~, We)

0
0

1.11803(Lf‘)2
LI)

0
_ ilk U}:
1.3 — ——

“-1- ‘Uk 2
1.93649 — —
(U1, )< vb)

1.55005(%/‘i)2 — 65.7441(%)10 +172.187(-:’/i)8
b b (J
—163.124(%)6 + 67.1689(%~)4 —11.1948(;7k-)2
b b b
we 9 Uk 7 ’Uk 5
17.4667 — — 40.4492 —— 32.1214 —

— 9.99334(:’/—")3 + 2.88269(%5)2(:—5)
b b ’b

—1.47321(;‘j—"')2 + 62.4849(:)/—k)10 —163.651(%)8
12 b b

+155.03s(5’f—‘)6 — 63.839(”—{“‘)"1 + 8.91533(%)2 + 5.17352(

‘ b Vb Vb

80

1152352

2.9 Appendix 2B

Equations (2.48) and (2.49) in section 2.3.2.2 are solved using the Secant method. The
method implemented is described here by assuming that the nonlinear term fz- in equation
(2.2) is a function of the modal positions only. The orthogonality conditions from equation
(2.47) are used in equations (2.48) and (2.49), the simplified equations are given respectively

by

0 = Ff“
: _Dg’~q+ Z A'gq‘l“"‘(C)Cf“"' (2.67)
(l+m)#2,3
0 = Fix“
= ..ch’~q—f§’*q(0)+ Z Kgq’l’m(C)D:’m, (2.68)

(l+m);é2,3

, where Kgq‘l‘m, Kgq‘l’m, and fip‘q are given by

r1).q.l.m _ ,p.q.l.m
AC _ AD
1 1 81/11”).
= / (—) (7) LAIDJ] “k TE
“k’vk b b llk
6LAI,
+ (“-1.01% “k + fk) —()i—lr£:l (Ink dvk (2.69)
’k
, l l
'“kd’k L'b b

Note that if either the conservative conditions (equations (2.50) and (2.51)) or both the

81

conservative conditions (equations (2.50) and (2.51)) and the odd-polynomial conditions

are used, then Kgq‘l’m 76 Kgq'l’m.

From equation (2.67), it can be seen that we can express the D’s as functions of the C’s.
Those expressions are used in equation (2.68). Hence Fix’p’q will be functions of the C’s

only and they are given by

F.X"’“’(C) = 0 . (2.71)

If C’“ is a solution of equation (2.71) in vector form, then equation (2.71) in vector form
FX ’1’"? (C) can be approximated using the Taylor series expansion about 00)) , which is

close to C", up to the first-order terms by

81?"

11M!
0 = FX’P’WC) z FX’qu(C(°)) + T(C(°))(C — 0(0)) . (2.72)

anip’q 0
Defining the Jacobian matrix J E —8—C_(C( )), the approximate C* is given by
0* a: —J-1 vapv4(c(0)) + 0(0) . (2.73)

If J is determined in closed form, then the method is called the N ewton-Raphson method.
If J is determined approximately using the finite difference method and its components

are given by

82

X,l.m thsm IMHO) ‘ _ X,l,m p,q,(0)
J’~"W~Q=——05 (C(0))zFi (Ci “:2 F2- (Cj ), (2.74)

 

where AC is a chosen small number, then the method is called the Secant method. For each

component Ji’JI-nip’q, it is required to recompute Kgq‘l’m and K $71,137:. at each step due to

the perturbed C's. Therefore. the computation of each component can be very expensive.

The following algorithm is implemented instead:

(i) Initialize n. = 0, tol 2 small number, norm > to], and 0“».

(ii) \Nhile norm > tol
(a) Compute Kg” = {Ix’gq‘l‘m}(”) and Kg) = {A’lqu’l‘m}(") using equation
(2.69).
(b) Compute FX’p’q(C(n)) using equation (2.71).

(c) Compute J(") : J(")(C(n),K(Cn),Kgl),FX’p’q(C(n))) using equation
(2.74). Note that here K gt) and K g) from step (a) are used instead of recom-

puting them with the perturbed C ’s.
(d) Compute 6C(") = —(J("))_1Fx’p’q(C(n)).
(e) Compute = || 60“” ”2
(f) Compute C("+1) = C(") + 600‘).

(g) Compute n = n + 1_

Similarly, equations (2.65) and (2.66) in section 2.3.2.4 are solved using the Secant method.

Therefore, equations similar to equations (2.67) and (2.68) are given by

83

0 = FY‘M
2

.,Im D1771 (,I,(m
= 2K6? +ZI<2P

[.771 I m
0 = FXPPPQ
7
:. Z(_w 12)A,pr11”C.I771 1,777

1,771

(1 I 771 A4). .1,,qI m A.p,(1,I,-m A,]).(1.I.In

,rp

$111,771 _ [pq./.772
I‘CJ _ ADJ

: —LIl’[[’m (11:31” 17;”)

A,p.q.l,m

x1) (12. I. 771
C2 AD

8L1“
: (d1; + U z)_ I. 771

).
f: q = fll(u’.

_le)~(qC

BLIl-I
0 + a, éfll
”k bk

C)CIP’” (2.75)

2

)+ Z Kp’ .7127 "’( C)Df.P'"‘, (2.76)
1,772
, and ff“) are given by
(2.77)
63(2) +112.) + ml (2.78)
J (ui2u‘z’p,vi=v£’q)
(2.79)

Note that the algorithm used to solve equations (2.75) and (2.76) can be developed in the

same manner as the one described above.

84

CHAPTER 3

Finite-Element-Based Nonlinear
Modal Reduction of a Rotating Beam

with Large-Amplitude Motion

3.1 Introduction

The dynamic analyses of helicopter blades, turbopropeller blades, wind-turbine blades and
robotic arms has provided motivation for investigations of the vibration of rotating beams.
To predict the dynamic characteristics of rotating flexible structures, the kinematics must
be carefully modeled, which leads to nonlinear coupling effects between degrees of freedom
(DOF) in different directions. These coupling effects can cause slow modal convergence,
thereby often requiring large system models for accurate dynamic representation. Simula-
tion of such large—scale models is a time consuming process, which slows parametric studies

and design cycles.

Much work has been done using finite elements (FE) to model the nonlinear, large amplitude
vibrations of rotating beams, including [44], [45], [46], [47], [48], and [49]. These models are
typically quite complicated due to their geometry. degrees of freedom (flap, lead-lag, axial,
and torsion), and nonlinear coupling effects. Furthermore, because of the nature of the

finite element approach, many elements are required in order to obtain an accurate model.

85

A common approach is to use linearization of the finite element model about the nonlinear
static equilibrium position and solve the eigenvalue problem of the resulting linearized
model to obtain the linear natural frequencies of the system ([44] and [49]). Bauchau and
Hong ([45]) also utilized finite elements in time to obtain nonlinear responses and stability
results of the rotating beam undergoing large deflections. However, the computational time
associated with obtaining the equilibrium solution was expensive, because all the spatial
degrees of freedom at all time steps are coupled. Perturbation modes ([50] and [51]) were
applied to the finite element model of a helicopter rotor blade in order to obtain a reduced
order model ([46]). Bauchau and Bottasso ([52]) applied perturbation modes to the space-
time finite element model of a beam subjected to a sinusoidal load in order to obtain a
reduced order model. Crespo da Silva ([53]) utilized a truncated set of eigenfunctions or
eigenvectors obtained from the linearized system of partial differential equations (PDEs)
or the linearized finite element model about the nonlinear static equilibrium position in
order to obtain a reduced order model of a beam in planar motion. Crespo da Silva
([54]) also extended his work to handle multi-beam structures in planar motion. In most
nonlinear structures, there is no simple expansion of basis vectors which decouples the DOF
(i.e., modes) in the frequency range of interest from those outside that range. Therefore,
some (potentially important) nonlinear effects will be ignored in the truncation process.
Generally, many linear modes must be retained in the nonlinear model in order to minimize

these effects.

Over the past decade, systematic procedures have been developed to obtain reduced order
models (ROM) via nonlinear normal modes (NN M) that are based on invariant manifolds
in the state space of nonlinear systems ([20], [21], and [74]). These procedures initially
used asymptotic series to approximate the geometry of the invariant manifold and have
been used to study the nonlinear rotating Euler-Bernoulli Beam ([3]). More recent work
has employed a numerically-based Galerkin approach to obtain the geometry of the N NM
invariant manifolds out to large amplitudes ([33]). These procedures can be applied to
more general nonlinearities over wider amplitude ranges, and have been recently applied to

study the vibrations of a rotating Euler-Bernoulli beam ([34]). In that study, the system

86

of PDEs governing the axial and transverse motions of the nonlinear rotating beam are
derived by Hamilton’s principle and discretized by a Rayleigh-Ritz method. These PDEs

are similar to those in [86], which were derived by Newtonian methods.

The study presented here uses the same PDE model considered in [34]. However, in this
study we use Hamilton’s principle as a basis for deriving a nonlinear finite element model
of the rotating beam. The primary goal of this effort is to demonstrate that the invariant
manifold NN M approach can interface with nonlinear FE models, thereby Opening the door
to the application of the approach to systems with more complex geometries and additional

degrees of freedom (e.g., lead-lag and torsion).

This chapter is outlined as follows. The nonlinear FE model is first generated and
converted into a truncated (but still large-scale) modal form that is convenient for
the NNM analysis. The invariant manifold equations are formulated, and a numerical
collocation method is used to obtain the solution of the NN M invariant manifold for the
fundamental flapping mode of the beam. This invariant manifold is used to construct a
nonlinear single DOF ROM, which is subsequently used for a simulation study. The linear
natural frequencies of the transverse motion, obtained by solving the eigenvalue problem
of the linearized FE model, are verified against analytical solutions from [1] and [2]. A
study is also carried out to determine the size of an appropriate reference model for the
full nonlinear system, which is used for comparisons with the ROM. Detailed results for
the fundamental nonlinear flapping mode form the bulk of the numerical results, including
the amplitude-frequency relationship, the nature of the NNM invariant manifold, the
associated master-slave modal relations, and the actual beam dynamics. In all cases, the
results obtained from the ROM are checked against the full reference model, and against
the collocation-based ROM determined in [34]. Some conclusions are drawn at the end of

the chapter.

87

3.2 Rotating Blade Formulation

A dynamic model for the vibrations of a uniform, cantilevered, Euler-Bernoulli beam,
attached to a rigid rotating hub is considered. The system is schematically shown in
Figure 3.1. In the formulation, the following assumptions are made: rotary inertia is
neglected; the motion is restricted to axial and transverse vibrations in a rotating plane,
i.e., twist and lead-lag motions are not considered; and, the nonlinear axial strain due to
element extension is included *. A finite element approach is adopted, which is based on
the element. kinetic energy, T, and the element potential energy, U which are expressed as

follows,

1 “+1 -2 -2 2 2
T = —2- m(u. + w )+ 7720 (h + :L‘ + u.) (11: (3.1)
Jig
1 ‘1'P+1 1
U = 2 / ‘ 131(wm)2 + 132107,, + §(w,3;)2)2d:r (3.2)
1'8

where u(;r, t) and w(;r, t) are the axial and transverse displacements, respectively, (-),x is a
derivative with respect to the spatial variable :13, an overdot represents a time derivative, h
is the hub radius, 9 is the constant angular velocity of the hub/ beam, m is the beam mass
per unit length, E, A, I are the usual beam material and geometric parameters, and are
and :66“ are the global coordinates at nodes 6 and e + 1, respectively, of a typical element.
By applying Hamilton’s principle to these two expressions, the weak formulation for the

equations of motion is obtained:

 

‘Note that other sources of nonlinearity, such as finite curvature effects, while potentially significant in
magnitude, are not considered here, since our primary interest is the axial/transverse coupling.

88

t2 .1“€+1 .. 2 ..
f / [—-mu + 7779 (h + :r +11)](5u + [—mw]6w +
II fife
[——EA(u,_.,; + %(Pu7,r)2)](6uvr + 16,751.31.) +

[—EI‘lU,1"r]6u/I9Ix (1.7:(11: : 0. (3.3)

A finite element model of the simplified rotorcraft blade is developed by introducing fol-

lowing expansions into the weak formulation

“gill t) R5 2 ¢1(I)ui(tl iue(:17,t) % Z lI’ihrlsift) (3-4)

.

721 721

where ue and we are u and in restricted to the spatial domain of element 6, (352(2) are
standard linear shape functions, 11,-(t) are nodal variables for axial displacements, 16,-(33) are
standard cubic shape functions, and 3,-(t) are nodal variables for transverse displacements
and rotation angles. By substituting these approximations into the weak formulation, one
obtains the linear element mass and stiffness matrices, the force vector due to rotational
effects, and the quadratic and cubic nonlinear force vectors, for both axial and transverse

deflections. These are given by the following expressions:

89

~71 PTe+1 . ,.
777,-]- : / mo,(a:)r3j(:r) dx, (3.5)
’l

 

 

 

 

C
, Ic+l ([0 (1G)
6:3 = < 7,— (,1 n Meme» dx (36)
Te '
u re+1 2 ‘
f, = mil (I7+:1')(,9.,P(.r) (1.7 (3 7)
1‘e
4 4
9:1 : 2211]],157377» (3.8)
[=1 71:1
”38+1 1 (1(5- du”) d773,,
Pl . I" = —EA—‘—£—'—'—— (in, .
“ 1e1e all” [.8 2 (11‘ (II (1.1: I (3 9)
. 1.6+]
and 777.21 : / 777.72’)k(:r)7.5',(.r) (1.7:, (3.10)
Te
:7) 6’ +1 (1121,71. (i211!) I
I78: = / E1 ' (1;, 3.11
A" 1'7: 611:2 (11:2 E ( )
4 2
g“: = Z Z ailmslum, (3.12)
[:1 77721
$e+l dwk (It?) d- 1.9777
VVllCI‘C (121," = A EAE—IZE— (11' 1, (3.13)
--e . .
4 4 4
and hi. 2 ZZZbilmslsnsr, (3.14)
[=1 n:1r=l
xe+l 1 (1717;, d717, (1112,, (IL/’7
where bZ'Inr : [L -2- fi-E—d—T— dT (11‘, (315)
e .. .. ..

where the indices take on values as follows {7' = 1,2}, {j = 1,2}, {k = 1,2,3,4}, {I =
1,2,3,4}, {777 = 1,2}, {77, = 1,2,3,4}, and {r = 1,2,3,4}. (-)“‘ denotes quantities in
the u direction. 777?]- is the ij component of the linear axial mass matrix, It]: is the ij

component of the linear axial stiffness matrix, f,” is the 7' component of the axial force

vector, y? is the 2' component of the quadratic nonlinear axial force vector, and ai‘m is the
In quadratic nonlinear coefficient of 913‘. ()3 denotes quantities in the .9 direction. mil is the
kl component of the linear transverse mass matrix, k2] is the kl component of the linear

transverse stiffness matrix, 9;: is the k component of the quadratic nonlinear transverse

force vector, (721m is the [777 quadratic nonlinear coefficient of 9):, I72, is the I: component of

90

the cubic nonlinear transverse force vector, and bfdm is the [777 cubic nonlinear coefficient

of hi.

M one-dimensional beam elements are used and assembled using standard procedures ([87]).
Since there are three degrees of freedom at each node, and accounting for the boundary
conditions, one obtains 3M nonlinear equations of motion in the nodal coordinates, which

can be expressed as follows:

MUU + KUU + GU(S) : Fa (3.16)

M3s+KSS+CS(U, S) +HS(S) = o (3.17)

where U is the global axial displacement vector and S contains the global transverse
displacement W and rotation angle vectors 6). Quantities in the U direction are denoted
(-)U, as follows: M U and K U are the linear axial global mass and stiffness matrices,
respectively, and GU( S ) is the quadratic nonlinear axial global force vector. F9 is the axial
global force vector due to rotational effects. Quantities in the 8' direction are denoted by
(0)5 , as follows: M S and K S are the linear transverse global mass and stiffness matrices,
respectively, and GS (U, S) and H S (S ) are the quadratic and cubic nonlinear transverse

global force vectors, respectively.

To include rotational effects in the transverse stiffness K S , thus accounting for centrifugal
stiffening, the axial displacement U is separated into static and dynamic components, as

follows:

91

where Us is the “static” axial blade deflection due to F9 and Ud is the dynamic axial
displacement relative to U3. U3 is obtained by solving for the axial deformation in the

case of no vibration in either the transverse or axial directions, i.e.,

KUUS = F9, (3.19)

where GU(0) = O has been used.

Figure 3.2 depicts the comparison between Us obtained by a 182-element model and an
analytical solution ([3]), indicating very close agreement. (In fact, good agreement can be

achieved with as few as five elements for this simple shape.)

Substituting equation (3.18) into equations (3.16) and (3.17) offers a convenient form of
the equations of motion. A linear transverse stiffness term in S, Kq(U3), which arises
from rotational effects, can be separated out from G5 (U, S). This represents the linear
transverse centrifugal stiffening due to rotation, which is combined with K S to provide the
total transverse stiffness. Also, the rotational force vector is cancelled by using equation

(3.19). The finite element model then takes the form:

MUU‘d + KUUd + GU(S) = 0 (3.20)

M53 + (KS + Kq)S + GS(Ud, S) + H5(S) = 0. (3.21)

These equations have the form desired for the application of modal analysis. In particular,
they have a zero solution, (Ud, S) = (0, 0). Equations (3.20) and (3.21) are then rear-
ranged such that the nodal variables at each node are grouped together, resulting in the

following equations of motion in (finite element) physical coordinates:

92

M)? + KX + G(X) + H(X) = 0 (3.22)

where

X = lUd,2W292l---|Ud.11+1I'l”.i1+19M+1lT- (323)

These equations of motion are next transformed into linear modal coordinates and trun-

cated using the following coordinate transformation:

X 2 (P77 (3.24)

where the 3M x Q matrix <1) is the collection of transverse and axial normal modes of interest
of the linearized rotating beam. Here Q = Nt + N0 is the number of kept modes, where
N)- is the number of kept transverse (flapping) modes and Na is the number of kept axial
modes. This truncation is used to remove unreliable linear modes from the full finite element
model. The model is still relatively large and is to be ultimately reduced to a single DOF.
The con'iponent form of the equations of motion in these linear modal coordinates is given

by

Q Q Q
7771+ 4'72. 7In + Z Z anjk 77)- 'le + Z Z Z (371)-17 '77)- Wk 777 = 0 (325)

j=l k2] j=l k=l [=1

for 77=1,2,3,...,Q

93

where am are the natural frequencies associated with the 77th mode, anjk are the coefficients
of the jk quadratic nonlinear terms associated with the 77th mode, and ,8”ij are the
coefficients of the jkl cubic nonlinear terms associated with the nth mode. The formulae
for anjk and finjkl can be found in Appendix 3. The determination of these nonlinear
coefficients has been computationally automated using the finite element formulation and

the coordinate transformations outlined above.

3.3 Modal Reduction

3.3.1 Galerkin-based Invariant Manifold Approach

Typically, one is interested in the dynamics of modes that exhibit large amplitudes and/or
lie in a certain frequency range, but not in the response of all system modes. To obtain
an accurate ROM of a system of coupled nonlinear ODES, one should not simply truncate
the linear modes that lie outside the frequency range of interest. The contribution of the
truncated modes can have important effects and should be accounted for in the ROM in
order to accurately represent the dynamics of the system. The nonlinear model reduction
approach based on N N M is a systematic procedure that accounts for the contributions of
all linear modes to the NNM of interest, without the direct dynamic simulation of these
linear modes. It does so by slaving the linear modes to the dynamics of the NNM of
interest in a particular manner. The ROM obtained by this procedure can be made to be
very accurate over a large amplitude range if one can accurately construct the invariant
manifold corresponding to the NNM of interest. This has been accomplished by Pesheck
et al. ([34]), by making use of a Galerkin-based numerical approach to solve the invariant
manifold equations. This approach was applied to the rotating beam problem, where a
Raleyigh-Ritz approach was used to create a discretized dynamic model of the blade ([34]).

Some results from that paper are used for comparisons in the present work.

94

A brief summary of the invariant manifold procedure is reviewed here. It is followed by
a description of a solution approach for the invariant manifold equations that is based on

collocation methods, which offers some advantages in terms of computational efficiency.

From equation (3.25), the mode of interest, i.e., the master mode, is taken to be the mth
mode. This linear mode is to be extended in order to generate the corresponding NNM.
The generalized master modal position and velocity, 77m and 77m, are expressed in terms of

a master mode amplitude, a, and phase, 45, using the coordinate transformation

77m : 0 005(6)) (3.26)

Tim : —(I, Wyn Sin(¢). (3.27)

Using these dynamic variables, the equation of motion for the master mode can be expressed

as two coupled lst order ODEs in a and (b, as follows,

— fm Sin (Q)

a = ————-——. (3.28)
Wm
<5 2 Wm _ fm COSfQ), (3.29)
0 Wm
Q Q Q Q
Where fm Z —( Z: amjk 773' III; + Z Z :78ij 7Ij "It 771 ) (3'30)
j:1k=1 j=1k=11=1

corresponds to the nonlinear terms in the 777th equation of motion, which contain coupling
effects from all linear modes. The remaining Q - 1 linear modal positions and velocities

are slaved to a and at through functions that are to be determined. These are expressed as

95

772' = Pifaa 45) (3.31)

77'.- = QM, <7) (3.32)

for 7:1,2,3,...,Q,i# m.

Equations (3.31) and (3.32) are substituted back into the components of the equations of
motion (equation (3.25)) associated with the slaved coordinates. The functions P,- and
Q,- are then obtained using a standard invariant manifold approach. The time derivatives
are removed from the equations of motion by the chain rule, resulting in the following
nonlinear, time-independent, partial differential equations which govern the geometry of

the manifold through the functions F, and Q,:

 

 

6P, —- m S. 8P, m
0.- = Eth-fw) + at m — L331?) (333)
Jaw.- = aagi(‘f";:“(f)>+gii(wn.—f—”;{f—:@) (3.34)

for i=1,2,3,...,Q, 274 m.

Equations (3.33) and (3.34) are PDEs in the PS and Q’s and are solved using an expansion
of basis functions. Via a Galerkin projection, each PDE, after the expansion is introduced,
is projected onto each basis function. These basis functions are products of selected shape
functions in the a. and (6 directions. Since 96 has period 277, the shape functions in the
(,0 direction are chosen to be a Fourier basis, {1, cos(77¢), sin(776>)}. The shape functions
in the a. direction are chosen to be piecewise linear segments. The desired domain in the

a direction is divided into K small sections, producing annular subdomains given by (n

96

E [0, 277] and a. E [a], aj + An]. Therefore P,(a, <75) and Q,-(a, qt) can be expressed over the

jth interval as:

P7(a.¢>) = :Cl’n7lm(a.¢)

1,77.

a—aj

7v.
‘ G9
‘ (l — (1' 23., . ,
: Z[Ci1'n(—El—J)+ Ci n(1— -—A—a—)] cos((n "1)QD) (3.35)
7721 '

 

Q7(a.e) = ZDfiP’iUmw)
1,77
Na
' a — a- n a — a- .
= Z]D:’"(_£a_3)+Di2’ (1— (13)] Sln(n(,f>) (3.36)
7721

for i=1,2,3,...,Q,-i7ém

where the C ’s and D’s are to-be-determined coefficients. (Note that only a particular subset
of harmonic functions in (7') are required for this conservative, gyroscopic system, since it
possesses synchronous modes [34].) Equations (3.35) and (3.36) are substituted into the
manifold-governing equations, equations (3.33) and (3.34), and each of these is projected
onto each basis function using a Galerkin projection over the domain of interest. This leads

to:

97

 

, 8T -— , ' ’
0 = /¢Up.q -aZDi-‘"Uz,n+ZCf’" gift fmasmmh
a.

 

 

 

lvn 1,71 Wm
. 6T.
1, Ln f7, cos(q§)
+ €01. n a¢ ((1. Wm, "' —”—’u}rm—) d0 dgb (337)
-3U7- —f a sin(¢)
0 z: T _ . 2 . CART. __ _ DI,n ,n m
-/a.¢> 17.71 617,012”: 2 13,, af2+l§g 2 8a( wm )
8U , , .
+ Z DéJl I I." (a w,” _ M) da d¢ (338)
In 696 Wm

for i=1,2,3,...,Q,77£ m;

Equations (3.37) and (3.38) are a set of 2(Q— 1) x 2 X My nonlinear equations in the C ’s and
D’s. Note that there are K sets of C’s and D’s, one for each Aa interval. These individual
solutions are assembled to construct the invariant manifold over the domain of interest.
Once all the expansion coefficients are obtained, the PS and Q’s , which are the slaved
linear modal positions and velocities, are known functions in terms of the master variables
(a, (1)). These functions dictate how the slaved linear modes must follow the master mode
such that the equations of motion are satisfied and the overall motion is invariant. These
known functions are used to express fm in equation (3.28) and equation (3.29) in terms
of only a and (f7, rendering the ROM single degree of freedom oscillator. The key to the
invariant manifold approach is that the solutions of this oscillator represent solutions of
the full equations of motion for a particular NNM, and they systematically account for
the dynamics of the slaved linear modes. The numerical solution of the invariant manifold

equations allows one to obtain NNMS that are accurate over a large amplitude range.

98

3.3.2 Collocation-based Invariant Manifold Approach

The computational cost associated with evaluating and solving equations (3.37) and (3.38)
can be quite expensive. To reduce these computational costs, the collocation method has
been adopted here ([77]). Instead of projecting each manifold-governing equation onto
each basis function, it is instead projected onto a set of Dirac delta functions in the master

coordinates. This amounts to an approximation of the integrals over the domain, and it

 

 

leads to:
_ . ,. Ln, 21,1187) n( “fm a Sin(¢)
99') [an
8T :
+ Z 0’; n 0’" (a w,” — N) da da (3.39)
Wm

1,71

12:: n X 1,871 Ln. fm a fink?)

1,11

 

)

+2040 ” —,—’—"( (a cum — f$94—$65 da dd) (3.40)

Wm
l. n

for i=1,2,3,...,Q, 3% m;

19:12,
611—1 ,Na:
(12 :11 'aNd)»

where (a.p,o,‘)ql) E [(Lj,(lj + An] x[0,27r] are collocation points associated with Qi(a,gb)
and ((lpszqoZ) E [an (1]- + An] x [0, 2n] are collocation points associated with 192-((1.46). This

projection, while an approximation, greatly simplifies the integrals.

99

3.4 Results

The rotating beam parameters used here are the same as in [34]: L = 9 m, m = 10 kg/m,
E1 = 3.99 x 105 N -m2, EA 2 2.23 x 108 N, Q = 30 rad/s, and h = 0.5 m, which are
feasible for a rotor blade with sub-sonic tip velocity. First, the convergence of the linear
natural frequencies is considered as the number of elements is varied, and the results are
verified with analytical solutions from [1] and [2]. Then the convergence of the nonlinear
amplitude-dependent fundamental flapping natural frequency is studied and verified against
the reference model and the collocation-based ROM of the model from [34]. Finally, the
time response of the ROM of the fundamental flapping N NM is studied and compared with

the reference model and the ROM based on results from [34].

3.4.1 Linear System Convergence

The finite—element model of the rotating beam is obtained by the procedures outlined in
section 2. There are 8 cases considered: 5, 10, 15, 20, 45, 90, 136 and 182 elements. The
linear natural frequencies of the rotating beam are calculated by solving the eigenvalue
problem of the linear part. of equation (3.22). The first four flapping linear natural fre-
quencies are shown in Figure 3.3 as a function of the number of elements. These natural
frequencies are compared with the natural frequencies from table 7 of [1] and from Table 3

of [2]. In [1], the following dimensionless parameters are defined:

‘1 ‘2
2 mL (2 El
’ z ‘ z __ t 3.41
7’ E1 T mL4 ( )

where a is the dimensionless hub radius, 7} is the dimensionless rotating speed, and T is
the dimensionless time. Since the parameters used here do not fall in the parameter range

used in [1] and [2], the frequencies must be extrapolated in 7} and interpolated in a. The

100

first four linear natural (flapping) frequencies in the T time scale are presented in Table
3.1. These frequencies are then converted to frequencies in the t time scale and compared

with the natural frequencies from the finite-element model in Table 3.2.

From Figure 3.3, it is clear that these four linear natural frequencies quickly converge as the
number of elements is increased. In addition, from Table 3.2, the maximum difference be-
tween the reference natural frequencies and the natural frequencies from the finite—element

model is 0.2%.

3.4.2 Nonlinear Model Development

The finite-element model of the rotating blade generated by M one-dimensional beam
elements has 3M degrees of freedom (with boundary conditions included). Due to the
fact that transverse displacement and element rotation degrees of freedom have the same
natural frequencies and mode shapes, the natural frequencies that lie in the frequency range
of modes 1.5M — 3M may be unreliable ([88]). To be conservative, we consider a frequency
range that includes only the first M natural modes (that is, the M with the lowest natural
frequencies), and we desire to keep Q g M retained modes for our reference model. For
this system it has been found that a reliable approach is to maintain an equal ratio of
transverse modes to axial modes when reducing from M to Q. If Nt, M and Na, M represent
the number of transverse and axial modes among the first M modes, respectively, then we
take Nt = Nt.M x (Q/M) and Na, 2 Na,M x (Q/M) I. These Q modes are used in equation
(3.24) to arrive at equation (3.25). The procedure outlined in section 3.3.2 is then applied

to the model in linear modal coordinates to obtain the single DOF ROM.

In particular, the first linear modal coordinate (first flapping mode) is chosen to be the
master mode of interest. The dynamics for this NNM, equations (3.28) and (3.29), are

numerically integrated with initial conditions for mode 1 (flapping) amplitude and phase,

 

lSince NM! + NM” 2 M, then N, + N,, = Q, as desired.

101

a(0) = 4.0 and (,b(0) = 0 I. For this amplitude, the NNM frequency is studied as the
number of linear modes and the number of elements are varied. For 90, 136, and 182
elements, the number of kept modes (Q) used to generate the frequency data are 15, 21,
27, 33, 39 and 45, respectively §. From the frequency data, it found that as the number of
elements is increased, the frequency is nearly converged for 182 elements, over a wide range
of values of Q. As the the number of kept modes, Q, increases, the frequency decreases.
For 182 elements and 45 modes, the frequency differs from the 182 element, 39 mode case
by only 0.0375 rad /s. However, it appears that more than 45 kept modes are needed to
demonstrate convergence in a convincing manner. At this point, we employ the 45-mode

model generated from 182 elements for use as the reference model.

The NNM manifold is obtained from this model over an amplitude range of a E [0,60],
which is generated using 120 piecewise linear manifold segments in a with width Aa =
0.05, and N¢ = 16 harmonics in (p'. This amplitude range, in physical terms, corresponds
to the beam tip moving with a peak—to—peak amplitude of about 2.4 meters (recall that the
beam is 9 meters in length). The single DOF ROM is obtained by restricting the dynamics
of the 45 DOF reference model to this invariant manifold. This amounts to restricting
the dynamics, which occur in a 90-dimensional state space, to a two-dimensional invariant
manifold that is described by the master-slave relations given by the Pis and Qi’s. Note

that there are 88 such relations in this case.

Figure 3.4 depicts the natural frequency of the first nonlinear flapping mode as a function
of the initial mode amplitude, a(0), with ¢(0) = 0. The + line is a result of the ROM (1
DOF) which is generated from the finite element reference model. The 0 line is a result
of the ROM (1 DOF) which is generated from the Rayleigh-Ritz model from [34], using
the same N3 and Na as used in the reference model. The solid line is a result of the full
reference model (45 DOF), which is obtained by searching for periodic solutions using a

shooting algorithm 1] The results from the three models are virtually identical over this

 

1T his amplitude corresponds to the beam having a peak-to—peak tip vibration amplitude of 1.59 meters.
§Due to computer memory limitations, the maximum number of kept modes we can achieve is 45.
1Note that all modal solutions based on the reference model require use of this shooting algorithm.

102

amplitude range.

3.4.3 System Response

We now have a single DOF ROM (equations (3.28) and (3.29)), which is generated from
182 elements and 45 kept modes. This ROM is simulated with initial modal amplitude
(1(0) 2 5.9 and phase 95(0) = 0, and the result is shown as the dashed line in Figure 3.5.
This Figure can be geometrically interpreted as the projection of the motion that occurs
on the NNM manifold onto the linear eigenspace of the first linear flapping mode. This
numerical solution is verified against the numerical solution of the ROM from [34] with the
same initial conditions, which is shown as a dotted line. These two solutions are nearly

identical.

Figure 3.6 depicts the corresponding time responses of the N NM displacement 771 obtained
from three different models. The dashed line is the time response of the ROM obtained
by applying equation (3.26) to the numerical solution shown in Figure 3.5. Obtained by
applying the same procedures, the dotted line is the time response of the ROM from [34],
while the solid line is the time response of the reference model (45 DOF) initiated on the

NNM manifold. These three responses are nearly identical.

Figure 3.7 depicts one of the attendant master-s]ave-constraint relations, specifically the
second linear flapping mode displacement as a function of the NNM amplitude (a) and
phase ((1)). Note that solutions of the ROM are used to obtain (curb), after which the
second linear flapping mode displacement is obtained by the corresponding trace on this
surface. Of course, there are 87 other such surfaces, 43 more slaved displacements and 44
slaved velocities, for the linear modes. Note also that the contribution of this (and all)

linear mode(s) start at zero at a = 0 and increase in amplitude as 0. increases.

Figure 3.8 depicts the time response for the second linear flapping mode displacement

103

obtained from the ROM, the ROM from [34], and the reference model. The dashed line is
obtained by applying the master-slave constraint (depicted in Figure 3.7) to the numerical
solution of the ROM (the dashed line in Figure 3.5). The dotted line is obtained by applying
the master-slave constraint from [34] (which is similar to Figure 3.7 but is not shown here)
to the numerical solution of the ROM from [34] (the dotted line in Figure 3.5). The solid
line is obtained from the 45 DOF reference model. Again, the results are virtually identical,
demonstrating that the invariant manifold accurately captures the effects of higher linear

modes.

Further sample results are shown in Figure 3.9, Figure 3.11, and Figure 3.13, which depict
the master-slave constraint relations for the tenth flapping mode, the first axial mode, and
the twelfth axial mode, respectively. Associated with these are Figure 3.10, Figure 3.12,
and Figure 3.14, which are similar in nature to Figure 3.8, and depict the slaved time
responses for the tenth flapping mode, the first axial mode, and the twelfth axial mode, re-
spectively. In all cases the N NM manifold accurately captures the contributions from these
linear modes. It is interesting to note that the first linear axial mode makes a significant
contribution to the fundamental flapping N N M, thus demonstrating the importance of the

nonlinear axial / transverse coupling.

The 88 master—slave—constraint relations will generate the NN M manifold for the first non-
linear flapping mode in the 90-dimensiona1 state space. From Figure 3.7 through Figure
3.14, we see that the responses of the slaved linear flapping modes have the same fundamen-
tal frequency, as that of the N N M. However, the axial modes move at twice the frequency

of the NNM. This arises from the physics of the beam motion, and is discussed below.

Figure 3.15 depicts time responses of the beam-tip transverse deflection obtained from the
present ROM, the ROM from [34], and the reference model. The dashed line is obtained
by applying equation (3.24) to the time responses of the linear flapping and axial modes of
the ROM, and plotting the result for the transverse deflection of the end node of the beam.

Obtained by applying the same procedures, the dotted line is the result from the ROM from

104

[34], and the solid line is the result from the reference model. Note that three responses
are very close and correspond to a tip-to-tip deflection of 2.34 meters. The discrepancy in
amplitude is, we believe, related to convergence difficulties with the Rayleigh-Ritz approach

near the beam tip [3].

Figure 3.16, Figure 3.17, and Figure 3.18 depict the spatial nature of the transverse, angle,
and axial motions of the beam in the NNM response for the same initial conditions as
the previous Figures. They represent the motion starting from t = 0 up to a quarter of
the period of the NNM. The time intervals between each deflection curve are equal. The
bottom curves of Figure 3.16 and Figure 3.18 and the top curve of Figure 3.17 are at the
initial time, corresponding to the peak NNM amplitude. The top curves of Figure 3.16 and
Figure 3.18 and the bottom curve of Figure 3.17 are one quarter-period later. The dashed
line in Figure 3.18 (close to the top curve in the Figure) is the static axial extension us due
to rotation of the beam. At the initial time, the beam experiences axial foreshortening due
to nonlinear effects through the transverse deflection, as can be seen by the negative axial
displacements in Figure 3.18. At the quarter period, the transverse deflection is at zero, and
the beam does not experience axial foreshortening, but deforms axially due to centrifugal
loads. Note that during the first quarter-period, the transverse and angle responses go
through a quarter-period of the periodic oscillation, but the axial response goes through
one half of its full oscillation. This is due to the symmetry of the axial response about the
zero transverse and angle deflections condition, that is, the axial foreshortening is the same
for —w and —6 as for +11) and +6. This is why the axial motion has twice the frequency of

the transverse and angle motions.

By combining the transverse deflection from Figure 3.16 and the axial deflection from
Figure 3.18, the deformed shape of the rotating beam on the first nonlinear mode manifold
can be generated. Due to the small scale of the axial motions, the beam shape will be very
close to the shapes shown for the transverse deflection (this is not true at all amplitudes,
but is a very good approximation here). It is important to note that the shape of the beam

changes in a periodic manner through the course of the NNM motion. This is due to the

105

contributions from the slaved linear modes, which are not fixed, but vary in time according
to the NNM invariant manifold. This is in distinct contrast to more typical reduction
methods in which the equations of motion are projected onto a single mode shape, resulting
in a response during which the shape of the beam remains the same throughout the course

of the motion.

3.4.4 Computational Considerations

An important issue related to this procedure is that of computational time. As a compar-
ison, we consider here the relative times used to generate the invariant manifold solution
and run simulations on it with those of generating the manifold using a shooting technique.
It takes about 1,600 seconds to generate one strip of the manifold, yielding a total of about
192,000 seconds to generate the manifold. This is not trivial, but with the manifold in
hand, a simulation for a set of initial conditions takes under 500 seconds for about two
periods of response. In contrast, if one generates the manifold directly by using a shoot-
ing technique to find the correct initial conditions, each strip takes nearly 8,400 seconds
to generate, a factor of five higher. And, note that this approach uses the master-slave
relations as starting points for the shooting technique, which are not generally available,

making this result quite conservative.

When parametric studies are required, it is possible that the ROM using invariant manifold
techniques can be extended to handle the situation. A proposed approach is to augment
the system by considering the parameters of interest as elements of the master mode co-
ordinates. This is the so—called “suspension trick” ([89]). By this approach, the single
nonlinear mode manifold of interest will also be a function of the parameters of interest.
Hence, the computational cost of solving the invariant manifold does not become a recur-
ring cost when the parameters of interest are changed, however the initial cost of generating
the “suspended” manifold will be higher. This should be feasible and efficient for a single

parameter.

106

3.5 Conclusions

In this chapter, the problem of obtaining nonlinear single—mode ROMS from a finite element
formulation of a rotating beam was addressed using N NMs. Accurate simulations of this
system typically require many DOF, due to the nonlinear coupling effects between axial and
transverse deflections. The problem of obtaining accurate ROM was solved using invariant
manifold-based NN Ms. The ROMS obtained govern the dynamics of the NNM of interest
(the master mode), and the approach allows one to accurately capture the dynamics of
the slaved linear modes. Hence, only a single nonlinear oscillator is required for accurate
simulation of the NNM out to large amplitudes. The numerical solution of the NNM
invariant manifold equations was facilitated by employing a collocation method. Excellent
agreement was found for the dynamics of the fundamental flapping mode of the beam, by
comparing results from the single DOF ROM obtained here with the single DOF ROM
based on a Rayleigh-Ritz discretization ([34]) and a full 45 DOF reference model. The
method can be similarly applied to any mode of interest, and can be generalized to include

dissipation effects, both linear and nonlinear.

The results presented demonstrate significant promise, since they combine procedures from
the versatile finite element method with the accurate reduction procedures via numerically
generated NN Ms. This will allow the NN M method to be applied to the large amplitude
vibrations of beams with more complex geometry and more nodal DOF, as required for
more realistic models of rotor blades and other structural members. Also, current work on
multi-mode manifolds shows promise for developing nonlinear ROM with two DOF in a

similar manner.

107

3.6 Tables

Table 3.1. Dimensionless linear natural frequencies. wl and tag were computed using results

 

 

 

 

 

 

 

 

a = 0.06
77 w1 w2 w3 w4
12.17 13.8248 38.9163 81.37 142.606

 

 

from [1]. (.123 and M4 were computed using results from [2].

 

 

 

 

 

 

 

 

 

 

 

 

 

h :2 0.5 in
Q w1 w2 w3 w4
(rad / s) Analy. FEM Analy. FEM Analy. FEM Analy. FEM
30 34.09 34.03 95.97 95.84 200.67 200.49 351.68 351.49

 

Table 3.2. Comparison of the linear natural frequencies obtained by power series and finite
element methods. The analytical results for an and UJ2 were computed using results from

[1]. The analytical results for w3 and (124 were computed using results from [2].

108

 

3.7 Figures

 

 

 

 

 

Q We(xe:t)
D I fees.»
Vii/Ci/r/ Ungt) 8:1, 2, 3,...,M
V///, J l l lLlell I I ll I]
1 U _ x. x...” \E,A,I,L,m
h 3:

 

 

Figure 3.1. Finite element representation of a rotating beam with Q = constant.

109

 

 

 

 

 

 

 

 

t I j j T I —l— AnaIytical Sfilution q
0012 + 182 Elements
0.01 r . ' .
0008* 4
:5.
3 0.003 -
U)
3
0.004- a
0.002[ ,
G L l i 1 1 l l 1
0 1 2 3 4 5 6 7 8 9

X (m)

Figure 3.2. Comparison of the static axial blade deflection, 'Us, obtained from an analytical
solution [3] and by the finite element method.

110

 

53

1st Linear (Flap) Natural Freq (rad/s)
8 .
2nd Linear (Flap) Natural Freq (rad/s)

 

co
.‘1
or
1

$3
8

96.5 ’

K
2

i 1 1
I r r

 

 

 

 

2%

50 100 150 200 0 50 100 150 200
Number of Elements Number of Elements

 

 

23
o
m
co
0:
'01

O

 

 

203

(rad/s)
c.)
or
x:

q

- 202.5 1 1

00
01
O)

202 f

3

201.5 1

201 [ +

0)
01
Q)

 

200.5 ' v if +

(a)
(II
N

fin-

l l l
I F I I

 

 

 

 

#A 4A

50 100 150 200 50 100 150 200
Number of Elements Number of Elements

 

§

 

4th Linear (Flap) Natural Freq. (rad/s)
8 a

3'“ Linear (Flap) Natural Freq

0

Figure 3.3. Convergence of the first four linear natural frequencies (flapping modes) of the
finite element model.

111

 

 

 

 

 

 

 

 

34.4 . , , . .
’0? + ROM from FE ‘/4>—4>
g 34.35» 0 ROM from Rayleigh-Ritz ,-a/ -
b —- Reference Model /
>‘ 34.3 » -
8 fl
8 ,/
,ar
g, 34.25 » // _
u. [,1
E 34.2 » ,/ -
2 ,ar
:5 /
E 34.1 5 l' ,1"! _,
8 ,/
g 34.1 r ,/ -
C ,/
0 /gr
.2 34.05% ,/ q
0’ _,_ -49
34 g ‘ . 1 .
O 1 2 3 4 5 6

Mode (1St Flapping Mode) Amplitude

Figure 3.4. The first nonlinear flapping natural frequency as a function of initial mode am-
plitude a(0) with ¢ = 0 obtaining using three methods: the FE—based ROM, the Rayleigh-
Ritz based ROM, and the reference model (182-element and 45-kept-modes).

 

 

 

— — ROM from FE
5.945 F ROM from Rayleigh—Ritz ‘
5.94» ,1 ‘\ / “\ ~
A / \\ / \
g 5.935- x \\ ,/ \\ _
Q) / ‘x / \
/ I
:3 5.93» ,’ ‘, ,’ \. 3
z: \ ‘
Q.
E 5.925r ,’ ‘, ,-’ ‘, -
< I \ I \
- . 2» ' ‘ ’ ‘ 4
a 5 9 1’ \ I’ \
g 5.915» ,’ l, I \\ ~
’I ‘ I, l
5.91 r- r ‘\ I ‘\ "
[I \ I' \
5.905— r \ I \ ~
I '\ / \
5 9 ' 1 1 “ 4' ‘ Q

 

 

 

Mode 1 Phase Angle ((1)

Figure 3.5. The projection of the motion occurring on the first nonlinear flapping mode
manifold onto the linear eigenspace of the first linear flapping mode, in amplitude and
phase coordinates.

112

 

- - ROM from FE

 

 

 

 

E
.g
8
g) 8— ROM from Rayleigh— —thz
'07
6‘» _
E
I- 4" '
2'3
3 2— -
2
05 or
8
2 _21
0"
.E
ca. _4r
3
LL -6‘
257-
: —8— ~
<1: . . . . n A
E O 0.05 0.1 0.15 0.2 0.25 0.3

Time

Figure 3.6. The time responses of the first nonlinear flapping mode displacement m for the
FE-based ROM, the Rayleigh-Ritz based ROM, and the reference model.

P2(a,¢) (2nd Flapping Mode)

 

Figure 3.7. The contribution of the second linear flapping mode 772 = P2(a,¢) to the
first nonlinear mode manifold, solved by the collocation method, as a function of the first
nonlinear mode amplitude and phase.

113

 

- - ROM from FE
ROM from Raylelgh-thz 4
——~ ~ Reference Model

.0
o:

 

 

 

.0
a

.0
m

I l
.o .o
A M

 

2"d Flapping Mode Deflection
O

 

l
.0
a:
I
J

 

 

 

0.05 0.1 0.15 0.2 0.25 0.3
Time

0

Figure 3.8. The time response of the second linear flapping mode deflection 172 =
P2(a(t), ¢(t)) on the first nonlinear mode manifold for the FE—based ROM, the Rayleigh-
Ritz based ROM, and the reference model.

 

P10(a,¢) (1 0th Flapping Mode)

 

Figure 3.9. The contribution of the tenth linear flapping mode 7710 = P10(a,¢) to the
first nonlinear mode manifold, solved by the collocation method, as a function of the first
nonlinear mode amplitude and phase.

114

 

 

— — ROM from FE
' ROM from Rayleigh—thz ‘
— Reference Model

 

 

 

 

 

,5 1.5 » -

13 -

a) _

8

2 ~~ _

U)

C —

‘a.

E -

LI.

5

o -l

.l
0.1 0.1 5 0.2 0.25 0.3
Time

Figure 3.10. The time response of the tenth linear flapping mode deflection rho =
P10(a(t), ¢(t)) on the first nonlinear mode manifold for the FE—based ROM, the Rayleigh-
Ritz based ROM, and the reference model.

P22(a,¢) (1“ Axial Mode)

 

Figure 3.11. The contribution of the first linear axial mode 7122 = P22(a,¢) to the first
nonlinear mode manifold, solved by the collocation method, as a function of the first
nonlinear mode amplitude and phase.

115

 

- - ROM from FE
- ROM from Rayleigh-Ritz
-— Reference Model

0.05

 

 

-0.05

—0.1 » /
—O.1 5 ~
—O.2 *
-O.25r / \
-O.3 r /

l \
-335.) . (Z

—o.4 “
o

I

3‘ Axial Mode Deflection

1

 

 

 

 

0.15 0.2 0.25 0.3

Figure 3.12. The time response of the first linear axial mode deflection 7722 = P22(a(t), ¢(t))
on the first nonlinear mode manifold for the F E—based ROM, the Rayleigh-Ritz based ROM
and the reference model.

1

x10
6\

 

P33(a,¢) (12th Axial Mode)

 

Figure 3.13. The contribution of the twelfth linear axial mode n33 = P33(a,¢) to the
first nonlinear mode manifold, solved by the collocation method, as a function of the first
nonlinear mode amplitude and phase.

116

 

I

[ - - ROM from FE -l
ROM from Rayleigh-Ritz
~—— Reference Model

 

 

 

 

 

 

 

 

8
7
5
z..- 6*
.33 \
E‘S’ 5*
5 4*
2 l
.5 3“ ,
E 2i- \
E
N
'r- 1,_ \
o» \v \«'
-1 l 1 J A A I
O 0.05 0.1 0.15 0.2 0.25 0.3

Figure 3.14. The time response of the twelfth linear axial mode deflection 7733 =
P33(a(t), ¢(t)) on the first nonlinear mode manifold for the FE—based ROM, the Rayleigh-
Ritz based ROM, and the reference model.

 

 

 

 

 

 

 

 

1.52 — - ROM from FE _
ROM from Rayleigh-Ritz
“\ —— Reference Model ,
[ \
1 \ a
\
E, 0.5[ .
c
_O
E o.
.9- -o.sl .
l— /
-1 >— / / -l
—1 .5 r 3
0 0.05 0.1 0.1 5 0.2 0.25 0.3

Time

Figure 3.15. The time dependence of the beam-tip transverse deflection, w(L, t), for the
FE-based ROM, the Rayleigh-Ritz based ROM, and the reference model.

117

 

 

 

 

l-
.—

1 I 1

0123456789
X(m)

r—
p—

—1.4 m

 

Figure 3.16. Transverse deflection ’lU(LlI, t) for a quarter-period of motion on the first non-
linear mode manifold. The motion starts at the maximum deflection (the bottom curve)
and moves as shown to the zero deflection at a quarter-period, after which it moves upward
in a symmetric manner about 11) = 0 and then repeats.

118

 

0.18
0.16“
0.14~
0.12’

0.11

9(X) (m)

0.08 -
0.06 -
0.041

0.02[ '

 

 

 

 

x (m)

Figure 3.17. Angle deflection 6(23, t) for a quarter-period of motion on the first nonlinear
mode manifold. The motion starts at the maximum deflection (the top curve) and moves
as shown to the zero deflection at a quarter-period, after which it moves downward in a
symmetric manner about 0 = 0 and then repeats.

119

 

 

0.02 l T I I I I I I
0.01

—0.01
—0.02
-0.03

UlX) (m)

-0.04
-0.05
-0.06
—0.07

 

 

 

l l l l

n—
l—

 

41080123456789

x (m)

Figure 3.18. Axial deflection u(:1:, t) for a quarter-period of motion on the first nonlinear
mode manifold. The dashed line denotes the static deflection, 113(23). The bottom curve
corresponds to the initial condition. The top curve from Figure 3.16 and the bottom curve
from Figure 3.17 correspond to the top u(:1:) curve here, all at one quarter-period. Note
that the axial motion occurs at twice the frequency of the transverse and angle motions,
due to symmetry. As time progresses beyond the quarter—period, the axial motion moves
downward again, and oscillates between the top and bottom curves shown.

120

3 .8 Appendix 3

The coefficients of the Jk quadratic nonlinear terms associated with the nth mode are as

follows:

Ill-F1
. LI , ‘1? , e
0an = Z (0(3e—2ln G'ejk + 99(3e—1)n Oejk + @315)” 063-1.) (342)
821
{n,J', It 2 1,2,3, Q}

(1W

eJk’ aejk and

\vhere 90,-]- is the Ij component of the modal matrix in equation (24), and 0U

,9 , .
061k are.

4 4
U U
aejk : :1 "2 Gel Injk (3°43)
1141112
W’ W
aeJk 21 "1:10 aelrnjk (3'44)
(41 m2
8 -
“BI/r :1 mzlae Qelrnjk (3.40)

N

m1,2,3, ...,M +1}; {J',k =1,2,3,...,Q}

PM
rs
ll

,9 . . .
elnjk’ 06:1ka and aelmjk (“9‘

121

,U
aelnjk

a

11'

eI

rnJ'k

(aglnIe ¢(38+I—5)j ¢(3e+n—5)k + (allInIE ¢(3€+I—2)j ¢(38+n—2)Ic

{1:12}; {n :12}; {e = 1,2,3,...,!v1 +1}; {13k =1»2131~-1Q}

u e—l . u e
(0211.) ¢(3e+l—5)j ¢(3e+n—4)l~ + (Gun) ¢(3e+l—2)j ¢(3e+n—1ll.~

{[21’2}; {71: 3’4}1 {8 :112131""A/I +1}; {jak : 1,2131°"1Q}

(a’é‘inle‘l ¢(3e+I—4)J' ¢(36+n—5)k + (aflrrle ¢(Be+I—1)J' ¢(3e+n—2)k (3-45)

{1: 3,4}; {71:12}; {6 =1,2,3,...,M +1}; {J',k =1,2,3,...,Q}

)e—l e .

U . ' '
(“2m C29(3e+l—4)j ¢(36+n—-4)k + (“(11111) ¢(3e+l—1)j ¢(3e+n—1)k

{1: 3,4}, {71: 3,4}, {8 : 1,2,3,-~1A1 +1}; {jk =112131'°'1Q}

(”firmly—1 ¢(3e+l—5)j ¢(3e—5)k + (aII~nz)e ¢(3e+l—2)j ¢(3e—2)k

{1:192}? {771:1}? {6 : 19213IH'1AI +1}’ {jfk :1’2’3’°"’Q}
. 3—1 ' ~

(”filmy ¢(:3e—+l—5)j @(3e—2lk + (ail-"116 ¢(3e+l—2)j ¢(3e+1)k

{121,2}; {m = 2}; {e =1,2,3,...,M +1}; {j,k = 1,2,3,...,Q}

(”153111118—1 ¢(3e+l—4)j ¢(3r—5)l~ + (aIImIe ¢(3e+l—1)j 9(3e—2)k

{I : 3,4}; {77121}; {e = 1,2,3,...“71/1 +1}; {j,k =1,2,3,...,Q}

I

s e—l s e
(”31:71) @(3e+I—4)J CTb(3(3—2)Ic + (allrn) ¢(3€+I‘~I)j Q75(3e+1)k

{I = 3,4}; {m = 2}; {e =1,2.3,...,.M +1}; {Ik :112I3I”'1Q}

122

Q ._ 3 0-1 3 e '
0(1ka — ((141171) ¢(3e+1—5)j ¢(3e—5)k+(“21m) ¢(3e+l—2)j ¢(36—2)k

{121,2}; {77121}; {e 21,2,3,...,]\II +1}; {j,k =1,2,3,...,Q}

._ —1 4 . . ,
= (“3sz $(3e+z—5)j ¢(3e—2)k + (aglmf ¢(3e+1—2)j ¢(3e+1)k

{1:1’2}; {771: 2}; {e =1,2,3,...,AI +1}; {],k =1,2,3,...,Q}

v'

c —1 .
= (031,”)6 ¢(3e+1—4)j @(3e—5)k+(031m)e ¢(3e+1—1)j¢(3e—2)k (3-48)

{I = 3,4}; {m =1}; {e =1,2,3,...,A'I +1}; {J'Jf- =1,2,3,---,Q}

‘3 ‘_1 . , .
= (“-317er ¢(3e+1—4)j ¢(36—2)k + (03177;)6 ¢(3e+z—1)j ¢(3e+1)k

{1: 3,4}; {m = 2}; {e =1,2,3,...,M+1}; {j,k =1,2,3,..-,Q}

vvhere (-)e—l denotes quantities associated with element 6 — 1, (~)" denotes quantities asso-

ciated with element 6?, and of“

”n and (121m are quantities defined by equations (9) and (13),

1‘ es pect ively.

The coefficients of the jkl cubic nonlinear terms associated with the nth mode are as

fEDIIOWS

Al+l
, , ,/ w .~ ,1 e
flnjkz = Z (0(3e—l)n flejkz + ¢’(3e)n fjejkl) (3-49)

621

{71,j,k,l = 1,2,3» --:Q}

n a o / ’1, /
w}1ere (bl-j is the ij component of the modal matrix In equation (24), and c.1523“ and .52“

flare :

123

4 4
33kt = Z 22:33,:li

vvhere [3le and [[33

e nrrjkl 3."er are:

124

(3.50)

(3.51)

,/ W
"361*nrjkl

__1 ,
[( 31*721')e ¢(3e+l*—5)j ¢(3€+n-—5)k ¢(36+r—5)l
+(b‘fp.,,,.)e ¢(3e+z*—2)j ¢(3e+n—2)k¢(3e+r—2)z}
{1* 21,2}; {n =1,2}; {r =1,2}; {e =1,2,3,.
.—1
[( gm")? ¢(3e+z*—5)j ¢(3e+n—5)k ¢(3e+r—4)1
+( ‘iz*n1~)e ¢(3e+1*—2)j ¢(3e+n—2)k¢(3e+r—1)z}
{1* 21,2};{71 = 1,2}; {1" = 3,4}; {6 =1,2,3,.
s .—1 -
}( 3m”)? ¢(3e+1*—5)j ¢(3e+n—4)k ¢(3e+r—5)1
+(bf1*n,.)e ¢(3e+z*—2)j ¢(3e+n—1)k¢(3e+r-2)t}

{1* 21,2}; {n = 3,4}; {7‘=1,2}; {e =1,2,3,.

[( 31“an ¢(3€+l*—5)j ¢(38+n—4)k ¢(3e+r—4)l
+(’)f[*,,r)e ¢(3e+l*—2)j ¢(3e+n—1)k¢(3e+r—1)1}
{1* 21,2}; {71 = 3,4}; {7‘ = 3,4}; {6 =1,2,3,.
, ,—l ,
[(bgfinr)? ¢(36?+l*-4)j ¢(3€+Tl*5)k gD(38-+-1"-—5)l
+(b}[*m‘)e ¢(3e+l*—1)j ¢(3e+n—2)k¢(3e+r—2)l}
{1* = 3,4}; {n = 1,2}; {7" 21,2}; {6 =1,2,3,.
. —1 1 .
[( §l*-nr)€ ¢(3e+1*—4)j $(3e+n—5)k ¢(3e+r—4)1
+1 'iz*m~)e $(3e+1*—1)j ¢’(3e+n—2)k¢(3e+r—1)l}
{1* = 3,4}; {7121,2}; {1‘ = 3,4}; {(3 =1,2,3,.
.' —1 . 1
[( Sunf 0(3e+1*—4)j <b(3e+n—4)k ¢(3e+r—5)1
+( .]S:I*nr)e $(3€+l*—1)j ¢(3c+n—l)k¢)(3e+r-2)l}

{1* = 3,4}; {71 = 3,4}; {r 21,2}; {6 21,2,3,.

,' ’—-l '
[(bglxmy ¢(3e+[*—4)j ¢(3e+n—4)k $(3€+1'—4)l

i

+(l)?[*,lv,-)e ¢(3€+l* —1)j @(3c+n—l)k¢(36+r—1)I}

{1* = 3,4}; {11: 3,4}; {7‘ = 3,4}; {8 =1,2,3,...,A’I +1}; {j,k,l=1,2,3,...

125

..,1W+1}; {j,k,l=1,2,3,...

.;M44x{¢h1=L2ew.

.WM44h{fihl=L2fivu

.,M44h{fih1=L2ew.

..,]W +1}; {j,k,l=1,2,3,...

"JJ+U;U$J=13fiW.

H,AI—+1};{j,k,l==1,2,3vu

as»

,Q}

,Q}

,Q}

,Q}

,Q}

,Q}

,Q}

,Q}

. G
”381*m‘jkl
—1
[(b4[*m-)e $(3e+l*— 5)j ¢(3e+n— 5)k ¢(38+T—5)l

+(b§l*,,,.)e ¢(3e+z*—2)j ¢(3e+n— 2) k¢( 3e+r— 2)1}

{1* 21,2};{71 21,2}; {7" I1,2}; {e =1,2,3,...,

[( 31*n.r)€_1 ¢(3e+1*—5))’ ¢(3e+n— 5)k ¢(3e+r-4)z

+(bgl*,,,~)e¢(3e+z*— —j2) ¢(3e+n— 2)k¢( 3e+r— 1)1}

{1* 21,2}; {71 21,2}; {7“ = 3,4}; {6 =1,2,3,...,

[(bjnm.)€—l ¢(3e+1*—5)j ¢(3e+n—4)k ¢(3e+r—5)1

+(b§1*m)e ¢(36’+l*—2)j ¢(3e+n—1)k¢(3e+r—2)l}

{1* 21,2}; {73: 3,4}; {1‘ 21,2}; {6 =1,2,3,...,

[( lizmfle—l ¢(3e+z*—5)j ¢(3e+n—4)k ¢(3e+r—4)z

+(b31*m-)e ¢(3c+l*— 2)j ¢(3e+n— 1)k¢(3e+r— 1)l}

{1* 21,2}; {71: 3,4}; {7‘ : 3,4}; {6 =1,2,3,...,

[(bj[*m-)e—1$(36+1*—j4) $(3e+n— 5);; ¢(3e+r—5)z

+( 3pm,.)e ¢(3e+1* 1)) ¢(3e+n 2)k¢(3e+r 2)z}

{l* = 3,4}; {n = 1,2}; {1‘ = 1,2}; {6 =1,2,3,...,

[( 21*m‘)€—1 ¢(3€+1*—1)j¢(3€+n—5)k ¢(l3e+-r—4)l

+( 21*72.r)e 90(36+l*—1)j¢(3e+n—2)k¢(3e+r—1)l}

{1* = 3,4}; {71 =1,2}; {r = 3,4}; {6 =1,2,3,...,

[( 31*,” )(J—l ¢(3€’+l*—4)j Cb(3e+n—4)k ¢(3€+1‘—-5)l

+(b2l*m‘e) Q‘)(3€’+l"‘-1)j(z)(3€+n l)k¢(3(+r— 2)l}

{1* = 3,4}; {71: 3,4}; {7“ 21,2}; {6 =1,2,3,...

[(I):l*nr)e—1¢(3€+l*—‘1)1¢(38+n—4)k ¢(3e!+r—4>l

+( 21‘2”)? ¢(3()+l*—l)j ¢(3e+n—1)k¢(36+7‘—1)1}

{1* = 3,4}; {n = 3,4}; {7~ = 3,4}; {6 = 1,2,3,

126

M +1}; {j,k,l=1,2,3,...

A] +1}; {j,k,l=1,2,3,...

M+1}; {3,3,1 = 1,2,3,...

M+1}; {j,k,l=1,2,3,...

M+ 1}; {3,132 = 1,2,3,...

M+1}; {j,k,l=1,2,3,...

,1” +1}; {j,k,l=1,2,3,...

,3! +1}; {Jk7,l=1,2,3,...

(3.53)

,Q}

,Q}

,Q}

,Q}

,Q}

,Q}

,Q}

,Q}

w11ere (-)€-1 denotes for quantities associated with element 6 — 1, (-)e denotes quantities

zlssociated with element 6, and bZl*nr are quantities defined by equation (15).

127

CHAPTER 4

Component Mode Synthesis Using

Nonlinear Normal Modes

4 - 1 Introduction

1\-I any complex structures are composed of several relatively simple substructures that
are assembled together. This occurs in trusses, bladed-disk assemblies in turbine rotors,
aerospace and ground vehicles, and many other applications. In such cases it is convenient
to develop a dynamic model for the overall structure by taking advantage of the dynamic
properties of the substructures. Component Mode Synthesis (CMS) was developed using
these ideas, in order to synthesize models that are described in terms of the component
Structures and to take advantage of model size reduction carried out at the substructure
leVel. There are two general types of CMS methods; they are known as the fixed-interface
and the free-interface approaches. The fixed-interface CMS technique, developed by Hurty
( [60] ,[61]) and simplified by Craig and Bampton ([62]), is widely used, since the reduction
procedure is straightforward and typically produces highly accurate models with relatively
few component modes ([64]). Free-interface CMS methods are more attractive than fixed-
interface CMS methods when the component modes are obtained from modal testing or
‘Vhen an experimental verification of the component modes is required ([65]). The free-
interface CMS technique developed by Craig and Chang ([66], [67] and [63]) is the most

accurate among the free-interface CMS techniques. It is a modified version of Rubin’s

128

Inethod [68] and MacNeal’s method [69]. It is superior to the CMS of Craig-Bampton in
terms of accuracy, but is more difficult to implement ([64]). An extensive review on CMS
can be found in [59]. CMS-type methods are well developed for linear structural models

alld they have been used extensively, especially in the aerospace industry, see ([55], [56],

[57] , and [58], for example).

T11e finite element (FE) method is used in CMS to model and characterize the behavior of
the individual substructures. Often, accurate models of the component structures require a
large number of FE degrees of freedom (DOF). For linear systems, model reduction can be
achieved using linear modal analysis, wherein one simply truncates the higher modes of vi-
bration. In fact, this is how one carries out model size reduction at the substructure level in
linear CMS. However, the dynamic analysis (e.g., the determination of natural frequencies,
rnode shapes, time responses) of such structures can require considerable computational

effort, and this is especially true for nonlinear structures ([5]).

The difficult issue of model size reduction for nonlinear structures continues to be a ma-
jor Challenge for computational vibration analysis. For relatively simple systems one can
use nonlinear normal modes (NNMS) for this task. For example, Shaw and Pierre have
developed systematic procedures to obtain reduced-order models (ROM) via NNMS using
invariant manifolds ([20], [21], and [74]). Asymptotic series were initially used to approxi-
mate the geometry of the invariant manifold, and this approach was applied to the study
Of a variety of systems, such as a nonlinear rotating Euler-Bernoulli Beam ([3]). Pesheck
e t al. developed a numerically-based Galerkin approach to calculate the geometry of the
NNlVI invariant manifolds out to large amplitudes of vibration ([33]). These procedures
Can be used for quite general nonlinearities over a wide range of amplitudes, and they have
been applied to many systems, including the rotating beam ([34]). Recently, these methods
h ave been shown to be applicable in conjunction with nonlinear FE models ([35]), which
Opens a new frontier for their application to more complex nonlinear structural systems.
Jiang et al. have recently extended the work of Pesheck et al. to calculate the geometry

3f Inulti-Inode invariant manifolds that are able to capture the dynamics for Situations

129

 

where more than one mode is active, for example, in the case of internal resonance ([42]).
However the computational cost involved in generating the multi-mode invariant manifold

is still quite high.

The present chapter describes recent research that is aimed at extending the fixed-interface
linear CMS method to nonlinear structures by making use of the fixed-interface NNMS
of the component structures. By synthesizing the reduced-order representations of the
substructures, one can obtain accurate low-order models of structures that are composed
of assemblies of nonlinear substructures, with significantly less computational cost than
by computing the ROM directly from the fully coupled system. It is found that such an
approach is valid, so long as the coupling between substructures is relatively weak. It
is also shown how this method relates to the usual linear CMS, and reduces to it under

linearization of the model.

The chapter is outlined as follows. We first review the development of N NMs, as needed for
the individual substructures. The associated invariant manifold equations, parameterized
by modal position and velocity, are formulated, and a numerical collocation method is
presented that allows one to obtain the solution of the NNM invariant manifold out to
moderate amplitudes. Then, the procedures for fixed-interface nonlinear component mode
synthesis are described. A five-DOF system and a forty-one—DOF system are used as

examples to demonstrate the method, and some conclusions are drawn to close the chapter.

4.2 Formulation of the N NM Invariant Manifolds

4.2.1 Invariant-Manifold Governing Equations

We begin with a general discrete representation of the vibrations of a nonlinear structural

system (in our case, a subsystem component), obtained either by a. finite element model

130

followed by linear modal expansion, or by a Rayleigh-Ritz approach using the linear mode
shapes. We assume that the system at linear order is undamped. (This assumption greatly
simplifies the problem, but can be relaxed, in principle.) In this case, the equations of
motion for a Q-DOF system are uncoupled at linear order and can be expressed in the

form:

Iii+An=f(n,7i) (H)

where I is the identity matrix, A is the diagonal matrix of squared linear natural frequen-
cies, f (17, 1'7) a vector of nonlinear forces, 7) the modal position vector, and 1'7 the modal

velocity vector. The component form of equation (4.1) is given by

772+w12 m = fina’lj) (4-2)

for i,j=1,2,3,...,Q

where w,- is the linear free vibration natural frequency of mode 2' and Q is the number of

retained linear modes.

In order to search for a particular individual NN M, it is assumed that the N NM manifold
is parameterized by a single modal position-velocity pair corresponding to the mode of
interest, referred to as the master mode. This is accomplished by using the fact that for a
N NM response all of the remaining modal positions and velocities are slaved (constrained)
to this master mode. For the kth nonlinear mode, we take uk 2 77k» and Uk 2 77)C as the

master states. The remaining slave states are expressed as

131

7h“ = Xifuka‘l’k) = Xi(77k)71k) (4.3)

'77: = W‘llka’l-‘H = Wnkflik) (4-4)

for 1: = 1,2,3,...,Q; 2' 7f k.

Equations (4.3) and (4.4) constitute a set of constraint equations that are to be determined.
The constraint functions in equations (4.3) and (4.4) are obtained by an invariant manifold
procedure that generates equations that can be solved for the unknown constraint relations.

The process begins by taking a time derivative of the constraint equations, yielding

 

ax,- ax,-
: ,. 4.
77! (91% “It avk I). ( 5)
.. BY, . 81’,- ,
- = —' . —— .’. 4.
7]? 011}: ”k + 8L1.“ ( 6)

for i=1,2,3,...,Q; 3% k.

The time dependence in these equations is eliminated by substitutions that make use of
the the equations of motion, as follows: 1'1). 2 Uk, bk 2 ijk 2 flag m. + fk(nj,1'7j), 7),- =
—w,2 7},- + f,(-r)j,'r'}J-). Then, the constraints (equations (4.3) and (4.4)) are substituted in the
resulting expression everywhere in place of the slave state variables, resulting in a set of
partial differential equations for the functions (X z(21k, ek), Yi(uk, 'vk)). This set of 2Q - 2,
time-independent, partial differential equations govern the geometry of the kth manifold,

and are given by

132

 

2 o
—wz X2+f2(Xja)/Jiuk9vk) : Uk+—'

for i,j=1,2,3,...,Q; 12¢ k.

These equations are not solvable in closed form (except in very special cases). Methods

for obtaining approximate solutions for X ,j and Y,- are described in the next section.

4.2.2 Invariant Manifold Solution

Asymptotic expansions can be used to obtain approximate solutions of the invariant man-
ifold equations, equations (4.7) and (4.8), for smooth nonlinearities, but such solutions are
only locally valid ([20], [21]). Numerical Galerkin-based solutions have also been developed,
wherein one can utilize either local or global basis functions to describe the invariant man-
ifold ([33]). In this work we employ a Galerkin-based procedure that relies on a patchwork
of local solutions to obtain a solution that is valid over a given domain: uk 6 [—Ub, Ub] and

“k E l-le Vbl-

We describe the development of the method for a local domain described by ui E [—ub, ub]
and v}: E [—v),,vb]. Once this procedure has been developed, it can be applied to each
patch, and the final result is obtained by collecting the results for all patches such that
the entire domain is covered. The relations between the coordinates in the global domain

, , 4 ° , ' . f? . (.9 ' ~
(uk, wk.) and those in the local domain (uh, Uk) are given by

133

u k
'U k

where du

and dU

du + “Z, (4.9)

(iv + 22):, (4.10)
2U

‘Ub + (TV—5X6“ — l) + ab, (4.11)
u
2V

—V:. + (7,?er — 1) + vb, (4.12)

where eu and en are the patch indices in the uk and U), directions, respectively, and N5 and

N5 are the number of patches used in the uk and vk directions, and du and dv represent

the shift from the ori 'in of u ., v. to u‘five, , res ectivel I. Here 6,, runs from 1 to N e
8 k A k 1, P l u

and ev runs from 1 to N5.

Substituting equations (4.9) and (4.10) into equations (4.7) and (4.8), the partial differential

equations governing the geometry of the kth manifold in the local coordinates (uz, vi.) are

 

given by:
BX' . (9X;
Y2. : 875% + vi) + 8125 (flag ((1,, + ui.)
+fk(vaYjad'u+uiadv+v2)) (4.13)
2 X' , _ , e e _ BY; 6 Yi . 2 , e
“’00)“ z+fz(X]aYJadu+Ukadv+vk) — 51"?de +vk)+51_'5(—wk (d‘u‘l‘uk)
k ’k

+ fk(X)-, Yjadu + ui..dv + 222)) (4-14)

for i,j=1,2,3,...,Q; 2',j 34 k.

The solution of equations (4.13) and (4.14) on the local domain is obtained by expanding

(X1, Y2) in terms of basis functions as:

134

Npm NW!

. l,
X,(u‘,:.,vf;) = Z: Z CimLMl,m(uZ,v}:) (4.15)
[:1 "1:1
Npm NW 1
WM) = Z Z Di‘mLA/Iz,m(uti,vi), (4.16)
1:1 m=1
where LAI),.,n(u‘Z,v;) = T1_1(uZ/ub)XTm_1(vf./vb), (4.17)

where T)_1(:L‘) and Tm_1 (1') are standard Chebyshev polynomials defined over :1: E [- 1, +1],
and the C ”s and D’s are the to-be-determined expansion coefficients. Equations (4.15) and
(4.16) are substituted into the local manifold-governing equations, equations (4.13) and
(4.14). Normally, each of the resulting equations is projected onto the basis functions, but
here we employ a collocation method, which is computationally more efficient, yet retains
very good accuracy. This is carried out by projection of the equations onto Dirac delta

functions in the local master coordinates over the local domain, as follows:

1, 1,, BLM),
0 : ff’ 6 6(212—ui‘pwi—vgq) — 2 DZ. mLIWLm + Z Cz- m(dv + ”fl—Efifl
U ”U; k
k It I m 1,771
1,. 3L1”) ,
+ 2 Ci m ave m (-wf.(du + ui) + fk.) d“): dvi (4.18)
l,m k
. .. , ,, 2 1. 1, (”Mam
0 = [6 e 6(‘112—‘ltz‘pyvz —- “1:21) a), Z CimLIl/Il’m — f,- + 2: Di "‘(dv + Up—Bue
U‘k‘vk l,m l,m k
8L3!
+ Z Dg'ln—Elél(—wg(du +212.) + fk) (in?C (it); (4.19)

l,m

for i=1,2,3,...,Q; 1'75 k;

_ T .
P— 13'“) 1).“)

q :1) "'9 Arne)

135

where (ufwwz‘q) E [—ub,ub] x[—vb,vb] are the collocation points, which are zeroes of
TNmei/“bl and TNpJ,(vf:,/vb), respectively (see [78], [81]). Equations (4.18) and (4.19)
constitute a set of 2(Q — 1) x Np,” X Np‘v nonlinear equations in the C’s and D’s. Note
that there are N; x NS sets of Cs and D’s, that is, a set for each patch. However, if the
system is conservative and the nonlinear terms are functions of solely the modal positions,
then only 0.25 x N3 x N5 sets of C’s and D’s need to be solved for, and the remaining
coefficients can be generated using symmetries in the NNM manifold. Once all of the
expansion coefficients are obtained, the X ‘s and Y’s, which describe the slaved modes, are
known functions of the master states ((12.1%). For 2' = k, these known functions are used to

express fk in equation (4.2) in terms of only 21E. and vi, rendering a single-DOF oscillator

as the reduced-order—model for the km N N M.

Once all of the coefficients have been determined for each subdomain, the entire manifold
is known, in a. patchwork form. It should be noted that a motion started on this manifold,
that is, one that satisfies the master-slave relations captured by the X’s and Y’s, remains
on that manifold for all time. by dynamic invariance. However, a given response will visit
many subdomains, and thus the solution moves across this two dimensional surface like a
curve tracing across a tiled floor. It should also be noted that these “tiles” are obtained in
a patchwork manner, without any continuity or smoothness constraints, so that the slave
constraints may exhibit jumps as the response moves across the tile boundaries. If the
subdomains are taken to be. sufficiently small, these jumps will likewise be small. Examples

of nonlinear modes constructed from a similar approach can be found in [33] and [34].

4.3 Fixed-Interface Nonlinear Component Mode Syn-

thesis

In this section, we extend the concept of the fixed—interface CMS of Craig and Bamp-

ton ([62]) by making use of the fixed-interface NNMS instead of the fixed-interface linear

136

normal modes (LNMs). The aim here is to account for nonlinearities at the substructure
level without having to resort to the retention of several linear modes. First, the nonlin-
ear structural system is partitioned into component substructures. Then, fixed-interface
dynamic representations of the substructures are constructed, where the nonlinear models
are expressed in terms of fixed-interface LNMS, which are coupled through nonlinear terms.
These substructure models are then reduced, each to a single nonlinear mode, using the
NNM constraint relations. and subsequently synthesized with linear constraint modes to

produce a ROM that describes the dynamics of the combined structure.

4.3.1 System Representation in Physical Coordinates

Consider a structure that is partitioned into two substructures, denoted by a and B. The
equations of motion of the substructures in physical coordinates can be written separately,

along with appropriate conditions on the common junction coordinates, as follows:

0 .- O’ O 0
M11 MIJ XI K11 KIJ XI
MJI MJJ Jh KJI KJJ XJ
(4.20)
0 ‘]Q
0100 _ 0
GJ(X) FJ
,1} .. ,3 l3 )3
M11 MIJ XI + - K11 KIJ XI
MJI MJJ XJ [KJI KJJ XJ
(4.21)
13 ,3
GI(X) _‘ 0
GJ(X) FJ

137

with junction boundary conditions on displacements and forces, as follows,

X3 X’B, (4.22)

F9+Ff : o, (4.23)

where ()0 denotes quantities associated with substructure a, and all terms defined below
have ()3 counterparts. The vector X 0: contains all coordinates for the a substructure
and has dimension N a, 3’ is the vector of interior (non-interface) physical coordinates
of dimension 71?, and X“; is the vector of junction (interface) physical coordinates of
dimension nC. G? is a vector of the nonlinear forces associated with the X? equations,

3‘ is the vector of nonlinear forces associated with the X3" equations, and F? is the
vector of reaction forces from substructure ,8 acting on substructure (1. Mg], Kf’I, etc.,

are mass and stiffness matrices, respectively, defined in an obvious manner.

4.3.2 System Representation in Linear Modal Coordinates

In this section, the (typically) large number of physical coordinates for each substructure is
first reduced to a smaller (yet still possibly large) number of coordinates by truncating using
the substructures’ fixed-interface linear modes. The physical coordinates of substructures
a and x3 are transformed into a truncated set of fixed—interface linear modal coordinates

using the following coordinate transformations:

0:

Xa : we: "a : l: (pa ‘1’0 :l "N (4.24)
N C a

17C

138

 

i0

nN
a- a a- B a
X _\11 n _ [‘I’N QC] "[3 , (4.25)

where (Pg, is the. 1 ’0 x n0 matrix of retained fixed-interface linear normal modes, 1710‘; is
the vector of retained fixed-interface linear modal coordinates (of dimension n“ ), \I'g. is
the N a X "C matrix of linear constraint modes, 778. is the vector (of dimension TLC ) of
linear constraint. modal coordinates, and similar terms are defined for substructure 3. The
procedures for calculating the retained fixed-interface linear normal modes and the linear
constraint modes are described below. Note that at this step, the dimension of the system

of equations for substructure a has been reduced from (N0 = n? + 72C) to (n0 + mg) with

n” < 71‘}, and similarly for substructure 3

4.3.2.1 Fixed—Interface Linear and Nonlinear Normal Modes

The concept of fixed-interface linear CMS of Craig and Bampton ([62]) is extended by
making use of the fixed-interface NNMS in place of the fixed-interface LNMS. Since the

procedures for substructures a and ,3 are the same, these superscripts will be omitted from

the development here.

Consider equations (4.20) and (4.21) with XJ = 0, that is, with the interface fixed. This

yields

MIIXI + KIIXI + GI(XIs XJ = 0) = 0. (426)

Using standard linear modal analysis (with the modes obtained for G I = 0), we obtain the

139

normalized fixed~interface linear modal matrix <1) I N associated with the interior physical
coordinates (of dimension n] X n, where n << 71,). Then, the retained fixed-interface linear

modal matrix <I> N is defined as

‘I’N = . (4.27)

The nonlinear equations of motion (4.26) are then transformed into fixed-interface linear

modal coordinates using the following coordinate transformation:

XI = ‘I’IN 771v. (428)

This yields the fixed interface nonlinear equations of motion expressed in fixed—interface

linear modal coordinates. as follows:

INN fiN+ANN 77N+GN(77N) = 0 (429)
where INN = @fNMuem (4.30)

ANN = @fNKHem (4.31)

éNWN) = @fNGfl‘PINnN’XJzol- (432)

This fixed-interface model is still relatively large and is to be ultimately reduced to a
single DOF using the nonlinear modal reduction procedure described in sections 4.2 and

4.2.2. Note that in terms of matching notation from those sections, the function G N and

140

dimension n are — f and Q, respectively. The NNM reduction provides the constraint
relations between the slaved states and the master states of the kth fixed-interface NNM

for a substructure, and can be expressed as follows:

n.” X471? .77.? ) (4.33)

, r N , r
n? = Yzlnk .721?) (4.34)

for i=1,2,3,...,n; iaé k.

After obtaining the constraints X,- and Yi, and doing so for each substructure, we have
single mode nonlinear models for the substructures, based on fixed interface models. We
now turn to the procedure for coupling these component structure models to one another

through the interface.

4.3.2.2 Linear Constraint Modes

The definition of a linear constraint mode from Craig and Bampton [62] is adopted here.
It is defined by statically imposing a unit displacement on one physical coordinate of the
set of junction coordinates and zero displacements on the remaining coordinates of the set.
This procedure is applied consecutively to all junction coordinates. Again, this process is
the same for the a and t3 substructures, and therefore these superscripts are omitted. The

collection of linear constraint modes is obtained from

we 2 = , (4.35)

141

where ‘I’IC has dimension n I X TLC and I JC has dimension nC X 720.

Note that nonlinear constraint modes are not considered in the present study, for a few rea-
sons. First. to do so would require additional steps involving the derivation of another set of
master-slave relations to describe nonlinear constraint modes (even if they are static). Also,
if one were to use nonlinear constraint modes in the synthesis procedure, the coefficients
of the nonlinear terms would be functions of the constraint mode amplitudes, which would
make the synthesis process cumbersome to the point of being impractical. Furthermore,
and perhaps most importantly, since fixed-interface methods work best for assemblies of
weakly coupled subsystems, for which fixed—interface (linear or nonlinear, as the case may
be) modes are well-suited to describe component motions accurately, it is appropriate to
use linear constraint modes to represent the small interface-induced motions in the com-
ponents. This restriction must be kept in mind, however, since it limits the method to a

class of systems with weak coupling.

Equations (4.20) and (4.21) are transformed into linear modal coordinates using equations
(4.24) and (4.25). The equations of motion for both substructures expressed in terms of

these coordinates have the form

(1 .0 C! a
INN MNC 771v ' + ANN 0 UN
' MCN MCC fie 0 K00 ’10
(4.36)
~ 0 O
GN(77Na770) __ 0
(30(77ch) ' F0

142

INN MNC fiN + ANN 0 7m
MON MCC fie 0 K00 no
(4.37)
~ B X3
GN(77N, 77C) _ 0
éC(’7Na 770) PC
with the following junction boundary conditions on the displacements and forces:
X31 = ng 2 n3 -_- Xfi, (4.38)
Fg = Ff; and pg = Ff,’ (4.39)
Fg + pg 2 0. (4.40)

Note that the stiffness matrices are block diagonal due to the orthogonality between <1) N
and WC with respect to K, which results from how (I) N and ‘110 are defined in equations

(4.27) and (4.35), respectively. We now turn to the synthesis procedure.

4.3.3 Synthesis with Nonlinear Modal Reduction

In this section, the model size of the substructures is reduced using the NNM constraint
relations. The substructure ROMS are then synthesized to obtain the final ROM that

describes the dynamics of the combined structure.

In summary, this nonlinear CMS process involves two steps of model reduction, one of

which is standard and uses linear modes, while the other involves the NNMS. First, we

143

use the linear modal reduction described in section 4.3.2 to reduce the system DOF from
(n? + n? + ac) to (n0 + n3 + 71.0). Note that this reduction is not central to the technique,
and may not be used in some systems (in fact, it is not done in the examples considered
in the next section). Then, the key step makes use of the nonlinear modal reduction based
on the approach described in sections 4.2 and 4.2.2, which reduces the system size from

(n0 + 72:3 + 72.0) to (2 + 720) DOFs.

Again, the superscripts a and ,3 are omitted in the derivation. The component forms of

equations (4.36) and (4.37) are given by

n 1
..N .NC--C 2 N ~N N N C
77,- + mis 713 +w, 77,: +9,- (77! ,7]k ,177.) = O (4.41)
3:1
”C
~N ’VCuC 2 N ~N N N C
m, +2774... Us +wkm, +91. (721 .77.. .771) = 0 (4-42)
321
n "C ”C
CV~V CC~C CC C ~C N N C J
7”er ”3' +anrs Us +Zkrs "ls +9r (771 ’77k inr) = Fri (4'43)
j=1 3-1 3:1

for i,l = 1,2,3,...,n; i,l;ék;

r = 1, 2, 3, ..., nC,

where k is the kth fixed—interface master mode defined in section 4.3.2.1 (typically taken to
be the fundamental mode). From equation (4.41), the expression for 77;“, is obtained , and

it is used in equation (4.43). Rearranging and grouping terms in equation (4.43) yields

144

”C
CC I I .. ~C 1 t N N
mfiN ijk+ 3: 771,3 “pm e 7),?' + 821 kCC ”SC + 9 up” 8(7), ,nk ,0?) = F,:](4.44)

where mgc‘wdate = 771C? + mac , (4.45)
Tl
777g.C * = Z —777,C.;-N m 79:0, (4.46)
j=ld¢k
g?"“”"““’ = a? + a? (4.47)
n
and 99* = Z 77123] (‘sz 7)]N — 9?), (4.48)
j=ld¥k

for r : 1,2,3,...,n

The modal constraints given in equation (4.33) are used in equations (4.42) and (4.44) to
reduce the equations of motion associated with the synthesized fixed-interface linear modal
coordinates (which are coupled to the constraint modes) onto the km fixed-interface NN M
invariant manifold (which is uncoupled from the constraint modes). This approximation
will yield an accurate ROM if substructures a and [3 are weakly coupled, that is, the actual
NN M invariant manifold of dimension 2 X (2+nC) of the combined structure has invariant 2-

kth

dimensional submanifolds that are close to the fixed-interface NNM invariant manifolds

of the substructures, which are each of dimension 2.

With the NNM reduction of substructures a and ,3, equations (4.42) and (4.44) can be

expressed in matrix form as

O (1 (1 r O
1 MKC ”iv “1% 0 n?
d t ..
MCK M31}; 0 e "C 0 K00 ”C
(4.49)
(JIM/[l]. ‘I'ij. 770) _ O
.. d t 7V —
GE? (1 30)}? ,77iV "0) F]

145

1 MKC 77,13’ + ...,: 0 77,1,"
d t ..
MCK Mg? 0 e no 0 K00 no
(4.50)
~ N V 3 fl
Gk(nk 777; an) __ 0
d t '— ’
Cup ”(77,, 77,. We) FJ

along with the junction boundary conditions from equations (4.38) and (4.40). The final
form of the synthesized ROM is obtained by imposing the junction conditions, equations

(4.38) and (4.40), on equations (4.49) and (4.50), resulting in

 

 

 

 

 

 

 

 

 

 

 

 

r 1 0 Mgc - ‘77]“0‘1
0 1 M13“; fifcv’fi
MCK MgK M32510“, L "C _
, (4.51)
"(4,2 0 0 q Fm?“- [ 53 - ’0-
+ 0 (of)? 0 77,1)” + (7]: = 0
_ 0 0 K00] _nc _ _c':‘g’d“‘8_ _0J
where Mupdate = (Mupdatea+Mupdate,fi) (4-52)
KCC = (KgC+KgC), (4.53)
and ézépdate : Gupdate,a(nllcV,a,77i:V,a,nC)
+G“pd“‘e’f’<n,, 675753.770). (464)

This is the desired model, which has (2 + 71C) DOF, one for each substructure and "C for

the constraint modes.

146

Note that these procedures have been computationally automated. Therefore, it is quite
practical to interface this fixed-interface nonlinear CMS with nonlinear finite element mod-

els of the form in equations (4.20) and (4.21).

4.4 A Five-DOF Nonlinear Spring-Mass System

A five-DOF nonlinear spring-mass system is used as a “proof of concept” example that
demonstrates the procedures and shows the effectiveness of the proposed method. The
system is schematically depicted in Figure 4.1, and naturally partitions into the three sub-
structures shown in Figure 4.2. The general features of this system are that the coupling
mass ml is connected to ground with a very stiff spring In, such that its isolated natu-
ral frequency is significantly larger than the fixed—interface natural frequencies of the two
attached subsystems (that is, when they are attached to ground instead of to m1). In
such a case, mass m1 will have a relatively small amplitude for typical responses, and the
fixed-interface linear modes of the two attached subsystems will be strongly reflected in
the linear modes of the combined system. The subsystems are assumed to vibrate at am-
plitudes such that significant nonlinear effects are encountered, and in a frequency range
such that their responses are dominated by their respective fundamental NNMS. Thus, the
procedure consists in developing single-DOF fixed-interface N N Ms for the attached subsys-
tems, and combining these using the proposed method, resulting in a three-DOF nonlinear
ROM that captures the dynamics (except for the high-frequency modes of the two attached
subsystems). However, by using NNMS to describe the subsystems, the essential nonlinear
interactions between the linear modes within each substructure will be contained in the

resulting ROM.

The equations of motion of the five-DOF spring-mass system are given by

147

 

ml 0 0 0 0 ‘ P {fl 1
0 7722 0 0 0 if2
0 0 "1'3 0 0 i3
0 0 0 m4 0 i4
_ 0 0 0 0 ""5 - _ f5 l
(M + k2 + *6) ‘1”? 0 4%
..,, (k2 + k4) -’~4 0
+ 0 —I.-4 A4 0
46 0 0 (It-6 + log)
L 0 0 O —f~8
—l.‘-3(:F2 *1171l3 — (97(174 — 11):;
k3(:172 — 171)3 — k5(1’3 _ I2)3
k5(;r3 — 11'2l3 :
k7(r4 - 1‘03 — WW5 — “)3
_ (79(15 - 1'03 l

 

 

 

 

 

 

 

 

 

 

x1
1‘2
333
234

$5

 

(4.55)

The equations of motion for the individual substructures and the junction boundary con-

ditions are given by

_ --a . ,.a _ 0.13 07
7771.7:1+ £1.11 — FJ + FJ

148

(4.56)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0 0 If k2 —kg 0 x,
0 m2 0 1"123 + —7.~,2 (k2 + k4) —k4 2:2
,.../3
L'0 0 m3- -173- h 0 A4 k4-bl‘3‘
(4.57)
' 73 ,3 q ' 3 '
-k3(r'2 — (1:1)3 Féa
l3 ,3 3 ,6 i _
+ k3(:r2 — I, )3 — k5(.r‘3 — 5:2 )3 — 0
3 3
,_ A)(l'3 _.I,‘2)3 _ L 0 _,
0 0 0 1“] k6 —7.-6 0 f 4:}
0 7714 0 117.3 + —k6 (k6 + k8) —kg :13;
x17 _ . . 7
-0 0 7775‘ _.1.3J _ 0 kg 1.8 _ L273—
(4.58)
7 ‘ H P a -
—A7(zg—x’{)3 F]
+ A7(.12 -— r])3 — k9(:rg - 13.2)3 = O
. kg(173 — 1‘3)3 d L 0 J
with junction boundary conditions on displacements and forces:
3 '
1'? = 1:1 = .73] (4.59)
(,3 30
F; + FJ = 0 (4.60)
Ff,” + F30 = 0. (4.61)

The system parameters used for this study are ml 2 5,7772 2 1,7713 2 1,777.4 = 1,7715 2

1,k1=50,k2 =1,k3 =2,k4 =5,k5 = 1,196 =4,k7 = 1,k8=45, and kg: 1.

149

The linear natural frequency of substructure a is w“ = 3.16 rad/s, the first linear natural

)3

frequency of substructure ,8 is w] = 0.69 rad/s, and the first linear natural frequency of

substructure '7 is a)? = 1.40 rad / 3. Therefore the substructures fl and ”y are coupled through

6 7

substructure (1 via weak coupling, since wa >> “’1 , “’1 .

Applying the procedures from section 4.3.2.1, the fixed-interface nonlinear mode manifolds
for the fundamental modes of substructures fl and 7 are obtained. In this simple case
the manifolds can be expressed as constraint functions in which the second linear modal
position and velocity for a substructure depend on the first modal position and velocity of
that substructure. The results obtained numerically for the modal position functions for the
two multi-DOF substructures are shown in figures (4.3) and (4.4). For substructure 6, the
boundaries of the surface along modal position and velocity are Ub = 2.6 and Vb = 1.79,
respectively, the number of Chebyshev polynomials along each u": and U]: are NP,“ 2 3
and NP,” 2 2 respectively, and the number of pieces along uk and vk are N5 = 100 and
Nf,’ = 100, respectively. For substructure 7, the boundaries of the surface along modal
position and velocity are Ub = 12.0, Vb = 16.78, respectively, the number of Chebyshev
polynomials along 21 k and along 17], are Np,“ 2: 3 and NW) = 2, respectively, and the number
of pieces along uk and U), are N5 = 100 and N5 = 100, respectively. Note that similar

surfaces exist for the slaved modal velocities, although they are not shown here.

Applying the procedures from section 4.3.3, a nonlinear three-DOF ROM is obtained. In
this case it describes the motions of the coupling mass and the first NNM for each of
the attached substructures. This model is simulated with initial conditions initiated on

the first fixed-interface nonlinear mode manifolds of substructures B and 7, as follows:

N , 73

7)] (0) 2: 1.5,77‘1N’7(0) = 0.4, along with a. constraint amplitude of 770(0) = 0.1, with

initial modal velocities taken to be zero.

For comparison purposes we use four models constructed in various manners. They are:

the original five-DOF nonlinear model given in equation (4.55), a three-DOF nonlinear

150

model obtained using linear CMS (that is, by projecting the nonlinear subsystems onto
their fixed-interface linear modes and the linear constraint mode), a three-DOF linear
model obtained using linear CMS (this is the linearized version of the previous model),
and the threeDOF nonlinear CMS model developed by the proposed approach. Figures
(4.5) - (4.9) depict the comparison of the time responses of the various physical degrees
of freedom obtained by simulations of these four models. In these figures the dashed line
labeled NLCMS corresponds to the three-DOF nonlinear CMS model, the dash-dot line
labeled LCMS corresponds to the three-DOF nonlinear model obtained using linear CMS,
the dotted line labeled LCMSLPrb corresponds to the three-DOF linear model obtained
using linear CMS, and the solid line labeled Original Model corresponds to the five-DOF
original model. From the simulation results it is clear that the fixed-interface nonlinear
CMS model outperforms the other low-order models, including the linear CMS model,
which is the approach most commonly used in this type of problems. In particular, the
NLCMS correctly captures the frequencies of the substructures, as well as the effects of

their coupling.

It should be noted that there is a range of system parameters where this method gives
satisfactory results. One particular trend worth noting is obtained by varying the stiffness
to ground k1. As k1 is reduced and approaches the same order as the stiffnesses of the
substructures, the fixed-interface approximation begins to break down and the accuracy of
the NLCMS model deteriorates. On the other hand, as [$1 is increased, the NLCMS model
becomes more accurate, but the amount of substructure interaction is reduced, leading to a

system which consists of essentially decoupled motions for the two attached substructures.

4.5 A Class of Systems

In order to illustrate the general applicability of the proposed method, a class of nonlinear
spring-mass systems is examined. This class of systems is schematically depicted in Figure

4.10, and naturally partitions into Na appendage substructures, as shown in Figure 4.11.

151

This system has the same general features as the system in section 4.4 except that each
substructure has (Nmasg + 1) DOFs. (Note that this is easily generalized to the case where
the substructures have different number of DOF). In the following sections, the model of
this class of systems is first developed and then a system having Na 2 2 and Nmass = 20

is used as an example.

4.5. 1 Model Development

The component forms of the equations of motion of the N = (Na Nmass + 1)—DOF system

are given by

N N
Z "LU .I‘j + Z kij .Tj ‘l' gi(X) : 0 (4.62)
j=l j=1

for '= 1, 2, 3, ..., N.

The formulae for mij. Isl-j and 9,- can be found in the Appendix.

The component forms of the equations of motion for the individual substructures and the

junction boundary conditions are given by

-r . . 5
m1 .1)? + k1 1'? = 2 F? (4.63)
SI;’3.‘7.5,...

(Nmass +1) (Nnmss +1)
615.65 + Z 165. 1:53 + g;9(XS)=f.S (4.64)
j=1 i=1

for i: 1, 2, 3, ..., (Nmass +1),

and S 2,13, 7, 6,

with junction boundary conditions on displacements and forces:

1‘? = .17? = :17? = 51:31; 2 (4.65)
F?” +Ff“ = Ff,” +F}"’ 2 F35 +F§a = = 0. (4.66)

The formulae for mg, kg, 925‘, and fl-5 can be found in the Appendix.

For bookkeeping of the indices, the procedures from section 4.3.2.1 are slightly modified.
Instead of applying the coordinate transformation from equation (4.28) to equation (4.26),
the coordinate transformation X = q’N'IN is applied to equations (4.20) and (4.21) with

the interface fixed (X J = 0); this step also lead to equation (4.29).

Therefore the nonlinear vectors of the fixed-interface nonlinear equations of motion, ex-
pressed in fixed—interface linear modal coordinates of substructures S, are given in their

component forms by

715 ”S n5

- E :j :E : N,S, N,S N,S N,S
p=1q=pr=q

for 2': 1, 2, 3, 723

and S = i3, 7, 6, ...,

where the formulae for (2327:” can be found in the Appendix. Then the fixed-interface non—

linear mode manifolds for the fundamental modes of substructures ,B, 7, are computed.

Next, the linear constraint modes of the substructures S are computed. Equation (4.64) is
then transformed into linear modal coordinates using the coordinate transformation defined
in section 4.3.2. The nonlinear vectors of the nonlinear equations of motion (expressed in

linear modal coordinates of substructures S) are given in their component forms by

S

(115+nC) (nS+nC) (11 +110)
- Z Z Z 5,...
])=1 (1:!) 7:0

for 1:1,293a”°3(n5 + 71C)

and S = ,3, ’7, 6,

The formulae for big], can be found in the Appendix.

4.5.2 Example

As an example we consider a system with Na 2 2 and Nnmss = 20. In the following

sections, the fundamental fixed-interface NNMS of substructures [3 and 7 are first

154

developed. Then, by applying the procedures described in section 4.3.3, a three-DOF
nonlinear CMS model is obtained which describes the motion of the coupling mass and
the first NNM for each of the attached substructures. For comparison purposes, four other
models are also developed. They are: the forty-one—DOF nonlinear original model given
in equation (4.62), a forty-one—DOF nonlinear model obtained by using linear CMS that
retains all the fixed—interface LNMs from individual appendage substructures fl and 7
(71/3 = 20, n7 = 20), a three-DOF nonlinear model obtained by using linear CMS that
retains only the first fixed-interface LN M from individual appendage substructures fl and
7 (n5 = 1, 127’ = 1), and a three-DOF linear model obtained by using linear CMS (this

is the linearized version of the previous model).

4.5.2.1 Substructure ,3

In this study, the system parameters used to construct the fixed-interface NNM of sub-
structure 6 are first chosen such that the system is similar to the equal—length FE model
of a rod under axial vibration, i.e., each block has the same mass and each spring has the
same spring constant. For the discrete system with equal parameters and under axial vi-
bration, the ratios of the higher linear natural frequencies to the fundamental linear natural
frequency are closed to 3:1 (mode 2 to mode 1), 5:1 (mode 3 to mode 1), 7:1 (mode 7 to
mode 1), etc., which are the ratios of the continuous system under axial vibration ( [90]).
Since the nonlinearities present in the system are cubic polynomials, the system will expe-
rience internal resonances. Moreover, the system parameters chosen in this way will result
in weak nonlinear couplings between the fundamental linear mode and the higher modes,
which causes the fundamental nonlinear mode shape to be close to the fundamental linear
mode shape. To remove internal resonances from the system, and to generate stronger
nonlinear coupling and modal distortion, the linear spring that connects the subsystem to
ground is taken to be softer than the other springs. This causes the fundamental nonlin-
ear mode shape to be very different from the fundamental linear mode shape at the large

amplitudes. Some details of this result follow.

155

For the first trial, the system parameters used to construct the fixed-interface NNM of

substructure 6 are 7712 2 m3 2 m4 = : ”7.21 = 1, k2 2 k3 2 k4 = 2 k4] = 1000.

The number of retained fixed-interface LN Ms (n3) is chosen to be 20. The first four fixed-
interface (linear) natural frequencies are a)? = 2.42 rad/s, tag 2 7.25 rad/s, w? = 12.04

rad/s, cuff = 16.76 rad/s. The first four fixed-interface LN Ms are shown in Figure 4.12.

Figure 4.13 depicts coefficients of the pqr cubic nonlinear terms expressed in fixed-interface
linear modal coordinates and associated with the first mode (master mode). Note that there
are 1540 nonlinear terms ( (n!3 x (n5 + 1) x (nfl + 2)) / 6 ) for each retained fixed-interface
LNM. The horizontal axis corresponds to the nth term (i.e. the first term is (17?“fi )3 , the
second term is (wy‘lw’fi)2 n‘zw’fl,” the last term is (n%’fi)3). Note that there are 20 envelopes
in the figure, and each envelope corresponds to a value of the first index p. We can see that
most of the coefficients are zero, which means that the first mode is not strongly coupled

with higher retained fixed—interface LNMs, and therefore they will not be reflected in the
N NM.

The first fixed-interface NN M manifold can be depicted as constraint surfaces depending
on the first modal position and velocity. A total of thirty-eight such constraint surfaces are
needed to describe motions on the first fixed-interface NNM manifold in the 40-dimensional
state space. Samples of such surfaces are shown in Figures 4.14, 4.15, and 4.16, which cor-
respond to the contribution from the second, third, and fourth fixed-interface linear mode
amplitudes to the first fixed-interface NN M manifold. The boundaries of these surfaces
along modal position and velocity are Ub = 40.0 and Vb = 96.9, respectively. The sur-
faces are obtained by using Chebyshev polynomials along ui and v}: with Np,” = 3 and
NW, 2 2, respectively, and the number of pieces along uk and vk to be N5 = 50 and

N5 = 50, respectively.

Figure 4.17 depicts the time responses of the first fixed-interface mode displacement ob—

156

tained from two different models. The dashed line is the time response of the ROM defined
N fl

at the end of section 4.2.2 with initial modal displacement 771 (0) = 33.9 and velocity
01“3 (0) = 0. The solid line is the time response of the original model (20 DOFs) initiated

on the fixed-interface N NM manifold. Figures 4.18, 4.19, and 4.20 depict the time responses
of the second, third, and fourth, respectively, fixed-interface linear mode displacement ob-
tained from the ROM and the original model. The dashed line is obtained by applying the
master-slave constraints (depicted in Figures 4.14, 4.15, and 4.16) to the numerical solution
of the ROM (the dashed line in Figure 4.17). The solid line is obtained from the 20-DOF
original model. The time responses of the ROM and the original model are almost identi-
cal at the beginning, and as time progress they begin to diverge. From the linear natural
frequencies, we can observe that rug z 3014}, w? x 2h)? —wf , and a)? z a)? +0)? —w1fi, which
indicates that there are internal resonances (to the first-order perturbation approximation
([79])) among these modes in the fixed-interface system of substructure 3. Therefore, en-
ergy is being exchanged among the modes involved in the internal resonances, which cause
the motion of the original model initiated on (in practice, very near) the first fixed-interface
NNM to eventually leave the manifold. In order to obtain the ROM that embeds the in-
ternal resonances, a multi-mode invariant manifold formulation is needed, as described in
[29], [71], and [42]. The multi-mode invariant manifold formulation is beyond the scope of
the current study. From Figures 4.17, 4.18, 4.19, and 4.20, we can see that the contribution
of the higher modes to the first fixed-interface NNM is less than 3% of the initial modal
displacement nifV’/3(0) 2: 33.9. Hence we expect that the one-mode nonlinear model, which
is obtained by projecting the fixed-interface subsystem onto the first fixed—interface LNM,

will yield a good time response, at least for a limited time interval.

Figure 4.21 depicts the time responses of the first fixed-interface modal displacement ob-
tained from the linearized one-mode model, the one-mode nonlinear model, and the ROM.
The dotted line is obtained from the linearized one-mode model, which is the linearized
version of the one-mode nonlinear model. The dashed line is obtained from the one-mode
nonlinear model, which is acquired by projecting the fixed-interface subsystem onto the

first fixed-interface linear mode. The solid line is obtained from the NNM ROM. We can

157

see that the time response of the one mode nonlinear model is close to the time response
of the ROM for many periods. This is due to the fact that the first fixed-interface linear
mode is not strongly coupled with the higher retained fixed-interface LNMs, as mentioned

above.

To eliminate the internal resonances from the fixed-interface subsystem and to yield strong
coupling between the first fixed-interface linear mode and higher linear modes, a new set
of system parameters are introduced. These will make enhance the nonlinear aspects of

the problem to more fully demonstrate the accuracy of the proposed approach. By observ-

f
A ’S and bN’S

mm. ,1qu from the Appendix and Figure 4.12, we can see that the

in g coefficients b

displacements of DOFs near the fixed-interface of the low-frequency linear mode shapes

N,x3,*

1qu coefficients to be small, or zero. In order to

are small, which cause many of b
yield stronger coupling between the first fixed-interface linear mode and higher modes,
a new set of system parameters are considered: m2 = m3 2 m4 = = mm = 1,

k2 = 10, k3 = 164 =2 = k4] = 1000, which allows DOF near the interface to have

significant displacements, in particular for the fundamental mode.

Twenty fixed-interface linear natural frequencies are presented in Table 4.1, and it is seen
that they do not possess low—order internal resonances among them. Figure 4.22 depicts the
first four fixed-interface LN Ms obtained from using the second set of parameters. Figure
4.23 depicts the coefficients of the pqr cubic nonlinear terms expressed in fixed—interface
linear modal coordinates and associated with the first mode (master mode), obtained using
these parameters. Comparing Figure 4.23 with Figure 4.13 indicates that the first fixed-

interface linear mode strongly couples with higher modes for this set of parameters.

Note that for these parameters the first fixed-interface linear mode is similar to a rigid-body
mode. Therefore one might argue that the spring next to the wall is soft and the other
springs are very stiff, which would imply that the fixed-interface subsystem would behave
like a one-DOF lumped-mass system. Therefore the one-mode nonlinear model obtained by

projecting the fixed-interface subsystem onto the first fixed-interface linear mode would be

158

adequate to capture the dynamics of the subsystem. However, Figure 4.23 indicates that
the first fixed-interface linear mode is strongly coupled with higher retained fixed-interface
LNMs, which indicates that such an argument is not justified. Hence, the concept of NN M
is needed in order to capture the correct dynamics in the large amplitude regime, which is

illustrated in following results.

Figures 4.24, 4.25, and 4.26 depict the contributions from the second, third, and fourth
fixed-interface linear mode amplitudes to the first fixed-interface NNM manifold. The
surfaces are obtained by using Ub = 3.0, V5 = 2.06, NW, = 3, Np,” = 2, N5 = 40 and
N5 = 40. Note that the maximum contribution from the second linear mode amplitudes
to the first NN M manifold is around 10%, which indicates significant coupling between the

first and second modes.

Figure 4.27 depicts the time responses of the first fixed-interface mode displacement ob-
tained from four different models. The dashed-dotted line, the dotted line, and the dashed
line are the time responses of the linearized one-mode model, the one-mode nonlinear model,
and the ROM (based on invariant manifolds), respectively, with initial modal displacement
TAM/3(0) = 1.20 and velocity fiiN’fiW) = 0. The solid line is the time response of the original

model with initial conditions initiated on the fixed-interface NN M manifold. The time

responses from the ROM and the original model are nearly identical.

Figures 4.28, 4.29, and 4.30 depict the time responses of the second, third, and fourth,
respectively, fixed-interface linear mode displacements obtained from the ROM and the
original model. The dashed and solid lines are the time responses of the ROM and the
original model, respectively. The time responses from the ROM are close to the original
model, and the errors are very small when compared with the amplitude of the overall
response (> 1). (From these figures, there exist discontinuities in the time responses
since continuity between patches of the constraint surfaces is not enforced. However if the

subdomains are taken to be sufficiently small, these jumps will be small.)

159

Figures 4.31 and 4.32 depict the time responses of the physical displacements 3’26 and 56%,
respectively, (the leftmost masses of each substructure) with the interface fixed for the lin-
earized one-mode model, the one-mode model, the ROM, and the original model. The time
responses from the ROM and the original model are nearly identical. The time responses
from the linearized one-mode model and the one—mode nonlinear model are significantly
different from the original model, since the contribution from the higher fixed-interface
linear modes is not accounted for in those two models. In other words, the motions of
those two models occur on a flat approximation of the invariant manifold (i.e., the linear

eigenspace), which is inadequate in the present case.

Figure 4.33 depicts the comparison of deflections of all physical coordinates as? (t) at three

different times: t = 0, the initial time, where the system is at its peak configuration; t = 7%

, one eighth of a period of the nonlinear system; and t = 1:; , a quarter of the period,
at which the system is at zero displacement (with nonzero velocity, of course). Two cases
are shown: the first fixed-interface LNM and the first fixed-interface NNM. Note that the
shape of deflection on the linear mode manifold (linear mode shape) does not change with
time, however the deflection shape on the nonlinear mode manifold (that is, the nonlinear
mode shape) does change with time, due to the time-dependent contributions from the

other retained linear modes, which is significant for this case. The distortion of the NNM

is evident.

Note that the computational time required to obtain the manifold for this case (with
N5 = 40 and N5 = 40)is about 30 minutes. The computational time associated with the
simulation of the twenty-one—DOF original model is about 13 seconds. The computational
time associated with the simulation of the ROM is about 2 seconds. Thus, if one is to carry

out many simulations, the generation of the ROM is worthwhile.

4.5.2.2 Substructure 7

The system parameters used for substructure 7 are mg? 2 m23 = 777.24 = = 77141 = 1,

160

 

[£42 2 42.5, 1643 = [€44 = = 1681 = 4250. This yields strong nonlinear coupling

between the linear modes, just as was done for substructure 6.

The results that are analogous to Table 4.1 and Figures 4.22 to 4.33 for substructure 6 are
shown in Table 4.2 and Figures 4.34 to 4.45. To avoid redundancy, detailed descriptions
are not repeated here. Note that since the ratio among the linear and nonlinear spring
coefficients are the same as for substructure 3, the surfaces shown in Figures 4.36 to 4.38,
obtained by using Ub = 3.0, Vb = 4.24, NW, 2 3, Np,” = 2, N5 = 40, N5 = 40, are
similar to those shown in Figures 4.24 to 4.26. However, the fixed-interface linear-mode

natural frequencies of substructure 7 are higher than those of substructure 6.

4.5.2.3 Synthesized Structure

The system parameters for the base structure (substructure a) are ml 2 1,500 and
k1 = 24,000, which yields Lac 2 4.0, which is close to the fundamental frequencies of
the substructures. However, since this coupling structure is very stiff, its response will

remain small, consistent with the assumptions needed to employ linear constraint modes.

A comparison of the linear natural frequencies of the three-DOF linear-CMS nonlinear
model and the three-DOF nonlinear-CMS model is shown in Table 4.3. Comparison of the
first sixteen Linear natural frequencies of the forty-one-DOF linear-CMS nonlinear model
and the forty-one-DOF original model are shown in Table 4.4. These two comparisons are
used as the first check to verify that the synthesized models are correct. The linear natural
frequencies from Table 4.3 are in agreement with those from Table 4.4, at least up to the
third decimal digit. Note that the forty—one—DOF linear-CMS nonlinear model is equivalent
to the forty-one-DOF original model since the modal matrices from substructures fl and 7
together represent a complete set of forty-one bases that are used to describe the original

model in the synthesized linear-modal coordinates.

Figure 4.46 depicts the time responses of the first synthesized fixed-interface mode dis-

161

placement "(V213 obtained from four different models. The dashed-dotted line, the dotted
line, and the dashed line are the time responses of the three-DOF linear-CMS linear model,
the three-DOF linear-CMS nonlinear model, and the three-DOF nonlinear-CMS model, re-
spectively, with initial modal displacements niw’fim) = 1.2, nil/“7(0) = 1.2, 770(0) = 0.03
and modal velocities fjiw’fi (0) = 0, nil/”(0) = 0, 770(0) 2 0. The solid line is the time
response of the forty-one-DOF linear-CMS nonlinear model with initial conditions initiated
on the fixed-interface N NM manifolds of substructures 6 and 7 (this is the full model). We
can see that the time responses of the three-DOF nonlinear-CMS model and the forty-one—
DOF linear-CMS nonlinear model are nearly identical over the time interval shown, and

the other two models show significant errors, especially the fully linear model.

Figures 4.47, 4.48, and 4.49 depict the time responses of the second, third, and fourth,
respectively, synthesized fixed-interface linear mode displacements (77$V ’fl , "11;! ’fi , and nfiv’fi),
respectively, obtained from the three-DOF nonlinear-CMS model and the forty—one-DOF
linear-CMS nonlinear model. The dashed and solid lines are the time responses of the
three-DOF nonlinear-CMS model and the forty-one—DOF linear—CMS nonlinear model, re-
spectively. They are nearly identical except that the forty-one—DOF linear-CMS nonlinear
model has some high frequency “ringing”. This is because the initial conditions for the
forty-one-DOF linear-CMS nonlinear model are initiated on the fixed-interface NNM man-

ifolds of substructures .13 and 7, which are only an approximation of the actual 6-dimensional

N NM invariant manifold of the whole structure.

Figures 4.50, 4.51, 4.52, and 4.53 are analogous to Figures 4.46, 4.47, 4.48, and 4.49.
Therefore, the descriptions similar to those given for substructure 13 are not repeated here.
The comments related to those analogous figures are also applicable to substructure 7.
Note that since the first fixed-interface nonlinear-mode natural frequency of substructure
7 is higher than that of substructure 6, the time windows in Figures 4.50, 4.51, 4.52, and

4.53 are adjusted in order to depict clear comparisons.

Figures 4.54, 4.55, 4.56, 4.57, and 4.58 depict the time responses of masses 7721, 171-2, 77121,

162

77122, and 777.41, respectively, obtained from four different models ( these are the coupling
mass and the end masses of each substructure). The dashed-dotted lines and the dotted
lines are the time responses of the three-DOF linear-CMS linear model and the three-DOF
linear-CMS nonlinear model, respectively, obtained by applying the coordinate transforma-
tions defined in section 4.3.2 to the time responses of the synthesized fixed—interface linear
modal coordinates of both models with nff = 1, n”7 = 1, and 110 = 1. The dashed lines
are the time responses of the three-DOF nonlinear-CMS model obtained by applying the
coordinate transformations defined in section 4.3.2 to the time responses of the synthesized
fixed-interface linear modal coordinates having n3 = 20, n7 = 20, and no = 1. The
solid lines are the time responses of the forty-one-DOF original model with initial condi-
tions initiated on the fixed-interface NNM manifolds of substructures fl and 7. It is seen
that the time responses of the three—DOF nonlinear-CMS model and the forty-one—DOF
original model are nearly identical. However, the time responses of the three-DOF linear-
CMS linear model and the three-DOF linear-CMS nonlinear model are quite different from
the time responses of the forty-one-DOF original model. This is due to the fact that the
contributions from the higher fixed-interface linear modes are not accounted for in those
two models. In other words, the motions of those two models occur on the flat approx-
imation of the invariant manifold of dimension 6, which is not a good approximation in
this case. (Note that such an approximation would be quite good for the original set of

parameters. for which the modal coupling was small.)

Figures 4.59 and 4.60 depict the time responses of the first synthesized fixed—interface

fl and "4171M, respectively, obtained from the three-DOF nonlinear-

mode displacements niv’
CMS model and the forty-one—DOF linear-CMS nonlinear model. The dashed lines are
the time responses of the three-DOF nonlinear-CMS model with initial modal displace-
ments niv’flm) = 1.3, ni\r’7(0) = 1.3, nC(0) = 0.04 and modal velocities Trim/3(0) :-
0, 7'11V’7(0) = 0, 7")C(0) : 0. The solid lines are the time response of the forty-one—DOF
linear-CMS nonlinear model with initial conditions initiated on the fixed-interface NNM

manifolds of substructures B and 7. It is seen that the time responses of the forty-one-DOF

linear-CMS nonlinear model starts to deviate from the three-DOF i‘ionlinear-CMS model

163

after several oscillations. This indicates that at these initial conditions, with the base-
structure parameters m1 = 1,500 and k1 = 24, 000, the actual NNM invariant manifold
of the whole structure deviates from the fixed-interface NN M manifolds of substructures ,8
and 7. Note that we also performed a numerical experiment by fixing the ratio between m1
and k1 and reducing their values. It was found that the valid time domains of the fixed-
interface N NM manifolds of substructures ,8 and 7 are reduced as well. This is expected to
occur, since this parameter trend allows more interaction between substructures fl and 7.
This causes the actual N N M invariant manifold to deviate further from the fixed-interface

NN M manifolds of the individual substructures, especially for large-amplitude motions.

Finally, the computational time associated with the simulation of the three-DOF nonlinear-
CMS model is compared with that of the forty-one—DOF linear-CMS nonlinear model.
It takes about 180 seconds to simulate the forty—one-DOF linear-CMS nonlinear model.
However it takes only 60 seconds to simulate the three-DOF nonlinear-CMS model, which
is a factor of three lower. The primary reason for this is that a smaller time step (a factor
of 2.5 smaller) is needed for the ode-solver to integrate the stiff forty-one-DOF linear-CMS
nonlinear model. Therefore, in terms of computational time associated with the simulation,
the three-DOF nonlinear-CMS model does not have much advantage over the forty-one-
DOF linear—CMS nonlinear model, as we might expect. This is mainly due to the fact
that one is required to compute the nonlinear update terms gg’uPdate in equation (4.44)
of both substructures, which represent nonlinear terms that arise from the slave fixed-
interface linear-modal-coordinate equations of motion. This is the same scenario as for the
forty-one-DOF linear-CMS nonlinear model, since all nonlinear terms in this model need to
be computed. This situation is in contrast to the individual fixed-interface substructures.
From section 4.5.2.1, it can be seen that the simulation time of the ROM is a factor of
6.5 smaller than the simulation time of the twenty-DOF original model. This is because
the nonlinear terms of the master mode EOM are only computed in order to simulate the
ROM, but all nonlinear terms in the twenty—DOF original model must be computed in

order to simulate the original model.

164

 

4.6 Conclusions

In this chapter we have shown how to extend the fixed-interface linear CMS technique of
Craig and Bampton ([62]) to nonlinear structures by making use of fixed-interface NNMS in
place of fixed-interface LN Ms. This approach allows one to build nonlinear reduced-order
models with improved accuracy for systems that are composed of assemblies of substruc-
tures. The first example system presented here is quite simple, and the model reduction is
not very significant (from five to three DOFs). A second system with many more DOFs
is also used, and it shows a significant reduction in model size (from forty-one to three
DOFs). In fact, the N NM method has been developed to the point where it can be applied
to systems with thousands of DOFs. The roadblock to pushing this method further is that
it is very computationally demanding to generate substructure ROM with more than one
DOF using a NN M approach ([42]). Current work is focusing on systems with many more
DOFs at the unreduced substructure level, and it includes substructures that are modeled

by nonlinear FE techniques.

165

4.7 Tables

 

 

 

 

 

 

4th Mode of 1th Mode of
1 0.68596232102650 11 44.72697600017000
2 5.06072722592600 12 48.09670186700000
3 9.94306090168100 13 51.17011958090000
4 14.79648130027000 14 53.92824229550000
5 19.56717359004000 15 56.35403805246000
6 24.22073003072000 16 58.43253131062000
7 28.72676467269000 17 60.15089310481000
8 33.05675986752000 18 61.49851874983000
9 37.18364689122000 19 62.46709231770000
10 41.08177243723000 20 63.05063732260000

 

Table 4.1. F ixed-interface linear natural frequencies of substructure B using the second—trial

 

 

parameters.
7th Mode 0217 7th Mode w?
1 1.41414755239200 11 92.20702318159000
2 10.43295644747000 12 99.15389102074001
3 20.49814516979000 13 105.4899039538000
4 30.50372764424000 14 111.1759195941000
5 40.33876175326000 15 116. 1768256602000
6 49.93231412311000 16 120.4617492830000
7 59.22174251389000 17 124.0042428732000
8 6814825628723000 18 126.7824443123000
9 76.65605183909000 19 128.7792098755000
10 84.69224352313999 20 129.9822187218000

 

 

 

 

 

Table 4.2. Fixed-interface linear natural frequencies of substructure 7.

166

 

 

 

 

 

 

 

 

4th Mode 47,1401” 5 4th Mode waCMS
1 0.68582383924810 1 0.68582383933910
2 1.41280504607300 2 1.41280505021300
3 4.00456599773800 3 4.00460940350200

 

 

Table 4.3. Comparison of the linear natural frequencies of the three-DOF linear-CMS

nonlinear model and the three—DOF nonlinear-CMS model.

 

 

 

 

 

 

7th Mode ijCMSAlDOF 1th Mode 018M
1 0.685823839247 1 0.68582383924720
2 1.41280504587300 2 1.41280504587300
3 4.00452879688600 3 4.0045287 9688500
4 5.06079466571000 4 5.06079466571000
.5 9.94306486412500 5 9.94306486412500
6 10.43301741537000 6 10.43301741538000
7 14. 79648235748000 7 14.79648235748000
8 19.56717401354000 8 19.56717401354000
9 20.49815228625000 9 20.49815228625000
10 24.22073023840000 10 24.2207 3023840000
11 28.72676478757000 11 28.72676478757000
12 30.50372969952000 12 30.50372969952000
13 33.05675993624000 13 33.05675993624000
14 37.18364693456000 14 37.18364693456000
15 40.33876259811000 15 40.33876259810000
16 41 .08177246556000 16 41 .08177246556000

 

Table 4.4. Comparison of the first-sixteen linear-natural frequencies of the forty-one-DOF

linear—CIVIS nonlinear model and the forty-one—DOF original model.

167

 

4.8 Figures

 

é

\‘ \\\\\\\\\\\
g g
a

 

 

 

 

Figure 4.1. A five-DOF nonlinear spring-mass system.

[i substructure

 

 

 

 

 

 

 

 

X‘l

r—>
2 an
.//
.4va m. ,
'7, ———> F:
///,
2,2 l
./ —’

Y
x1 x5 X3
or substructure y substructure

Figure 4.2. System substructures.

168

               

          

   

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itude 772

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Xfi( N’fi,7'/[V‘fi) tot e
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Figure 4

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169

0.25 I I r 1 1

 

 

 

 

 

 

 

 

I L
- - NLCMS. 3 DOF
(12. --LCNEL3[M3F a
LCMSLPrb. 3 DOF
Original Model. 5 DOF
0.151 ‘ H 1‘ I‘
’ 1
o 1 r " '1 .1 '1 ‘ '1 «
o ‘ .
\ I 1 ‘ I /‘ I 1;“ " 1" I1 I ‘1‘,“ I 1 If.
1 1 I” ‘ '1‘” ,6" 1.1 ‘1 ‘I‘."‘ I [,11
0‘05 b' i {I ' ‘ 1| "1“ | ,.>‘ f \1\’ f .l 1 1 ' V '11 '1 ' '11“ 11.1
“ I)1” " 1| I . lr"1\’|\ I, ”h If.“ [1‘ I‘1!"‘1 ’ f :\1=.1I, [ ’1’ I“
‘ t l’ 3 I" "' 7 . ,‘ .‘ ‘ I .
xv— OL \' I, 1‘ “| ;’.‘\ I I\ [ll 1 ’1‘ '1 I “ ‘ '1' \II “(7 i“ V" ,fl I l H I J I I 1‘ .-
N l "‘l " “‘11 ' I" ' ‘ ' ‘ f1 -‘ 4‘ 1 ‘I 1 I" il I
._\‘ I .1 ‘1‘]11 1' ‘1‘12'1 1‘), 1' {1: ‘1‘} [i '11,,” ‘ \11” ,1f
-0‘05 ’- ‘ ' l I ' l ‘ | 1 J I." ‘ ) [I J . I 1‘1!» 1‘ 1 I 1 1’ ’7'
\I 1 . 1 \ 1
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Pl‘ ‘ II | 1’ x" 11‘; ‘ '1 " 1'
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l) f] 1’ [I
I
—O.15 h ...1
—O.2 - a
_O.25 EL 1 1 1 1 1 1
O 5 1O 15 2O 25 3O 35 40

Figure 4.5. The time response of the displacement $1 of mass m1 from Figure 4.1, which

corresponds to x?, :13? , and x] from Figure 4.2, for the nonlinear-CMS model, the linear-

CMS nonlinear model, the linear-CMS linear model, and the original model.

 

 

 

 

 

 

 

 

2 r - - NLCMS. 3 DOF ~
- - LCMS. 3 DOF

1 5, LCMSLPrb, 3 DOF _

' Original Model. 5 DOF
1‘ 1:}. I,“ 1,.‘\’\ ' ”“1 A" 17‘1 'A d

1 :I 1‘ ’-’ 11 I‘ ‘ I ’l \ I 1 ‘ , .., \

1 II 1‘, ' 1 (I ‘ \ 1 ' ‘ 1 ’ I ‘ 1‘ ,‘
05"I 1' I, I 1! 1‘ 'I1\ I '1‘ ‘ I 11‘ "‘

O ‘s H‘ [I ‘ ‘ , 1 \ ' I ‘ 1 I , I ‘ I ‘1 ‘ 1;.
‘ I " I, 1 ‘ f i ‘ ‘ ' ’I ‘ ‘ I I ‘ “ ’ ," ‘ 1|
N I ’ I ‘ 1 ‘ 1
x O ’- ‘i I ‘ [I ‘1 \ I 1' ‘ l ’ I ‘ ‘1 ,I ,l ‘ 1 ’ ; ‘ 1'1
I 1 I1 ‘ ‘ ‘ , 1 \ 1‘ ‘ .‘ I . . ,
1 I ‘ . f . I

I, (I I," 1‘ I ‘ 1 1‘ ' 1 ‘ ‘ I ' \ \ I i \ ‘
-O-5 _ \ 1 \\ I; \ \ I ‘ \ 1‘ ’ I \ ‘I \ f

\ I I‘. 1/ ‘1 1 I ‘ ‘ I I ‘ "I l ‘ ' ‘

\ d I ' I/ ‘ \f I \ ’ / ‘ H ’I \
_1__ ‘J ‘\_g {\1 \,\/ ,\ \/ \ [L
—1.5 7 a
_2 >- _.

1 1 1 1 4 L 1
O 5 1O 15 20 25 3O 35 40
t

Figure 4.6. The time responses of the displacement 3:2 from Figure 4.1, which corresponds

to 32/23 from Figure 4.2, for the nonlinear-CMS model, the linear-CMS nonlinear model, the
linear-CMS linear model, and the original model.

170

 

 

 

 

 

 

 

 

2 *- fi r Y I' L If L
-5 - - NLCMS. 3 DOF
- - LCMS. 3 DOF
2 * LCMSLPrb. 3 DOF ~
1 5 Original Model. 5 DOF
. f" r ,\ 1‘ II/\ ‘ ’6"
ll f" \ : ‘ ‘2 ‘
1" If "1 /’\\ 'I ‘ I ‘1 l I V \ I \7 f ‘
. , H I \ \ \ I y/\ . A I 7 i
‘ I! I‘ ‘ ’ I 1 1 l 1 \ ’ I 1 I 3 '1
05*) H u ,, “ "f i I 1‘. "\ ‘\ l :1 I...
\ 1, ‘ 1' H I . l l I 1 ‘ | , i l .\ ‘ "i
m [l N ‘) '5 ‘i 'f “ " 1] 1' ‘ l ' ’1 ‘
\l '.\
x O ' ,1 1, .t ‘1‘ , I 1 1‘ I '1 ‘ 1‘ 1 ’1 1 1 I I ‘ 1‘—
' 1 '. I -
[ 1 q! H. II 1‘. ,’ I ‘ l , I I \ ’ 1' \ i , .5 1 I
—0.5 " \ [I \\ I ' \ ‘ I ‘ ‘ I ' \ ‘ (q ‘ 1
1 \i ,l \ i 1 ‘ I ’ ‘ I ' ‘ I, I ‘
1 L f 1 \l I \l f ' ‘ U ’ \ V I \ } l 4
_ \ '1 \ I \\/ ’ \ ) I \ /\ \/§ I \
VI \ ’ ., \/ .\’ \Jl -’
-1.5 ~ ~
_2 - -4
—2'5 >- 1 4 1 41* P J 2_L ‘
O 5 1O 15 20 25 30 35 40
t

Figure 4.7. The time responses of the displacement :173 from Figure 4.1, which corresponds
5

to $3 from Figure 4.2, for the nonlinear-CMS model, the linear-CMS nonlinear model, the
linear-CMS linear model, and the original model.

 

 

 

 

 

 

 

 

- ~ NLCMS. 3 DOF
06* — — LCMS.3DOF _
' . LCMSLPrb. 3 DOF
Original Model, 5 DOF
0.4 F .1
q I} I \ I“ I
. . I ‘ 1,
11 I ‘ 'I 1 I \ ‘l \.
0.2 ~. ,’ .\ v I ‘. 1r 1‘ I \ 1 l
‘ /'\_ .1 ‘1 ‘ 1 1! \1 l.\\ ’ \ g ‘
f l I’ \ j \l ' ‘ i 11 i I ‘ j / ‘ 1 1‘.
Vt ' ‘ II \- A 1! ‘1 ' ill If f 1 ’f : A“
X 0’" I f ‘1 I, \\ / l I; i- ‘ 1‘ ’1 1 ’7 K I] ‘1 -[
. ‘. I I , 1 1 1 1
1 I ’ (ll ‘\ I 1; ‘1 1 \. I ‘ ‘I ‘ If H
a f 1 '/ 1 . ,t ‘1 ,' 1‘ I «(I ' I, 11
-—O 2 k- : , s/ \./ I ‘l 1' \ I \\ . ‘\ / I 1‘ .4
l '1 I 11 1 \-l I ‘ 1’ 1“
L- f l ,J' 1 ’ \l
‘J \‘l 1 I \
—O.4 ' a
-0.6 ~ -
O 5 1O 15 20 25 30 35 40
t

Figure 4.8. The time responses of the (:lisplacen‘ient .174 from Figure 4.1, which corresponds

to 1'3 from Figure 4.2, for the nonlinear-CMS model, the linear—CMS nonlinear model, the
linear—CMS linear model, and the original model.

171

 

 

 

 

 

 

 

 

— - NLCMS. 3 0015
0.6 t -- - LCMS. 3 DOF F‘
LCMSLPrb, 3 DOF
Original Model. 5 DOF
0.4 r . . .1
I" - 9, I“. I 1 ‘7 " \1"
II I" I] ‘1 ‘1 i '1‘ 5' IA‘ : {Tl
0.2 r I A j .\ 1, 1, 1/1‘ _, ' ;~ \1 5 '1- r
. , I 11 1 i \l I \\ ' ' .‘
‘, I l ‘ \‘ . )l 1 i / 3 I \l
l I II \ I { R. I ‘I '1’ ‘ 1 . I 1
to ‘ , I, , I , 1‘, l 1“ I l , II ‘ 1“
x 0 ” ‘1 f ,, ,1 I ,, 1; I 1‘ ' I i I! ,I 11 ‘
I I I . 1. 1 \ I \ 1 \. I
l . ‘ I \|\ ., l’ ll / .1 / I \ \ I } ’I ii.
'1 ,I 1 , 1x, \ ,2 1 , I 1 1 1 «1 - 1 11 11
_O 2 1- ‘ 1 ‘1 \J’ .’ i, ’ ll 4 \ I ‘\ l I \l -4
' -. I \ II. “‘ ’ . \ I V l I ‘1‘“
'1 ,1 1 ’ "J ” " 1 I w
J ‘1‘) ‘i,’ ‘u‘ ‘ A
—o.4 - , «
—O.6 - J
O 5 1 O 15 20 25 3O 35 40
t

Figure 4.9. The time responses of the displacement 9:5 from Figure 4.1, which corresponds
to :rg from Figure 4.2, for the nonlinear-CMS model, the linear-CMS nonlinear model, the

linear—CMS linear model, and the original model.

 

 

 

 

 

 

 

XNmass meass+1

I“?

-~+
'kz Nmass’lfz Nmass +1
X

2Nmass x2Nmass+1
,—>

i? k

4 Nmass’ , 4 Nmass + 1

xNa Nmass XNa Nmass + 1
1’ 1 b
l t k
IkzNaNmassI 2NaNmass+1

WIND

Figure 4.10. A class of nonlinear spring-mass system.

172

+22

x9x2x9xfi x2

4

mass xamassfl

[—>
kg Nmass’k2 Nmass ¢ 1

If} h.

:ke. ‘ m kiwi

1

u

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

7 ' F043 F30: I m m a £7 7 W Bsubstructure
J J / , /, » /
Xlrtmass meassH
+ r”
ngmass'k; Nmasu1
, /
/
m? t + F3” L7 L4 7 substructure
// /
p i D b j" + v
i i 1
‘ ‘ j 1 ,I /
__’ ‘ A ‘ /7l 7
z / /
a substructure
Figure 4.11. System substructures.
—~—— Fixed-Interfage Linear Mode Shape 1
0-5 “ — — — Fixed—Interface Linear Mode Shape 2 fl
- Fixed—Interface Linear Mode Shape 3
0 4 _ — Fixed—Interface Linear Mode Shape 4
0.2L s
ax... O i‘ J
_0.2 _ -
-O.4 ~ ‘
—0.6 ~ -
O 5 10 15 20 25

im Degree of freedom

Figure 4.12. The first four fixed-interface LNMs of substructure ,3 using the first-trial
parameters.

173

 

 

A

1 500

n

1 000

 

60

40*

 

-40-

 

500

—60

m term

n

Figure 4.13. The coefficients of the pqr cubic nonlinear terms expressed in fixed-interface
linear modal coordinates and associated with the first mode (master mode) of substructure

3 using the first-trial parameters.

          

s

s é

é sé ss

\ssxss ésss‘

~ é s ss

0. ss é ss s 0

\s\ x s s~¢s~ss

s

~§~s~s§s§ \\\§
us

         

 

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x. ¢ 5
s. s~ s ~s -
é ys L é ~ ss ~
s s fé -- s ‘ \~
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s 9

ss s s

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reevewgvfi...
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NMB

Figure 4.14. The contribution of the second fixed-interface linear mode amplitude 712

X 5 ( {V46 , niv‘fi) to the first fixed-interface nonlinear mode manifold of substructure B using

the first-trial parameters.

174

 

Figure 4.15. The contribution of the third fixed-interface linear mode amplitude "91,5 =

X g (”{Vfl , fiiv‘fi ) to the first fixed-interface nonlinear mode manifold of substructure fl using
the first—trial parameters.

 

Figure 4.16. The contribution of the fourth fixed—interface linear mode amplitude rig/fl =
X :3 (niv fl , fiiv’fl) to the first fixed-interface nonlinear mode manifold of substructure B using

the first-trial parameters.

175

 

50

 

 

 

 

rLtlLllllldl ' a 4 a
Inn-UH.” lllll

 

NNM. 1 DOF
—-—- Orlglnal Model. 20 DOF
A

 

 

 

 

 

 

 

 

50

40

30

 

 

20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1O

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.17. The time responses of the first fixed-interface mode displacement "{Vfi for the

ROM and the original model of substructure {3 using the first-trial parameters.

iflwfl) for the ROM and the original model of substructure fl using the
176

The time responses of the second fixed-interface linear mode displacement

H N J3
first-trial parameters.

ure 4.18.

5.12

1
772

Fi

 

 

 

0.5- 1

«a 111 111 I 111111 1 11111111 1111111 1'1'11111

°‘* 11111 11'111111111111111"" 1111111.
0 1O 20 t 50 4O 50

Figure 4.19 The time responses of the third fixed-interface linear mode displacement
n3 N1fl=3X (nN’ 5,119”) for the ROM and the original model of substructure fl using the
first- trial parameters.

 

" ' ' — - N.M 1oor=
—— OI'l lnal Model, 20 DOF
1.5“
" 1
0.5‘
1 1 11 1' , 1
a; O " ‘51.1'1’11"1‘1""1 : 1 1-1'1'1 ' '1‘1'1111'1 1 :1 "',""'11,"1 ' 111‘
S—‘v ‘1"1'1'1' "1 ‘ ' ' ""1" "1" " ' '11'1111"""'1 "‘1'
—o.5- 11
1
—1.5r
-2 -
0 1O 20 30 40 50

 

 

Figure 4. 20. The time responses of the fourth fixed-interface linear mode displacement
N—fl X f ( N fl, N’fl ) for the ROM and the original model of substructure 3 using the
first- trial parameters.

177

60 , , r

 

 

LNMiPrb. 1 DOF
- — LNM. 1 DOF
—— NNM. 1 DOF

 

 

 

   

 

 

 

0 1 O 20 30 40 50

Figure 4.21. The time responses of the first fixed-interface mode displacement 721V ’16 for the
linearized one-mode model, the one-mode model, and the ROM of substructure 5 using
the first-trial parameters.

 

 

 

 

 

 

 

 

 

 

 

0.6 —~— Fixed-Interface Linear Mode Shape 1 ~
— -— Fixed-Interface Linear Mode Shape 2
- 1 Fixed-Interface Linear Mode Shape 3
0.4 _ ~ Fixed—Interface Linear Mode Shape 4 H
l I I 1 f .— r0
0.2 '- / / f I >\:’ .1
tax" 0 » , » ’ '1 ~
' / ’ / I. \ \
\ .l /'
..0.2 ~ ‘\ j , " A / 4 1% 1
1 _. ’ T 1 \
—O.4 ~ -
—-O.6 r 1 J 1 . ‘
O 5 10 15 20 25

im Degree of freedom

Figure 4.22. The first four fixed-interface LNMs of substructure 3 using the second-trial
parameters.

178

 

600 800 1 000 1 200 1 400 1 600
nm term

400

 

 

4G

 

 

—4c
0

Figure 4.23. The coefficients of the pqr cubic nonlinear terms expressed in fixed-interface

linear modal coordinates and associated with the first mode (master mode) of substructures

5 using the second-trial parameters.

 

Nfl =

Figure 4.24. The contribution of the second fixed-interface linear mode amplitude n2
X g (”{Vfl’ nil/fl) to the first fixed-interface nonlinear mode manifold of substructure B using

the second-trial parameters.

179

 

Figure 4.25. The contribution of the third fixed-interface linear mode amplitude név’fl =

X g (niv ”[3 , 7'7?” ) to the first fixed-interface nonlinear mode manifold of substructure B using

the second-trial parameters.

 

Figure 4.26. The contribution of the fourth fixed-interface linear mode amplitude nil/fl =

X fluff} , nil/'5 ) to the first fixed-interface nonlinear mode manifold of substructure fl using
the second-trial parameters.

180

2.5 l w 1

 

 

 

 

 

 

 

 

- - LNMLPrb. 1 DOF
2 _ LNM. 1 DOF
- - NNM. 1 DOF
1 5 _ Original Model, 20 DOF
1:3.
\ \ \\ r I //
0.5F \ \'\ / / .1
c: \ \ \ / I //
Z 1— 0’. \ \\ / / 'l
\ / \ I /
\ I /\ /
“0.5 h \ \ / / / T
1‘ I. \. l
-1 i- \ l/ \ \ I I I \\ l/ "
—1.5~ «
-2 _ q
_2.5 1 4 A 4
0 2 4 6 8 10
t

Figure 4.27. The time responses of the first fixed-interface mode displacement nil/fl for
the linearized one-mode model, the one-mode model, the ROM, and the original model of
substructure ,3 using the second-trial parameters.

 

V Y I

— — NNM. 1 06F
0,15 _ *—* Original Model, 20 DOF H

 

 

 

 

 

I
l

—O.15

 

 

)-
p—
.-
.—

 

Figure 4.28. The time responses of the second fixed-interface linear mode displacement
77‘?)V ”3 = Xgmiv’fl, 1'79“} ) on the first fixed-interface nonlinear mode manifold for the ROM

and the original model of substructure ,8 using the second-trial parameters.

181

 

YY)—T

.0
O
.5

l
.0
o
w

 

L

 

 

— — NNM. 1 DOF
—— Original Model. 20 DOF

 

 

 

 

 

Figure 4.29. The time responses of the third fixed-interface linear mode displacement
név’fi = X :43 (nfV ’fi , fiiv’fi ) on the first fixed-interface nonlinear mode manifold for the ROM

and the original model of substructure fl using the second-trial parameters.

10

 

0.015

 

 

 

 

 

 

 

5— NNM. 1 05F
——- Orkflml Model. 20 DOF
0.01 M\\ I f A
\\ II I,
0.005l \\ _
n \
2“ v o~ \ ‘ p _
:
—o.oosi ‘
\ ‘\
—o.01 L ; ‘7’ ‘
—o.o15 ‘ ‘ l
o 2 4 6 8

Figure 4.30. The time responses of the fourth fixed-interface linear mode displacement
) on the first fixed—interface nonlinear mode manifold for the ROM

Nfi .Nfl

N, '
774 H=Xfil(nl .721

and the original model of substructure [3 using the second—trial parameters.

182

10

 

 

 

 

 

 

 

 

0.5 T T L L
'— ‘ - LNMLPrb. 1 DOF
0.4L- LNM, 1 DOF H
- - NNM, 1 DOF
0 3 - Original Model. 20 DOF
0.2 " l/I, w ~- \\I I I .\>\ ‘ Ill/,... “\\ . -1
0.1 L (I / \\ \'\ I \'\.
/ " 1‘ /\ \. -l
/ / I r" // '\ \.
CORN O T j/ / _ \y / '\’ \\ 1
I I \. / \ \ \
-O.1 r /,I l.’ \y [,1 \ \\ _
If I \I\ II" \\ ‘ \.\
—o.2~r ,’ “" . "t :
—0.3[- -
‘0'4i a
—O.5 1 L 1 L
O 2 4 6 8 10
t
5

Figure 4.31. The time responses of the displacement $2 from Figure 4.11 with the interface
fixed for the linearized onemode model, the one-mode model, the ROM, and the original
model of substructure fl using the second-trial parameters.

 

 

 

 

 

 

 

 

0.6% ' —-- LNMLPrb, 1 DOF ~
- LNM, 1 DOF
- — NNM. 1 DOF
0,4 r- Original Model, 20 DOF H
II/ \ 3 I , _ § ‘ \ I, / \ \
02* // ,‘1/ \\ x/ \\ J
I” 1.] \ \I/ \l\
v— ,/ I \ f \ \ \‘
QXN o l" / ’ / \\ \ \\ _,
/ I \ / \ \
,/ x/ \ // \ \
_02P I, / I / \\ ’1’ \\‘\ \\ :
J4 I \~\> ‘ I ‘ — ‘1
—0.4 " .4
—O.6 r L l l l a
O 2 4 6 8 10
t

Figure 4.32. The time responses of the displacement xgl from Figure 4.11 with the interface
fixed for the linearized one-mode model, the one-mode model, the ROM, and the original
model of substructure ,3 using the second-trial parameters.

183

 

 

 

 

    

 

 

 

O ‘, ' ; '
—e— Fixed-Interface Linear Mode Shape at t = 0
_+_ Fixed-Interface Nonlinear Mode Shape at t: 0
—O 05 _ _O Fixed—Interface Linear Mode Shape at t = 0.125 Tm _
- , Fixed—Interface Nonlinear Mode Shape at t = 0.125 T"l
O Fixed-Interface Linear Mode Shape at t = 0.25 Tnl
—0.1 r ( Fixed-Interface Noninear Mode Shape at t = 0.25 Tn. J
—0.1 5 - _, g .
cn.x._ 0 Q4}? 956 0‘9 O—GQ-O 0—9- 0-0- O-GO
—o.2~ ‘ " l
\K\
—o.25 ~ *\~\ J
\‘K*.\
—0.3 " *‘fi‘w‘ *Hfi '
-—0.35 ‘ 4 t ‘
0 5 10 15 20 25

ith Degree of freedom

Figure 4.33. Comparison of deflections :L‘fU) for a quarter-period of motion on the first
fixed-interface linear mode manifold (linear eigenspace) and on the first fixed-interface
nonlinear mode manifold of substructure B using the second—trial parameters. The motion
starts at the maximum deflection (the bottom curve) and moves as shown to the zero

5

deflection at a quarter-period. Note that linear mode shape is normalized such that $1 is

’3 . .
the same as ar’l of the nonlmear mode shape at each time.

 

 

 

 

 

 

 

 

 

0.6 r —-— Fixed-Interface Linear Mode Shape 1 ~
— - Fixed—Interface Linear Mode Shape 2
Fixed—Interface Linear Mode Shape 3
0.4} Fixed-Interface Linear Mode Shape 4 .
/ a g , , - ’ '1'“
0.2 - , , ’ _ ~
/ / /' \
>‘x—O __ i I / . \ J
I / 7" \
—o.2 - ‘, , - ’ ” - _ j J
-O.4 r ~
-0.6 l- 1 l l l d
O 20 25

, 10 1 5
Im Degree of freedom

Figure 4.34. The first four fixed-interface LNMs of substructure 7.

184

 

 

600 800 1000 1200 1400 1600
nlh term

00

200

 

 

200

150’

 

-150-

—200

Figure 4.35. The coeflicients of the pqr cubic nonlinear terms expressed in fixed-interface
linear modal coordinates and associated with the first mode (master mode) of substructure

 

Na =

Figure 4.36. The contribution of the second fixed-interface linear mode amplitude 172

'7) to the first fixed-interface nonlinear mode manifold of substructure 7.

.N
37]]

185

 

Figure 4.37. The contribution of the third fixed-interface linear mode amplitude 17?,” =
X ; (niv’l, 779]”) to the first fixed-interface nonlinear mode manifold of substructure 7.

 

Figure 4.38. The contribution of the fourth fixed-interface linear mode amplitude nil/'7 =

XZ(n{V’l,17{V‘7) to the first fixed—interface nonlinear mode manifold of substructure 7.

186

 

 

 

 

 

 

 

 

— - LNMLPrb, 1 DOF
2" LNM. 1 DOF ~
- - NNM, 1 DOF
1.5 P Original Model, 20 DOF
\\ I \ 1" ~ / N
\ ‘ , I I T \ ‘l
1 ” \ \ ‘/ \\_ ’ C
\ I ./
\ \ f /
0.5- ‘ \ \ I / J
\ / I
)1. \ \ _’ /
z r- — \ \ I I _
c. 0 \ z \\ , I I/
\ / /\ /
—0.5— X ’ \ / ‘
I \~
L /" \ / I \\ //
-1 \ // \ ‘ _ I / \\ x z/ T
-1.5* 1
-2» 7
O 1 2 3 4 5
t

Figure 4.39. The time responses of the first fixed-interface mode displacement "{Vn for
the linearized one-mode model, the one-mode model, the ROM, and the original model of
substructure 7.

 

7 I

- — NLNM, 1 DOF
—- Original Model, 20 DOF H

 

 

 

 

0.1

0.05

 

 

 

>-
p—
1..
1-

 

Figure 4.40. The time responses of the second fixed-interface linear mode displacement
; 7
n3}7 2 X; (17fV ’7, 17?“) on the first fixed-interface nonlinear mode manifold for the ROM

and the original model of substructure 7.

187

 

 

 

 

 

 

 

 

 

0.04 1 fi
- — NLNM. 1 DOF
~—- Original Model. 20 DOF
0.03 t _
"\
0.02r \‘
0.01 i
>:
Z or) Ol-
1;:
-0.01 ~
—o.02~ \A '
—0.03 r a
—0.04 4 L ‘ g
0 1 2 3 4 5

Figure 4.41. The time responses of the third fixed—interface linear mode displacement

I)?” 2 X37 (niN’l, nil/‘7) on the first fixed-interface nonlinear mode manifold for the ROM
and the original model of substructure 7.

 

I

- — NLNM, 1 D]OF

 

 

 

 

0.015 r — Original Model, 20 DOF
r\ C: 3—3
\‘\ I’ I,
0.005 r ‘\ \ 1

N"!
4

O ..,_,u "s
c \ /
—0.005[ \\ \ n
-o.o1 l \..// .. / .

—0.015r 7

 

 

 

Figure 4.42. The time responses of the fourth fixed-interface linear mode displacement
; r . .

1):,“7 = X](niv’7, 7'19”) on the first fixed-interface nonlinear mode mamfold for the ROM

and the original model of substructure 7.

188

 

. Y -—I — LNMLPrb, 1 DOF
0_4L ‘ LNM. 1 DOF a
-‘- — NNM, 1 DOF
Original Model, 20 DOF
0.3 r

0.2 r , , , A \ a

I \\/ ,./ ‘\

I .\ \ /
0.1 L \/ \»\ ‘1
l ,, ‘x \
>$<N 0 I. / \. \ d
[I l x \
)l I \ \
-O.1 r- // \ \ X“ -1
/ \ \
// / \ ‘.\

—o.2 % I I $\ ,_ ./"<
-0.3 r 1
—o.4 - «

_O.5 1 1 m 1
0 1 2 3 4 5

0.5

 

 

 

 

 

 

 

 

Figure 4.43. The time responses of the displacement at; from Figure 4.11 with the interface
fixed ( x? (t) z 0 ) for the linearized one-mode model, the one-mode model, the ROM, and

the original model of substructure 7.

 

 

 

 

 

 

 

 

0.6 l- — ~ - LNMLPI’b, 1 DOF ”
LNM. 1 DOF
‘ - NNM, 1 DOF
0.4 ~ Original Model, 20 DOF s
I/' \\ I -' — , 1 l/r“ \\
0.2» /’ \‘ ’ I \ x // x. l _
I, / /\\ \ / \
I I x \l \
v— ) l l I \ \,
>‘>< 0 ’- 1 / / I \ / \ \ \'\ d
/ / \- / .\ l\
’ I \\ \ \
/ I l [I \ \ \\
—O.2 ’- "I I f \\ 1’ \ \ '\ z
4’ I \ J ‘ _ ,.\’ ./
—O.4t -
—O.6 l“ 4 J 1 _
0 1 2 3 4 5

Figure 4.44. The time responses of the displacement $31 from Figure 4.11 with the interface
fixed for the linearized one-mode model, the one-mode model, the ROM, and the original
model of substructure 7.

189

 

 

 

 

 

 

 

 

 

O \ '
l —-e— Fixed-Interface Linear Mode Shape at t = O
l -+ Fixed-Interface Nonlinear Mode Shape at t: O
—0.05 _ l O Fixed—Interface Linear Mode Shape at t = 0.125 Tn. ,
l , Fixed-Interface Nonlinear Mode Shape at t = 0.125 Tnl
\ O Fixed—Interface Linear Mode Shape at t = 0.25 Tm
—O.1 r \ Fixed-Interface Nonlinear Mode Shape at t = 0.25 Tnl r
—O.1 5 ~ \3 _Q a
fig- l @ ? TH? 0‘0“0‘0-G“O»e—o—e~o—e-o~e—o
.412? -
—O.25 r -
-413- *
-O.35 ‘ ‘ ‘ ‘
O 5 10 15 20 25

ith Degree of freedom

Figure 4.45. Comparison of deflections $170) for a quarter-period of motion on the first
fixed-interface linear mode manifold (linear eigenspace) and on the first fixed-interface
nonlinear mode manifold of substructure '7. The motion starts at the maximum deflection
(the bottom curve) and moves as shown to the zero deflection at a quarter-period.

 

 

 

 

 

 

 

 

2.5 7 I I I
'- - LCMSLbe, 3 DOF
2 " ' LCMS. 3 DOF ..,
' ’ NLCMS. 3 DOF
\ ’ f ‘ fl IA - \.
1’; 1‘, " “ll [\ Fu‘ 1 ”,4, , 1" i’\ 1,", -
\\ ‘4 I ;I “ll ‘ 1‘ ill! ‘. I’ll .: N l’ \ , ‘il
0.5 _ l 1‘ l \ l , )I \f l‘ l 1» ‘ y l‘ II I l.‘ , i “I , f? l. .i
l l ’I ll “ ‘ l I. \ l I i , $ ’ .
n. \ l l l ’ ‘\ I l‘ l [’4 \ ‘ 1 ‘l f ’ . I H I, | x
2 v- 0 ~ ' ,I l |\ z , l l x, l I ’ \\ l '7 \ .-4
:- \ I ’l l ‘l I i ‘ ‘ ‘ ‘l l" , ll I, ill I \ \
II \ I ‘ I l l l “ l g _ . ‘ . ‘ \
_0 5 ’- l. ( l x l \ l l' l I l ‘l I;‘ l A b I l‘ (r! a l K ll, '|-.i
'. (l \l l l A} \ ‘1' ‘ / '- ‘z‘ ’ \ l A} j
‘ (I ‘ r l 'I ’1 ‘ f,- I ‘1’ \ _’l' I -‘ I \ I l‘ , “j" ’ V \
—1~ . , , , ,Jg, ”1,, a ‘, , \j i, p v
—1 .5 r .
—2 d
—2 5 4 . . x
0 10 20 30 40 50
t

Figure 4.46. The time responses of the first synthesized fixed-interface mode displacement

179/fl for the three-DOF linear-CMS linear model, the three-DOF linear-CMS nonlinear
model, the three-DOF nonlinear-CMS model, and the forty-one-DOF linear-CMS nonlinear

model of substructure B.

190

 

0.05

 

 

 

 

— - NLEMS. 3 DOF
—— LCMS, 41 DOF

 

 

 

 

 

 

 

0 10

Figure 4.47. The time responses of the second synthesized fixed-interface linear mode
displacement név‘fi for the three-DOF nonlinear—CMS model and the forty-one—DOF linear-

CMS nonlinear model of substructure fi.

30

4O

50

 

0.04 1
- - NLCMS. 3 DOF
—— LCMS. 41 DOF
0.03 h '-
0.02 ' -

—0.02

—-0.03

—0.04

i
i

 

. l
ll
(3:030- ll
-o.o1{ W K Mr} :

 

 

 

 

l

1

 

 

0

1O 20

3O

4O

50

Figure 4.48. The time responses of the third synthesized fixed-interface linear mode dis-

placement név’fi for the three-DOF nonlinear-CMS model and the forty-one-DOF linear-
CMS nonlinear model of substructure 5.

191

 

0.015 . r ,

 

- - NLéMS. 3 DOF
—— LCMS. 41 DOF

0.01 n [l .’ ~

 

 

 

0.005 [\f

NB

_1

-o oos~ l '[
-ool V V .. -

—o.01 5 . . g %

 

 

 

Figure 4.49. The time responses of the fourth synthesized fixed-interface linear mode

displacement niv’fi for the three-DOF nonlinear-CMS model and the forty-one—DOF linear-
CMS nonlinear model of substructure i3-

 

 

 

 

 

 

 

 

— - LCMSLPf‘b. 3 DOF
2 * LCMS. 3 DOF t
'— NLCMS. 3 DOF
1-5t LCMS. 41 DOF g
l‘ \ / f
1»\ ’1‘ i, ,x ’l ’8 I,“ A Mn
'. I y l ‘ f ' l- \ I ’ \\ I \
il i ll | / I x‘ [ , l I‘ ‘ ‘ I \
. I ‘ \l ‘I i 'I“ \ ‘ l \1\ j ‘ \ J
0-55‘ ’ I" ,4 i g l ‘ l l I! ‘ ‘ ' l ' ' (l l .
\ ’ ‘“ I I H " 1" l ‘I’' ‘ U \‘ I l ' ’H I
>: \ l l . '- l ‘l \ , 'l~ \. ' \l (
:- \ l \ i \ l I, l l‘ l 'p \ l l y ’ 4 ;
I l | , l ' l l l l l ‘ r ,
-0.5 '— \ I \\ I l \ l l, l l \ 4' l if I‘ll I ‘ l.,
‘ I l \ "l l H I i I ‘ I
\ \ I \ \ l I y ‘ l ‘ .
L ‘ I ( I ‘ ‘ \ j \ \/ \_ I \l I \VI ‘1‘,
—1 \’ ll 4 “I U ‘W \’ {LN
-1.5F fl
_2>- _‘
0 5 10 15 20 25
t

Fi ure 4.50. The time responses of the first synthesized fixed-interface mode displacement

r); ’7 for the three-DOF linear-CMS linear model, the three—DOF linear-CMS nonlinear
model, the three-DOF nonlinear—CMS model, and the forty-one-DOF linear-CMS nonlinear
model of substructure '7'.

192

 

 

0.2 r t — - NLCIMS.3DOF ‘
—-—- LCMS. 41 DOF

 

 

 

 

 

 

 

Figure 4.51. The time responses of the second synthesized fixed-interface linear mode

displacement 779/” for the three-DOF nonlinear-CMS model and the forty-one-DOF linear-
CMS nonlinear model of substructure 7.

 

0.04

 

— - NLCrMS. a DOF
w— LCMS. 41 DOF

All '.

 

 

 

0.03

0.02

0.01

N91

—0.01

f _.__I___...-—+—r r

M‘

‘2.
~52:
v}
1"

‘7'

-o.02L ‘

-0.03 -

 

 

 

—0.04

Figure 4.52. The time responses of the third synthesized fixed-interface linear mode dis-

placement 179’” for the three-DOF nonlinear-CMS model and the forty-one-DOF linear-
CMS nonlinear model of substructure 7'.

193

 

0.015 . . 1

 

- - NLéMs. 3 DOF
—— LCMS. 41 DOF

 

 

 

0.01 l '

0.005 ‘ ‘

My
114
o
Kr...
"/

 

 

 

 

-o.oos ~ l
-o.o1 - l4 ~'
“0'0150 5 1o 15 20 25

Figure 4.53. The time responses of the fourth synthesized fixed-interface linear mode

displacement n?” for the three-DOF nonlinear-CMS model and the forty-one-DOF linear-
CMS nonlinear model of substructure 'y.

 

 

 

 

 

 

 

 

- - LCMSLPrb, 3 DOF
0.06 ~ LCMS, 3 DOF W
a- - NLCMS, 3 DOF
O 04 , Original Model, 41 DOF
. A 1 1 1‘ ' I \ D , .\

ll ' \l‘ l . {l r" ‘3 PE ll fl P 1"; [la-I rlll [A (m (ill, Ill-.1 If
0.02". '1 11. [l 1" l‘ .13 ll '? 1" .l '1 '1‘! a? ',‘ 3" 5' l, Ml. IT
I 1’ '1 .’ l f l : l ,.' i j l f l I} l ,' 3 l l‘ ‘4 ll 1. I' If” “ l, “ l' "'I
-. 1'. 91 n1; 111. s 1*{“1H.11H‘:.tl".l'w
— o~"’i=ifi'!'i!;i s!ir-mu..."!.w".,«"..1"~u'u
" 1’ «w fluvial-1:: szwzwwm"w
* l l I l l l ’ | " l i i l l ”I i 'I ’5 L “ '1 ‘ u ‘ l l l. ‘- l' l l‘

‘1 ~: I' 'l *‘ " ‘r’l‘l’l'l‘m‘ L4! 'fll'li-‘J

' 1 l I l ‘ , | .3 P 1 - J l ‘l l l. l il l 1‘ ‘I ‘l
-002 _ 3‘ ‘1 l l l 1' l l, l. 1 '1 l .., 'l 3‘ 1" l {:1 l" l! '3 11“ 5 1‘ u ‘11" ‘11:" “14‘ a

if “v >4 if v a" u" 4 M ‘3’ “1’ ll ‘. ’ iv” ‘o"
-0.04~ ‘
-0.06>— 1

O 5 1O 15 20 25
t

Figure 4.54. The time response of the displacement 11:1 of mass m1 from Figure 4.10, which

corresponds to x?,xlfi, and :13? from Figure 4.11, for the three-DOF linear-CMS linear
model, the three-DOF linear-CMS nonlinear model, the three—DOF nonlinear-CMS model,
and the forty-oneeDOF original model.

194

 

 

T

 

 

 

 

 

 

0.5 ' - - LCMSLPrb. 3 DOF
LCMS. 3 DOF
0.4 *‘ - - NLCMS. 3 DOF H
O 3T Original Model. 41 DOF
0,2 I‘ . \ I A \ / I/ \ r _\ / f
I I ’ I I‘ I ’l‘ \ ‘ 1‘ I, \ 3’ , I
O 1 P I H \\ I ,I' " I. ,l \l‘ I/ ll ‘9‘ \\ ’I ‘ \ I, \ I1
' I I l \ , ‘ I 1 I‘ ’ ‘l I I I ‘ ‘I t \ I l I ’
\ I , II ‘ l ' I‘ ’\ . ' l ‘ ‘ .
N O P l I ‘ I I l I I u I ‘ -‘ II \ \ [ \ .I I
x "I I " I ' ‘ ’ ‘-’ I" V“ I I, ‘ ‘ ‘ l ‘1 l: I ’ 'H
I I .' l I l ‘1' I,‘ I ' '\ I \ ‘ f \ , II ’A
-01 I." I " ‘ \I I I I l .7 \l I! I, ‘ I] ‘ l / l I! ‘ ‘ I "
J ’ x ’ " I x ,I' "x. I ‘w ‘ ‘\ \, \ I ,2
“0.2 F, \ , ’ \ x, \.\ I \ I _,
_O.3 *- .4
"'O.4L -1
—o.5 » . 1 A A .
0 1O 20 30 40 50
t

Figure 4.55. The time response of the displacement 1:2 of mass m2 from Figure 4.10,

from Figure 4.11, for the three-DOF linear-CMS linear model, the
three-DOF linear-CMS nonlinear model, the three-DOF nonlinear-CMS model, and the

which corresponds to a:

forty-one-DOF original model.

 

 

 

 

 

 

 

 

— - LCMSLPrb. 3 DOF
0.5 * LCMS. 3 DOF ‘
—- - NLCMS. 3 DOF
(Ml Original Model, 41 DOF .
0.3 ~ -
1.4 ,‘1 J ’1“ ’-

\ I ‘ ‘ 1 '~ ’ \ I ’ - .
0.2 " 1., ’ i 1‘ I’ K. 'I ‘ I ‘ I ‘, I ‘\ ‘\ v\ 'H

I \ I ‘ , J l I
0 1 _ I ’. 1 I ‘ ’ \‘l ' ‘I ‘1 'l l \ ' | I I
. I \f I I I y 3 l I \ I | I
I I \ ‘I I I. , , \\ , ’I. 1 .1 ‘ ‘ I ' I
& O_I, ‘ ’\ ‘ " I ll ’ \l ’ 1‘ I” I " l ‘I .1 ' I"
x ’ l I ‘ I I , f I \ I ' ‘ ‘ I 5 5 ', I I
I ’ I ‘ \ I H I j I ‘ I ‘ l 1 'I
-0.1_” I ll! ’l l ‘ I l I \l ,—

II ’ \ ' 5 A ' .‘ “ I" I ‘ " h 1 ‘1 I, I I‘ I

, , C ' \ I l 1 ‘ I I \ I .
—o-2 I \ /‘\ ‘ I “\I / ‘J \ l I" \,’r \ ,I ‘ ’ -<

4. / I f A " I ’

\ U \ \
-O.3 r ‘
—OI4 _ -
—O.5 * ‘
I l I l
0 1O 20 3O 4O 50
t

Figure 4.56. The time response of the displacement 3:21 of mass "121 from Figure 4.10,

which corresponds to 33% from Figure 4.11, for the three-DOF linear-CMS linear model,
the three-DOF linear-CMS nonlinear model, the three-DOF nonlinear-CMS model, and the

forty—one—DOF original model.

195

 

 

 

 

 

 

 

 

o 4. - - LCMSLPrb. 3 DOF I
LCMS, 3 DOF
-‘ - NLCMS, 3 DOF p
0'3 b Original Model, 41 DOF
_ ‘N I /. -_‘> '\-‘
0.2 I, I \ \ A U‘ I \\ 1“ I \\ f I \ / .I
I. \I \ I r . _l \ r I ‘ I ‘I‘\ V ‘ \ ’N
' ' I ’ ’ ‘ ' \ I ‘ A l I \ , f l l
01—” “‘~ ~* M,‘ "I, u, u, 1,,
. . [I \ I 1 | ‘ I I I II ‘I I .I I ll ‘ II I
. I I ‘l ‘ I I .I I h I ‘ l I, . 1 I I ‘ II \
I I I ‘ I. I I l, I I ' I4 I I I, i .
g . I l I l I l 1‘ I I f I . ' I I l I; l
0 — ’ l , I - .I ' ’ I I ‘ I H
X I I s i H l ' . '\ I \‘ I I \ I l

I I ‘7 l I. I I II I I I I I I I. . I ‘

I I , I‘ I II l. Io I II 5 II I I‘ l I I. I \ I‘ II 5.
‘0'1 "7 I i I ‘ ‘ III " A 7 ‘I- I ’ I1 I n‘ I '~ ‘ i i- '. ’I 4+

"I \ I \ \I ,, i I “I I ‘1' I.‘ I" \I \ \II

I I l ‘ I - I k ' \ . \ /

I K \ ‘ l \ I] i I J \\,/l 1 ‘ L
-0.2I7 \ I \I I \ I \‘l' \ ‘1 _4
-0-3I I
—o.4L _

O 5 1O 15 20 25
t

Figure 4.57. The time response of the displacement 3:22 of mass mm from Figure 4.10,
which corresponds to at; from Figure 4.11, for the three-DOF linear-CMS linear model, the
three-DOF linear-CMS nonlinear model, the three-DOF nonlinear-CMS model, and the
forty-one-DOF original model.

 

J

J

 

 

 

 

 

 

 

0.5 r- - - LCMSLPrb, 3 DOF F
LCMS. 3 DOF
0.4 ~ - - NLCMS, 3 DOF ~
Original Model. 41 DOF
. -J
O . 3 r- \ I" / I~ I‘ ! {‘3 F & I‘r‘
02L- " ’ \ III yo‘ ‘i IfiI‘I II“ ”II III" \ (II III II
' ,, "I l i f ’. I ' ‘ l - \ ‘. I 4
I I I ‘ I I ' I! l I ‘ I Q ' ‘ ‘ I
II I‘ \ yr 1 I l ,‘ I .‘ I I ‘ I \ I I ‘ I, In \‘ {I I
O.1~ . ,. I g , x | , e I , ,. \. I I l: l ,3 r
I ‘I l '1' i ‘ I . I « ‘ ' I ' I l I I ‘, I I f l
._ ,~ . . ,, . \ z I I I \ t I . . I '4 ,
V o I' I I I I ' I. ‘ I I l\ 1 I ' l l I I ‘ I‘_ l II LI
>< ‘ \ l I \ \ l I , l I \ I I i I I
,I I 1. ,I‘ l I! . \l I I V II , ‘ l \I 'I’ ‘ ‘I‘ 3‘ I: ‘
"0-1 ”I I ‘ \ ‘. I l I I ll ‘ ,' l I \I i . 1 ‘ “a, I . 1
h I . 1 I, V. : I I l I l I ‘ I
— HI, ‘ I I‘ J I I l $ l. I \ I , ‘ vI- _
0.2} I) \ gyI 1‘ _ \\_l " ‘ \I ‘I v] ‘I \l \
i . \ ‘1 k‘ l_- ll \’ l; ‘I
J . 4
—O.3 r _
-0.4 r ~
—O.5 * -*
L 1 1 l
0 5 10 1 5 20 25

Figure 4.58. The time response of the displacement 11:41 of mass m4] from Figure 4.10,
which corresponds to 11:31 from Figure 4.11, for the three—DOF linear-CMS linear model,
the three-DOF linear-CMS nonlinear model, the three-DOF nonlinear-CMS model, and the
forty-one-DOF original model.

196

 

 

— — NLICMS, 3 DOF
1.5— — LCMS. 41 DOF

 

 

 

 

 

I
.A
0‘

 

 

O 1 O 20 3O 4O 50

Figure 4.59. The time responses of the first synthesized fixed-interface mode displacement
77;”) for the three-DOF nonlinear-CMS model and the forty-one-DOF linear-CMS nonlinear

model of substructure 5 with niv‘fim) = 1.3, ”{mm = 1.3, 170(0) = 0.04, 7796(0) =
.N, .
0, 721 7(0) = 0, Wm = 0.

 

 

 

 

 

   

 

 

 

- — NLéMS. 3 DOF
1.5~ -—~— LCMS, 41 DOF
f l
1 , '
\
>- \
2 v- O*-\ \ ‘
l \
—o.5r \‘
‘ \
_1_ J
—1.5~ 1
o 5 1o 15 20 25

Figure 4.60. The time responses of the first synthesized fixed-interface mode displacement

nix/‘7 for the three-DOF nonlinear-CMS model and the forty-one-DOF linear-CMS nonlinear

model of substructure 'y with niv’flm) = 1.3, ni’vq(0) = 1.3, 77C(O) = 0.04, 77?,”8(O) =

0, 7'2{V”(0) = 0. THO) = 0.

197

4.9 Appendix 4

The component forms of the mass and stiffness matrices and the nonlinear force vector of

the class of spring-mass systems from section 4.5 are given by

 

m- if j = 2'
7710' "—2 2 (4.69) .1
0 otherwise '

198

 

(Na-1)
k1 + Z k(23Nrnass+2) if i: 1’ j: 1
3:0
—k(2-9Nmass+2) If 2: 1, j : SNma33 + 2,

S : 0,1,2,...,(Na *1)

—k(231lvfll(188) If I: S Nrnass +1, J :2"1,
3 :1,2,...,Na
k(291v'l7l(188) if 7: 2 S anass + 1, j Z 7;,
S :1,2,...,Na
_k(2sifvn1088+2) if I = S NTTIOSS + 2, j = 1,

s =1,2,...,(Na,— 1)

k(2anmss+2) + k(2staSS+4) if i = SNmass + 2’ j = i,
s = 1,2, ..., (Na — 1)
(4.70)
412an....” +4, if i = s Nnms + 2, j = 2' + 1,
s =1,2,...,(Na — 1)
_k-‘(2z'—2) if (2 g 2' S Nnmss)
or (s Nmass + 3 S i S (s + 1) Nmass),
j=i— 1, s: 1,2,...,(Na— 1)
k(2,_2, + km, if (2 S i S Nmass)
0r (8 Nmass + 3 S 2' S (8 + 1) Nmass),
j: 2', s =1,2,...,(Na — 1)
—k'(2,-) if (2 S i _<_ Nmass)
or (s Nmass + 3 gig (8 +1)N7TIGSS)7
j =i+1, s =1,2,...,(Na — 1)

0 otherwise

199

f (NW—1)

4 ,. _ 3
— Z: A(2 8 N77103:; +3) (’L(«5' 4N‘TI‘L(£.S’8+2) $1)
320

if i: 1

[C(23Nmass+1)($i - $(i—1))3
if i : SNmass +1, 5 =1,2,...,Na

”2‘ MID

, 3 3
k(231\77nass+3)(xi — I1) — k(2sta33+5)(~T(i+l) — 12,)
if l28N771088+23 S 21,2,...,(Na_1)

k(2i—1)($i — 1*(i—1))3 “ k(2i+1)($(i+1) _ $223

if i = 2, “'3 Nmass, 01' i = SNmass + 3, ..., (3 +1)Nmass,

 

s = 1,2, ..., (Na — 1)

L

for i,j I 1, 2, 3, ..., (Ara [\r'NULSS +1).

The component forms of the mass and stiffness matrices, the nonlinear force vector, and

the reaction force vector for substructures S are given by

0 fiizszl
my: 7% iszi M73

0 otherwise

200

 

 

 

1:25 if 2f: 1, j =1
—k§ if i: 1, j =2
_kig2Nmass) if i : Nmass + 1’ j : i - 1
k5 nizN..+Lj=i
1?} = 1 (”ma“) (4.73)
kéi—Q) + kg?) if (2 S 1 S Nrnass), j :2
‘kén if (2 S i S Afmass): .7: i +1
k 0 otherwise
r
—k§(1‘§— 1‘?)3 if i=1
5' ' . .
{12' = klséNmass-l-l)(rig _ xii-UP 1f 1 = Nmass + 1 (4'74)
.5 ,.S S 3 S .S' S 3 ' - __
\ A‘(2i—1)(lz’ — 10.4)) - k(22‘+1)(l(i+1)— 51:2.) 1f 2 —— 2, ..., Nmass
( F5" 1f 2:
J
f? = (4.75)
0 otherwise
for 77,1 -_—. 1, 2, 3 (NW, + 1), and s = a, 7, 5, . Note that mff,’ :-
. 1’3 __ , .13 _ f3 _ ,. .13 _ ., 5 ...
mg, 7713 — m3, mleassd'l) —— m(2’Vnm.ss+l)’ k2 — k2, k3 —— k3, ..., k(2Nmass+l) _

4 '7 _ .7 _ ,. '7 _
A’(2N7nass+1)’ "L2 — "2(Nmass+2)’ m3 _ mmeass—f3)’ m’(N-mass+1) _ ”1(2Nma33‘f1)’

7__,
3_.

.7 _ , . ,.7 _ ,
k2 *— k(2Nmass+2)‘ A“ k(2Nm(133+3)’ "" k(2Nmass+l) _ k(4Nmass+1)’ and SO on.

The coefficients of the pqr cubic nonlinear terms expressed in terms of fixed-interface linear

modal coordinates and associated with the ith mode of substructures S are given by

201

 

 

bfV,S

ipqr
N,S N,S N,S
N,S.xr __ bipqr + biprq + birpq
I‘M" _ N,S N,S N,S
bipqr + biqpr + biqrp
N,S N,S N,S N,S N,S N,S
bipq’r + biprq + biqp'r + biqrp + birpq + birqp
N,S ‘
where bipqr are
N S (Ninass+1) N S N S
ipfzr Z (phi, bh-Pb"
h=2
for S = 13, 7, 6, ...,

where (1)295 is the hi component of the modal matrix defined in section 4.3.2.1, and b

are given by

202

if p = q = 7‘
if p = q
(4.76)
if q = r
otherwise,
(4.77)

N,S
hpqr

7N.S

h pqr —

The coefficients of the pqr cubic nonlinear terms expressed in terms of the linear

N,S N,S .N,S

f
‘5 .N,S ,N.S N,S 5 «NS .N,S [NS .3 -
[A C93p @311 993,, + 3 k5 $31) Cb3q CZ)21'

3 02 ¢2q ¢2r — 4‘5
.S .N.S N. .N,S _S (N,S N,S N,S
-3 A5 993,, go'Zq (927' + [‘5 ¢2p cb2q d)27‘

if h=2

N,S N,S N,S

,3 .N,S .N,S N,S ,5
[k )Cbhp Cbhq Qbhr —3A'(2h—1)¢hp Cbhq ¢(h—1)1‘

(2h—l
.S .N,S .N,S .N.5 _ ,S .N,S N,S N,S
+3 A'(2h-—l) Cohp ¢(h—l)q GUI—UT A(2h—1) <D(h—-1)p ¢(h--l)q ¢(h—l)r
_ S' .N.S NS N5 ,5 .N,S N,S .N,S
A(2h+1) $(h+1)p ¢(}z+l)q $(h+l)r + 3 A(2h+l) ¢(h+1)p ¢(h+1)q ¢hr
.8 INS 1N5 NS _5 WES .N,S N,S
-3 A'(2h+l) d)(h+l)p C{Dhq (9hr + A(2h+l) th whq (ph'r ]

. __ r f
1f (L — 334309U'9A771088

(S .N,S (N,S ”N,S _ ‘S N,S -N,S .N,S
[kwNnmsS'l'U 97);”, @hq 0hr A(2Nmass+1) ¢hP 9bhq $(h-1)7‘
..S' N,S N,S N,S _ ,5 N,S N,S
+3 A(2A"nlass+l) (bill) <b(h-—-1)q ¢(}l—1)7‘ (2Nnu133-i-1) ¢(h—1)p $(h—l

 

L if h : jV'IILUSS + 1

N,S

(h—1)r

modal coordinates and associated with the ith mode of substructures S are given by

w here ()5

S

1W!“ If p Z q : r
S S S . _
5.* __ bipqr + biprq + birpq If P — q
ipqr — S S S . —
bipqr + biqp'r + biqrp If q — T
S S S S S 5 .- .,
bipqr + biprq + biqpr + biqrp + bz'rpq + bin”) OtherWlbea
zpqr are

203

(4.79)

l

(4.78)

(Nmass+1)

 

S __ S S
bipqr '" Z whi bhpqr (4'80)
h=1
for S = ,3, 7, 6
where “Ki",S, is the hi component of the modal matrix defined 1n section 4. 3. 2, and bhpqr are
given by
[41.5 111.59,, 1:15,, 111,: + 3 k5 115,, 11125,, 11115,.

-3 ’13 32,. v1.1 1’151~+k3 v15 11% 151.]
if h = 1

S ,S S s s ,s s ,5

(”7211— 1) ‘I’hp 1th ‘I’hr " 3 k(2h— 1) Mp “(I111 111(11— 1)r

.1 ;,S ,S 1,5 _ .5 IS I,S
+3 k(2h—1) ‘I‘hp 701—1).) If(I1—1)r k(2h—1) lbw—1);; II’(I1—1)q‘P(I1—1)1~
S = S {'S [S ,S .S ,S
bhpqr i k(2h+l)1*(h+1)p 2*(h+1)qz*/’(h+1)r_+_3 k(2h+1) ¢(I1+1)p ’(Sh+1)q 1(’hr (4'81)
5 . , ,3 .s .

3 "(2I1+1) 1*(h+1)p ‘(Izq 15hr + A(2h+1) ‘(hp ‘(hq V’hr

if h = 29 3? 4’ ..., ‘N’,l(l843

.8 1,5 ,5 S _ .5 , s ,5 ,s

[k(21vn.a..s+1) 3),, 11),, 11hr 3 h<2Nmass+11 f’hp f'h-q ‘(m—nr

s s ,3 _ s 1,5 ,5 ,3
+3 k(2Nmass+1) ‘(Iw («h—1)., ”(h—w k(2Nmass+1) zNIH») fez—1m w(h—1)r]
it If h 3: Nynass + 1

 

204

 

CHAPTER 5

Conclusions and Future Works

Within this work techniques are developed that can be used to construct reduced order
models (ROMS) of nonlinear structural systems. The techniques are numerical in nature
and are all based on invariant manifold-based nonlinear normal modes. The methods consist
of NNM reduction from large-scale models, including those generated by nonlinear FE
analysis, and a component mode synthesis (CMS) technique that incorporates these models
for substructures. This latter method allows one to use NNM ROMS of substructures to
build ROMS of the combined structure through a CMS technique. The contributions made

in these subjects are summarized as follows:

0 In Chapter 2, two new methods to numerically solve for single NN M invariant man-
ifolds are developed. In both of these methods the manifold is parameterized by a
modal master displacement and velocity pair and the resulting invariant manifold
equations are solved using Galerkin approaches. The first method solves for the man-
ifold using global polynomial basis functions over the entire domain of interest. For
the second method, the domain of interest is subdivided into small pieces and the
manifold is solved using low-degree polynomial basis functions over these sub do-
mains, which are then pieced together. When compared with similar methods that
utilize amplitude and phase variables (Pesheck et al., [33]), the new methods are

found to be superior in terms of computational time but inferior in terms of accuracy.

0 111 Chapter 3 a specific application is considered; a N NM ROM of a. rotating beam is

205

 

developed from a nonlinear finite element formulation, wherein the invariant manifold
equations are numerically solved using the collocation method. The NNM formula-
tion allows one to systematically account for the nonlinear coupling effects between
axial and transverse motions, which are the source of slow modal convergence for this
type of problem when attacked by conventional approaches. The results demonstrate
promise for the field of nonlinear structural dynamics, since they combine the versa-
tility of the finite element method with the accurate NNM model reduction technique.
However, there is a trade-off between acquiring the accurate ROM and the computa-
tional effort involved, since solving for a single-NN M invariant manifold with many

retained linear modes requires significant computational time.

o In Chapter 4, a model reduction technique is develOped that combines numerically
generated NNMS and the CMS technique. This allows one to build ROMS at the
substructure level and assemble these to formulate a ROM of the assembled structure.
The fixed-interface linear CMS technique of Craig and Bampton ([62]) is extended to
nonlinear structures by making use of fixed-interface NNMS in place of fixed-interface
linear normal modes. This approach is suitable for systems composed of assemblies
of substructures, and it allows one to build nonlinear models with significant model
reduction and improved accuracy. By this approach, engineers can independently
design each nonlinear substructure and then build an approximate multi-mode model
of a complex structure that is the assembly of these substructures. This approach is
much less computationally intensive than a brute force approach that directly uses the
multi-mode models based on the multi-N NM invariant-manifold approach described
in sections 2.1.2 and 2.2.2 ([42]). This is especially true when the number of retained
linear modes of the assembled structure is large. However, the proposed nonlinear
CMS approach generally provides an accurate model only when the couplings between

substructures are weak.

The methods developed in this dissertation are numerical techniques geared toward simple

structural elements and systems that are composed of assemblies of substructures. Much

206

work in this field remains to be done. Some ideas that require further investigation, using

both analytical and numerical approaches, along this line of work include the following:

0 An instability of single N N M manifolds in terms of vibration amplitude is suspected
to exist in several of the systems considered. This suspicion is based on the fact that
when the amplitude of the master mode of interest goes beyond a certain point, the
family of trajectories on the exact single-N NM manifold is very difficult to numerically
find using the shooting method. This causes the time response of the original model
to have two components: a component having the nonlinear natural frequency of
interest and a component having a high frequency. This indicates that the NNM
of interest can not be uncoupled from the other NNMS. An analytical approach to
this issue is possible only if the original model size in terms of modes is limited to 2
or 3 degrees of freedom. A numerical approach using a simulation based variational
method and Floquet theory could address this problem for large—scale problems. Such
investigations would provide an estimate of the amplitude range of validity of the
NNMS. If known analytically, such information could be used as a guide to the
computational generation of the N N Ms, since one would know a priori the range over
which the formulation is valid. The works of Rosenberg and Atkinson ([6]), Rand et
al. ([15]), and Vakakis et al. ([19]) would be good starting points for guiding such an

investigation.

0 The continuation of a single N NM manifold in terms of a system parameter could be
studied by using the “suspension trick” ([89]), wherein the parameter(s) of interest is
(are) included as dynamics states (with trivial dynamics). Using such an approach,
the NNM of interest will be of higher dimension (2M + p, where 1V! is the number
of NN1\'ls and p is the number of parameters), and will depend on the parameters of
interest. Galerkin-type methods can be used to solve for the parameterized family of
NNM manifolds, and would provide useful information that can be used in the design
process of structural systems. It is unclear, however, that such an approach would be
computationally more efficient than simply generating the family of NNM manifolds

by direct con‘iputation over the parameter range of interest.

207

 

c As mentioned in Chapter 2, the drawbacks of using 11 — v coordinates might be
eliminated by first applying the invertible van der Pol transformation ( [22]) to the
master u — v coordinates. This would result in a new rectangular coordinate system
which rotates at the linear natural frequency of the master mode. In this manner it
would mimic more closely the amplitude and phase formulation, since the variables
would be varying on a slower time scale. In addition, this method uncouples the u and
v equations from one another up to linear order, which may simplify the formulation

of the NNM manifold equations.

It is well known that for systems composed of substructures, if the finite element
mesh of the substructures is refined, the number of generalized constraint coordinates
increases accordingly. This can result in a cumbersome CMS model in which there
are many more constraint modes than substructure modes. For linear structural
systems, Castenier et al. ([91]) have developed an effective technique to reduce the
number of constraint coordinates using the concept of “linear constraint modes”. It
may be possible to develop a similar technique for nonlinear structural systems using

invariant manifolds as the basis for the reduction of the constraint coordinates.

An approximate inulti-mode model of a system possessing an internal resonance can
be developed through the NLCMS technique. This idea can be explored by subdivid-
ing the whole structural system of interest and then synthesizing the substructures
using the NLCMS technique in such a way that the modes participating in the inter-
nal resonance are incorporated in the synthesized model. This idea is appealing due
to the large computational efforts required for attacking such problems directly using

the multi—NNM approach described in sections 2.1.2 and 2.2.2 ([42]).

When the component modes are obtained from modal testing, or when an experimen-
tal verification of the component modes is required, the free-interface CMS methods
are more attractive than the fixed-interface CMS methods. Therefore, it is worthwhile
exploring the idea of extending the free-interface CMS technique developed by Craig
and Chang ([66], [67] and [63]) to nonlinear structures by making use of free-interface

NNMS in place of free-interface linear normal modes (LNMs). The extension of the

208

 

l .I'
f
.

 

 

method of assembly to the nonlinear case appears to be challenging and remains for

future work.

 

 

 

209

BIBLIOGRAPHY

210

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