Aeroelastic Analysis of Super Long Cable …stoa.usp.br/osvaldonakao/files/2970/16498/Thesis.pdfAeroelastic Analysis of Super Long Cable-Supported Bridges Zhang, Xin SCHOOL OF CIVIL

Aeroelastic Analysis of

Super Long Cable-Supported Bridges

Zhang, Xin

SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING

NANYANG TECHNOLOGICAL UNIVERSITY

2003

Aeroelastic Analysis of

Super Long Cable-Supported Bridges

Zhang, Xin

SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING

A Thesis Submitted to Nanyang Technological University

in Fulfilment for the Degree of

Doctor of Philosophy

2003

CONTENTS

Acknowledgement i

Abstract ii

List of Tables iii

List of Figures iv

Nomenclature vi

Chapter 1 Introduction 1

1.1 Long-Span Bridges 1

1.2 Motivation for the Study 2

1.3 Organization 3

Chapter 2 Aeroelasticity and Aerodynamics of Bridge Decks 4

2.1 Introduction 5

2.2 Thin Airfoil Aeroelasticity 11

2.3 Aeroelastic and Aerodynamic Forces on

Long-Span Cable-Supported Bridge Decks 14

2.3.1 Formulation of the Self-Excited Forces 15

2.3.2 Buffeting Forces 20

2.4 Analytical Method in Frequency Domain 22

2.4.1 Analytical Method for Flutter Analysis 22

2.4.2 Governing Equations of Flutter 25

2.4.3 Buffeting Analysis 29

Summary 30

Chapter 3 Wind Tunnel Experiment to Extract Flutter Derivatives 31

3.1 Introduction 32

3.1.1 Similitude in the Experiment 33

3.1.2 Other Model Types 37

3.2 Extraction of Flutter Derivatives 37

3.3 The Experiment 40

3.3.1 The Wind Tunnel 40

3.3.2 Sectional Models 41

3.3.3 The Experimental Setup 44

3.3.4 The Experimental Procedure 46

3.3.5 Calibration 47

3.4 Basic Measurements 48

Summary 51

Chapter 4 Method Used to Identify Flutter Derivatives 52

4.1 Introduction 53

4.2 Basics of ERA 53

4.2.1 History of ERA 53

4.2.2 The Method of ERA 54

Summary 60

Chapter 5 Experimental Detection of Nonlinearity in

Self-Excited Forces 61

5.1 Introduction 62

5.2 Relative Amplitude Effect 63

5.3 Physical Significance of Flutter Derivatives with

Different Relative Amplitudes 65

5.4 Use of Output Covariance as Markov Parameters 67

5.5 Numerical Considerations for the Computation of

Output Covariance 68

5.6 The Experiment 69

5.7 Results and Discussion 70

Summary 79

Chapter 6 Numerical Flutter Analysis 80

6.1 Introduction 81

6.2 The Suspension Bridge and Modeling 83

6.3 Method to Solve the Aeroelastically Influenced

Eigenvalue Problem 89

6.4 Approximating the Impedance Matrix 90

6.5 Description of the Analysis 93

6.5.1 Analytical Cases 94

6.5.2 Effect of Relative Amplitude 97

6.5.3 Effect of Lateral Flutter Derivatives 98

Summary 99

Chapter 7 Time Domain Formulation of Self-Excited Forces

on Bridge Decks for Wind Tunnel Experiments 104

7.1 Introduction 105

7.2 Relative Amplitude Effect on the Transformation of

Flutter Derivative Model to Time Domain 107

7.3 State Space Model for SEF Generation System 108

7.3.1 The Model 108

7.3.2 Relation to Flutter Derivative Model 113

7.3.3 The Transformation in Modal Coordinates 116

7.4 Suggestions for Future Experiments 117

Summary 119

Chapter 8 Errors in the Identification of Flutter Derivatives 121

8.1 Introduction 122

8.1.1 Errors Due to Non-White Noise 122

8.1.2 Errors Due to Nonlinearity in the Self-Excited Forces 124

8.2 Evaluation Based on Block Hankel Matrix 125

8.3 Data From Experiment 129

Summary 132

Chapter 9 Conclusions and Future Work 133

9.1 Conclusions 133

9.2 Suggestions for Future Work 137

Appendix I 138

Appendix II 139

Reference 140

i

Acknowledgements

After going through almost three years of hard work it is time to thank all those who

have pulled me through this period and made my stay at NTU a pleasant one.

I would like to express my sincere gratitude and thanks to Prof. James Brownjohn

for his invaluable guidance and moral support.

My special thanks go to Dr. Piotr Omenzetter for the inspiring discussions and

valuable suggestions.

I take this opportunity to thank Mr. Tay Lye Chuan for the help in operating the

wind tunnel and setting up the experimental devices. My Thanks also go to Mr.

Phua Kok Soon for the help in manufacturing the sectional model and suspension

system.

My special gratitude is due to all my friends for making my time spent at NTU an

unforgettable memory.

I would like to thank the School of Civil and Environmental Engineering for the full

financial support and the research facilities they provided during my study.

ii

Abstract

A study on properties of interactive wind forces on bridge sectional models is

presented in this thesis. Two and three-dimensional sectional model tests in the

wind tunnel were carried out to detect nonlinearity in the self-excited wind forces.

The transformation of a frequency-time domain hybrid flutter derivative model to

either time or frequency domain usually requires the linearity assumption of the

self-excited wind forces, which has not been investigated thoroughly.

The self-excited wind forces on a bridge deck can be nonlinear even when the

vibration amplitude of the body is small. Through the concept of “relative

amplitude”, i.e. the amplitude of the externally triggered free vibration relative to

the magnitude of the ambient response of an elastically supported rigid sectional

model, nonlinearity in the self-excited wind forces is studied. The effect of relative

amplitude on flutter derivatives and on the flutter boundary reveals, from the

structural point of view, a complex relationship between the self-excited forces and

the “structural vibration noise” due to buffeting forces relating to signature

turbulence. Although the aeroelastic forces are linear when the body motion due to

an external trigger is not affected significantly by the turbulence, they are nonlinear

when the noise component in the vibration due to the turbulence is not negligible.

The effect of lateral motion related derivative on flutter boundary is also studied by

using flutter derivatives identified from 2 and 3 degree of freedom (DOF)

experiments.

A time domain model for the self-excited forces generation mechanism is suggested

with the objective in view to offer more flexibility for experimental studies of the

self-excited forces. This expression can be linked to the frequency-time-domain

hybrid flutter derivative model. A transform relationship between the two models is

suggested.

iii

List of Tables

Table 3.1 Intensity of Lateral Turbulence

Table 3.2 Experimental Information

Table 3.3 Derivatives of Respective Static Force Coefficients

Table 6.1 Material Properties of the Humber Bridge

Table 6.2 Dynamic Properties of the Bridge

Table 6.3 Flutter Speeds & Frequencies in Different Combinations

Table 6.4 Participation Factors of Major Modes at Flutter

iv

List of Figures

Figure 2.1 Damping Driven Flutter

Figure 2.2 Coalescence Flutter

Figure 3.1 Conventions

Figure 3.2 Power Spectral Density of Lateral Turbulence U=17.4m/s

Figure 3.3 Twin Deck Bluff Model

Figure 3.4 Streamlined Box Girder Model

Figure 3.5 Set Up for Free Vibration Test

Figure 3.6 Set Up for Static Force Coefficient Measurement

Figure 3.7 CL of Model A

Figure 3.8 CM of Model A

Figure 3.9 CD of Model A

Figure 3.10 CL of Model B

Figure 3.11 CM of Model B

Figure 3.12 CD of Model B

Figure 5.1 The Definition of Relative Amplitude

Figure 5.2 Non-Stationary Flutter Boundary

Figure 5.3a. Transient Signal of Model B at U=17.5m/s

Figure 5.3b. FFT of Transient Signal at U=17.5m/s

Figure 5.4a. Ambient Vibration of Model B at U=17.5 m/s

Figure 5.4b. FFT of Ambient Vibration at U=17.5 m/s

Figure 5.5. Output Covariance of Model B at U=17.5 m/s

Figure 5.6a. 2DOF H (Model A)

Figure 5.6b. 2DOF A (Model A)

Figure 5.7a. 2DOF H (Model B)

Figure 5.7b. 2DOF A (Model B)



Figure 5.8c. 3DOF P (Model B)

Figure 6.1 Plot of the Bridge

Figure 6.2.a Structural Modes of the Bridge Deck

v

Figure 6.2.b Structural Modes of the Bridge Deck

Figure 6.2.c Structural Modes of the Bridge Deck

Figure 6.2.d Structural Modes of the Bridge Deck

Figure 6.2.e Structural Modes of the Bridge Deck

Figure 6.3 Sensitivity of E-matrix to Damping Ratio

Figure 6.4 Singular Values at Flutter (2D FD Case), 1st Mode

Figure 6.5 E-Matrix of 2D FD

Figure 6.6 E-Matrix From 2D FD By Deleting P Related FD

Figure 6.7 E-Matrix From 3D FD

Figure 6.8 First Flutter Mode from 2D FD

Figure 6.9 Second Flutter Mode from 2D FD

Figure 6.10 The 1st Flutter Mode (2D FD from 3D FD)

Figure 6.11 The 2nd Flutter Mode (2D FD From 3D FD)


Figure 7.1 Indicial Functions of Different Kinds

Figure 7.2 Simulation Diagram of the SEF Model

Figure 7.3 FRF Matrix of the HB Bridge Section via Flutter Derivatives

(Transient)

Figure 7.4 FRF Matrix of the HB Bridge Section via Flutter Derivatives

(Ambient)

Figure 8.1 Singular Values at U=14m/s

Figure 8.2 Error Index for 3D Transient and Ambient Vibration Testing (HB)

Figure 8.3 Error Index for 2D Transient and Ambient Vibration Testing (HB)

Figure 8.4 Error Index for 2D Transient and Ambient Vibration Testing (TK)

vi

Nomenclature

A State Matrix of Discrete State Space Model

cA State Matrix of Continuous State Space Model

*mA , *

mH , *mP Flutter Derivatives

)(KAij Variables in ijE

fss AA , State Matrix of Rigid Body System

B Input Matrix of Discrete State Space Model or Width of the

Bridge Deck

cB Input Matrix of Continuous State Space Model

fB , sB , covB Input Matrix of SEF, Rigid Body and Covariance Dynamics

System

)(KBij Variables in ijE

C Output Matrix

[ ]strC , [ ]aeroC , [ ]effC Structural, Aeroelastic and Effective Damping Matrix

)(kC Theodorsen Circulation Function

)(kCi Output Covariance

fC Output Matrix of SEF System

MDL CCC ,, Static Wind Force Coefficient

sC Output Matrix of Rigid Body System

Cov Covariance Estimation

D Feed Through Matrix

aeaeae MLD ,, Aeroelastic Forces

aebb MLD ,, Buffeting Forces

dm Infinitesimal Mass

E Impedance Matrix

[ ]•E Expectation Operator

ijE Element in Impedance Matrix

vii

)(nEr Error Signal Matrix

)(sf State Vector of SEF

bufff Buffeting Force

seff Self-Excited Forces

F State Matrix of SEF System

)(),( kGkF Functions in Aerodynamic Coefficient

G Input Matrix for Covariance Dynamics

ji srG Modal Integral

α,, ph Displacements of the Rigid Body in Vertical, Lateral and

Rotational Direction, Respectively

ih , iα , ip thi Vertical, Rotational and Lateral Mode, Respectively

)(kH Block Hankel Matrix

[ ])(kH Flutter Derivative Matrix

[ ]K , [ ]aeroK , [ ]effectK Structural, Aeroelastic and Effective Stiffness

iI Generalized Inertia

K Reduced Frequency

l Bridge Deck Length

ML, Lift and Moment Forces of Wind

[ ]M Structural Mass Matrix

aeM Aeroelastic Moment

αP Observability Matrix

)(tp Buffeting Force

βQ Controllability Matrix

)(•r Rank Operation

[ ]R FRF Matrix

s Dimensionless Time

)(•Tr Trace Operation

U Wind Speed

viii

)(tv Measurement Noise

X State Vector

)(sX State Vector of Rigid Body Motion

Y Displacement

iY Markov Parameters

)(sY Output Vector of Rigid Body State Space Model

)(τZ Structural Function

21 ,ΞΞ Power Matrix of the Error Signal

Σ , nΣ Singular Value Matrix

Θ Signal Power Matrix

)(kΘ Sears Function

21 ,ΩΩ Ratio of Error to Signal Power

τ Time

iω Circular Frequency

ξ Participation Factor Vector of Structural Modes at Flutter

iη The Ith Full Bridge Mode Shape

)(sφ Wagner Function

ψ Kussner Function MDL χχχ ,, Admittance Function

Aeroelastic Analysis of Super Long Cable-Supported Bridge

1

CHAPTER ONE

Introduction to the Research

1.1 Long-Span Bridges

Long-span suspension bridges or cable-stayed bridges are highly susceptible to wind

excitations because of their inherent structural flexibility and low damping ratios. The

collapse of the center span of Tacoma Narrows Bridge in 1940 at a relatively low wind

speed of 42 mph is the most dramatic incident of wind-induced failure of bridges. This

incident caused investigators to examine many of existing suspension bridges built in

the same area for the possibility of excessive wind-induced vibrations.

Up to now, the driving force to build bridges of this kind is still obvious due to its

elegant appearance and economy. Fast developments in the state-of-the-art design over

the last two decades have brought about a new stage of the construction of such

structures. The ambitious Akashi-Kaikyo Bridge has a center span up to 2000 m. The

Strait of Messina Bridge with a center span of 3.3km will stand as the landmark bridge

of 21st century.


2

1.2 Motivation for the study

To have a better design, the study of the wind load on bridge decks is of vital

importance. The wind load is classified in two categories: motion dependent (self-

excited) forces and motion independent forces. One of the main tasks of bridge

aeroelasticity is to formulate the wind load on the structure when the body is in motion.

With increasing length of bridges, the structure becomes more flexible when the span is

longer. There is a transition of the analytical method from frequency domain to time

domain to overcome the difficulties in dealing with structural nonlinearity. Other

researchers also used the time domain approximation of self-excited wind load to study

control of bridge vibrations. The transformation of the frequency-time domain hybrid

flutter derivative model to either time or frequency domain usually requires the linearity

assumption of self-excited wind forces, which, unfortunately, is yet to be proven either

by theoretical or experimental means.

Furthermore, current analytical methods for the buffeting analysis use flutter derivatives

identified experimentally for flutter instability analysis, assuming that in these two

cases, the interactive forces are the same in their properties. It is also based on the

linearity assumption of self-excited forces.

The possible existence of nonlinearity in the self-excited forces could have a

fundamental impact on the state of the art understanding of the interactive wind load.

Experiments in this research are efforts to test whether or not the self-excited wind

forces can be treated linearly.

Because usually aeroelastic analysis is meant to predict the structural behavior when the

structural vibration amplitude and the angle of attack of the oncoming wind are both

small, it is important to detect the existence of nonlinearity in the self-excited forces

under the small amplitude condition. Previous tests (Scanlan, 1997; Falco, et al. 1992)

did not take this factor into consideration.


3

The existence of nonlinearity in self-excited wind forces will demand more efforts to be

exercised in the future to formulate the interactive wind load. It must be recognized that

a frequency-time domain hybrid flutter derivative model works as a linear model under

specific conditions. Any extension of the model to perform analysis under other

conditions will need experimental verification.

1.3 Organization

After a brief review in Chapter 2 on background literatures on bridge and airfoil

aeroelasticity, the design for experiments to extract flutter derivatives in 2 and 3

dimensions is presented in Chapter 3. The identification method selected in this thesis is

eigensystem realization algorithm (ERA); it is introduced in Chapter 4. Through the

concept of relative amplitude effect, the detection of the nonlinearity in self-excited

wind force by experimental means is described in Chapter 5, where most of the

experimental results are presented. Flutter boundary prediction is subsequently

described in Chapter 6 to illustrate the effects of the nonlinearity in the self-excited

wind force and the lateral flutter derivatives on aeroelastic instability analysis. Because

of nonlinearity in the self-excited force, new considerations on the interactive force

modeling is needed and an alternative model is proposed in Chapter 7 with the objective

in view to offer more flexibility to manipulate the experiment and the empirical model.

In Chapter 8, a new error index is presented to evaluate the identification of

experimental results. Conclusions and suggestions are given in Chapter 9.


4

CHAPTER TWO

Aeroelasticity and Aerodynamics of Bridge Decks

Abstract

This section is devoted to a review of the past work on aerodynamics and

aeroelasticity of bridge decks relevant to the present work. The discussion begins

with a few definitions followed by the classification of aeroelastic phenomena. After

a brief introduction of thin airfoil aeroelasticity, current methods for analyzing the

aeroelastic and aerodynamic behavior of bridge decks are reviewed.


5

2.1 Introduction

Researches are booming in the area of aerodynamics of civil structures, which are

not usually designed to influence or accommodate the airflow over them, but rather

with other objectives in view. The aerodynamics of such structures is characterized

by separated flow and turbulent wakes exhibiting widely varying degrees of flow

organizations.

A body immersed in a fluid flow is subjected to surface pressures induced by the

flow. If the oncoming flow is turbulent, this will be one of the sources of time

dependent surface pressure. If the body moves or deforms appreciably under the

induced surface pressure, these deflections, changing as they do the boundary

conditions of the flow, will affect the fluid forces, which in turn will influence the

deflections. Aeroelasticity is the discipline concerned with the study of the

phenomenon wherein aerodynamic forces and structural motions interact

significantly (Simiu and Scanlan, 1996).

If the body in the fluid flow deflects under some forces and the initial deflection

gives rise to successive deflections of oscillatory and/or divergent character,

aeroelastic instability is said to be produced. All aeroelastic instabilities involve

aerodynamic forces that act on the body as a consequence of its motion. Such forces

are termed self-excited.

A body is said to be aerodynamically bluff when it causes the wind flow around it

to separate from its surface leaving a significant trailing wake. In contrast, wind

flow around a streamlined body remains tangential and attached to its entire

surface, leaving a narrow trailing wake. Most civil engineering structures, including

the bridge sections of the long span bridges qualify as bluff bodies, while the shapes

of an airfoil belong to the category of a streamlined body.

The fundamental aspects of aeroelastic phenomena that need to be taken into


6

account in the design of certain structural members, towers, stacks, tall buildings,

suspension bridges, cable roofs piping system and power lines are not completely

understood. In most investigations empirical models are set up because pure

theoretical computations based on CFD can hardly produce reliable results. The

corresponding analytical models usually include just enough parameters to match

the strongest observed feature of the phenomena. Such models are minimally

descriptive, but not explanatory in the sense of revealing basic physical causes;

subtle but important details of the actual fluid-structure interaction may in certain

cases be left unattended. According to these models, aeroelastic phenomena fall into

the following categories:

1 Vortex Shedding and the Lock-in Phenomenon.

Under certain conditions a fixed bluff body sheds alternating vortices (Ehsan 1988;

Hartlen and Currie 1970; Iwan and Blevins 1974; Nakamura and Nakashima 1986

and Ongoren and Rockwell 1988). The primary frequency of the vortex shedding is

according to the Strouhal Relation:

SU

DNs = (2.1.1)

where the Strouhal number S depends on body geometry and the Reynolds number,

D is the across-wind dimension, U is the mean velocity and sN is the primary

frequency of the vortex shedding.

If the body is elastically supported and being driven periodically by the vortices

shed in its wake, it will experience small response unless the Strouhal frequency of

the alternating pressure approaches the across-flow mechanical frequency of the

structure. At this stage, the body interacts strongly with the flow. The mechanical

frequency controls the vortex shedding even when variations in flow velocity

displace the nominal Strouhal frequency away from the natural mechanical

frequency by a few percent. This phenomenon is known as lock-in.


7

2 Galloping

Galloping (Novak 1972 and Van Oudheusden 1995) is an instability of typically

slender structure with a special cross section shape such as a rectangular or a “D”

shapes. The structure exhibits large amplitude vibration in the direction normal to

the flow at frequencies much lower than those of vortex shedding from the same

section. In the across wind galloping, the relative angle of attack of the wind to the

bridge section depends directly on the across wind velocity of the structure. Mean

lift and drag force coefficients of the cross section obtained under static condition,

as functions of angle of attack, suffice as a basis upon which to build the analytical

description. Wake galloping is due to the turbulent wake of the upstream cylinder,

and may occur only under conditions where the frequencies of response of the

downstream cylinder are lower compared to its vortex-shedding frequencies and to

those of the upstream cylinder. It is also governed by parameters describing mean

rather than instantaneous aerodynamic phenomena.

3 Torsional divergence

Under the effect of wind, the structure will be subjected to drag, lift and pitching

moment. As the wind speed increases, the twisting moment may also increase,

twisting the structure further. This condition may also, by increasing the effective

angle of attack, further increase the twisting moment. Then additional deflections

occur. Finally, a velocity is reached at which the magnitude of wind-induced

moment together with the tendency for twisting to demand additional structural

reaction creates an unstable condition and the structure twists to destruction.

4 Flutter

In the context of bridge engineering, flutter is usually of single aeroelastic mode, a

typical self-excited oscillation (Jain et al. 1996; Matsumoto et al. 1994; and Chen et

al. 2000). The flutter mode changes from a pure structural mode to an aeroelastic

mode incorporating the effect of aeroelastic coupling. The structural system by


8

means of its deflection and time derivatives taps off energy from the wind flow. If

the system is given an initial disturbance, its motion will either decay or diverge

according to whether the energy of motion extracted from the flow is less than or

exceeds the energy dissipated by the system through mechanical damping. The

dividing line between the decay and divergent case, namely, sustained sinusoidal

oscillation, is recognized as the critical flutter condition, the threshold of negative

damping. Therefore, the criterion for the flutter to occur is based on eigenvalues, i.e.

whether or not one or more eigenvalues move, as functions of aerodynamic

parameters, from the left-hand to the right-hand side of the s-plane (Figure 2.1). In

the figure, ωσ is += is the complex eigen-frequency of the aeroelastic system.

Figure 2.1 Damping Driven Flutter

Together with the change of eigenvalue(s), there could also be a change of flutter

mode(s) from the pure structural mode(s) to aeroelastic mode(s) due to the fact that

the aeroelastic coupling could be strong to change not only the modal damping but

also mode shapes. This is classified in some literatures as coupled mode flutter.

However, the term “coupled” essentially means the coupling of pure mechanical

modes not the aeroelastic modes.

However for coalescence flutter, two modes of vibration are required and damping

is not necessarily present in the system. When coalescence flutter occurs, the energy

required to drive the instability is extracted from one of the stable modes, and this

energy is fed in a “non-conservative” manner to the other mode, which then

becomes unstable. It can be classified as a “non-conservative problem”. In contrast

to the more familiar conservative problems, the “non-conservative problems have

σ

ω

ωσ is +=


9

non-self-adjoint characteristics and are inherently unstable. If we define the set of

aeroelastically influenced stiffness matrix as a family of matrices depending on

parameters e.g. reduced frequency, coalescence flutter is defined by the bifurcation

position of the matrix family in a matrix bundle1.

The frequency degeneracy is not a sufficient condition for coalescence flutter. If the

degenerated eigenvalues are encountered, one has to inspect the corresponding

eigenvectors or, equivalently, the eigenvalue matrix at the point of eigenvalue

degeneracy (Figure 2.2). If the eigenvectors are not linearly independent, i.e. the

angle between two eigenvectors becomes zero as shown in the figure or,

equivalently, the eigen-matrix is a Jordan matrix, then coalescence flutter occurs

(Afolabi, 1994). The stiffness matrix, in this case, can no longer be diagonalized

with the eigenvector matrix. There is a shortage of eigenvectors. If generalized

eigenvectors are used, the resultant diagonal matrix appears in Jordan canonical

form. It is well known that this operation is not stable.

Coalescence flutter instability has not yet been studied extensively in bridge 1 A set of matrices is called a bundle if all the matrices belonging to it have Jordan normal forms

differ only by their eigenvalues, but for which the set of distinct eigenvalues and the order of the

Jordan blocks are the same. For example, all the diagonal matrices with simple eigenvalues define

one bundle. Families of matrices are in general position if they are transversal to all the bundles and

in exceptional position if they are not. The matrices in general positions are called generic, while

those are not in general positions are called degenerate. Corresponding to the decomposition of the

space of matrices into bundles, the parameter space of the family decomposes into sub-manifolds. In

a family in general position, almost all the matrices have simple eigenvalues. The exceptional

parameter values to which there correspond matrices with multiple eigenvalues define a subset of the

parameter space. This is called bifurcation diagram (Arnold, 1971). A generic matrix has structural

stability, and does not change its qualitative properties or behavior under small perturbations. A

degenerate matrix, on the other hand, is structurally unstable. An arbitrary small perturbation will

cause it to bifurcate into two or more generic matrices. Coalescence flutter happens on such

bifurcation point. As a result of this instability, degenerated objects are unobservable, and are

“almost always” not encountered in engineering practice. If they are encountered in mathematical

model, it is only because one has made a theoretical assumption, which is not qualitatively valid in

the actual physical problem.


10

engineering. Therefore, most of the past and current works concentrated on the

prediction of the negative damping boundary with or without considering the

change of pure structural modes to aeroelastic modes.

Figure 2.2 Coalescence Flutter 5 Buffeting

Buffeting is defined as the unsteady loading of a structure by velocity fluctuations

in the oncoming flow. If these velocity fluctuations are clearly associated with the

turbulence shed from the wake of an upstream body, the unsteady loading is

referred to as wake buffeting. However, the buffeting force is usually due to the

atmospheric turbulence.

In this study, the work is focused on flutter instability, i.e. the identification of the

flutter derivative model for self-exited forces on the bridge deck and the prediction

of the damping driven (single mode) flutter boundary. In the following part, some

related work based on flutter derivative models done in the past decades on long

span bridges are introduced. First, there is a review of some historical studies on

thin airfoils.

900

00

Aerodynamic Parameter

Non

-Dim

ensi

onal

Eig

enva

lues

A

ngle

Bet

wee

n Ei

genv

ecto

rs


11

2.2 Thin Airfoil Aeroelasticity

The aerodynamic forces acting on a thin airfoil undergoing complex sinusoidal

motion h and α : tiehh ω

0= (2.2.1)

tie ωαα 0= (2.2.2)

in two-dimensional incompressible flow are given by Theodorsen (1935) from basic

principles of potential flow theory. The expressions for hL and αM are linear in h ,

α and their first and second derivatives:

))21()((2)(2 ααπρααπρ &&&&&&& abhUkUCbahUbL −++−−+−= (2.2.3)

−+++

+

−++−−=

ααπρ

ααπρ

&&

&&&&&

)21()()

21(2

)81()

21(

2

222

abhUkCaUb

hababUbabM (2.2.4)

where Ubk /ω= is the reduced frequency, b is the half-chord of the airfoil, ab is

the distance between the mid chord and the rotation point, ρ is the air density, U is

the flow velocity and ω is the circular frequency of oscillation. The complex

function )()()( kiGkFkC += is Theodorsen’s circulation function. The coefficients

in the expression, referred to as aerodynamic coefficients, are defined in terms of

two theoretical functions )(kF and )(kG ,

[ ] [ ][ ] [ ]201

201

011011

)()()()()()()()()()()(

kJkYkYkJkJkYkYkYkJkJkF

−++−++

= (2.2.5)

[ ] [ ]2012

01

0101

)()()()()()()()()(

kJkYkYkJkJkJkYkYkG

−+++

−= (2.2.6)


12

in which 10 , JJ are Bessel functions of the first kind, 10 ,YY are Bessel functions of

the second kind.

This equation is in frequency and time domain hybrid format. There were also

efforts to transform the expression from unsteady aeroelastic force to time domain.

Wagner (1925) showed that the lift evolution with dimensionless time bUts /=

acting on a theoretical flat airfoil given a step function change 0α in angle of attack

is given by

)()2)(2(21

02 sbUL φαπρ= (2.2.7)

where )(sφ is the Wagner function:

∫∞

∞−= dke

kkC

is iks)(

21)(π

φ . (2.2.8)

)(kC is the Theodorsen circulation function and k is the reduced frequency.

For arbitrary motion, the lift force is given as

σσφσαπρ dsbUsLs

∫ ∞−−′−= )()()2)(2(

21)( 4/3

2 (2.2.9)

where )(4/34/3 sdsd αα =′ and 4/3α is the effective angle of attack,

−++=

Uab

Uh ααα

&&)

21(4/3 . (2.2.10)

h& is the vertical velocity and ab is the distance from the mid-chord to the reference

point at which deflection and rotation angle are measured.


13

Jones [1940] introduced rational approximation of the unsteady loads on a typical

airfoil section in incompressible flow in order to ease the difficulties in flutter

stability analysis. In dimensionless time domain,

ss ees 300.00455.0 335.0165.01)( −− −−≅φ . (2.2.11)

Kussner (1936) considered the problem of an airfoil with forward flight velocity

U penetrating a uniform vertical gust of infinite downstream extent and vertical

velocity 0w . He determined the lift due to this circumstance to evolve according to

the description:

)()2(21)( 02 s

UwBUsL ψπρ= (2.2.12)

with L and 0w considered positive upward, and )(sψ is the Kussner function

defined approximately by Jones(1941)

ss ees −− −−≅ 500.0500.01)( 130.0ψ . (2.2.13)

For the gust of arbitrary velocity distribution )(sw , the lift generated by an airfoil

advancing through it will be given as

∫ ∞−−′=

sdswUBsL σσψσπρ )()()( . (2.2.14)

For a gust velocity distribution that is sinusoidal of the form iksewsw 0)( = , Sears

(1941) derived the corresponding oscillatory lift on the airfoil in the form

iksekUwBUsL )()2(

21)( 02 Θ= πρ (2.2.15)


14

where )(kΘ is a complex frequency-domain function known as Sears function. It is

clear that the Kussner function and Sears function are a Fourier transform pair:

∫∞ −=Θ0

)()( σσψ σ deikk ik . (2.2.16)

It was further shown by Fung (1955) that Sears function )(kΘ is related to

Theodorsen circulation function )(kC as follows

[ ] )()()()()( 110 kiJkJkJkCk +−=Θ (2.2.17)

where 0J and 1J are Bessel functions of argument k .

Spectral forms of )(sL are also available, but will not be reviewed.

In the case of bridge engineering, however, because of the complexity of the bluff

body aerodynamics, special considerations are needed for the formulation of self-

excited forces on bridge decks.

2.3 Aeroelastic and Aerodynamic Forces on Long-Span

Cable-Supported Bridge Decks

A basic task in the study of the bridge aeroelasticity is to formulate the forces of

wind on the structure. The total lift force L, drag force D and moment M are

decomposed to motion dependent force and motion independent force: aeroelastic

forces (ae) and buffeting force (b):

bae LLL += (2.3.1)

bae DDD += (2.3.2)

bae MMM += (2.3.3)


15

2.3.1 Formulation of the Self-Excited Forces

Special considerations are needed for the formulation of self-excited force for

bridge decks. The signature turbulence, in the case of efficient airfoils in smooth

flow, is intentionally reduced by careful streamlining with notable attention to the

introduction of a sharp trailing edge. For bluff bodies, however, the situation is

different. The use of Theodorsen aerodynamics for such bluff bodies is not

guaranteed correct. In view of this, the formulation of self-exited forces on civil

engineering structures, such as a bridge deck, is more experimental than theoretical.

Scanlan and Tomko (1971) suggested reduced frequency dependent flutter

derivatives be used in the modeling of self-excited wind load on bridge deck. This

is the counterpart of Theodorsen theory in the experimental bridge aerodynamics.

The flutter derivative format representation of self-excited wind forces, after being

expanded from two degree of freedom to three degree of freedom to take into

consideration the lateral vibration, now takes the form:

+++++=

BpHK

UpKH

BhHKHK

UBKH

UhKHBULae

*6

2*5

*4

2*3

2*2

*1

2

21 &&&

ααρ (2.3.4)

+++++=

BhPK

UhKP

BpPKPK

UBKP

UpKPBUDae

*6

2*5

*4

2*3

2*2

*1

2

21 &&&

ααρ (2.3.5)

+++++=

BpAK

UpKA

BhAKAK

UBKA

UhKABUM ae

*6

2*5

*4

2*3

2*2

*1

2

21 &&&

ααρ (2.3.6)

In which ph ,,α are deck deflection components in vertical, torsion and horizontal

direction, respectively. 6,5,4,3,2,1,,, *** =mAPH mmm are reduced frequency dependent

aerodynamic flutter derivatives, UBK /ω= is the dimensionless frequency. ω is

circular frequency, B is deck width, U is wind velocity, ρ is the density of air.


16

The flutter derivative model works only for sinusoidal or exponentially modified

sinusoidal motion with decay rate less then 20%. Among the flutter derivatives, *1H , *

2H , *5H , *

1A , *2A , *

5A , *1P , *

2P and *5P describe aerodynamic forces in phase

with bridge deck velocity. Therefore they are damping terms. *3H , *

4H , *6H , *

3A ,

*4A , *

6A , *3P , *

4P and *6P describe aeroelastic forces in phase with the bridge deck

displacement. They are stiffness terms. A better understanding of the flutter

derivatives is due to quasi-static theory in the following way (Simiu and Scanlan,

1996). The static wind force coefficients are defined as non-dimensional numbers:

BlULCL 2

2ρ

= , BlU

DCD 2

2ρ

= , lBU

MCM 22

2ρ

= (2.3.7)

where MDL CCC ,, are mean lift, drag and moment force coefficients respectively

L is lift force, D is drag force, M is moment, B model width, U is wind speed, l is

model length, ρ is the density of air.

For small angle of attack α ,

Uh&

=α or UBαα&

= (2.3.8)

the typical term in equation (2.3.4~2.3.6) can be viewed in the classical patterns of

expressions for aerodynamic lift force per unit span.

αα

ρρddCBUCBUL L

L )2(21)2(

21 22 ≅= (2.3.9)

Formally, term *1KH is analogous of the lift coefficient derivative αddCL / . These

flutter derivatives should be referred to as motional derivatives and they go over

into steady-state derivatives only for zero frequency, i.e. 0→K . The general

expressions of flutter derivatives in the form of quasi-static theory are as follows:


17

'*1 2

1LC

nBUH

=

π (2.3.10)

'2

2*3 4

1LC

nBUH

=

π (2.3.11)

LCnBUH

−=

π1*

5 (2.3.12)

'*1 2

1MC

nBUA

=

π (2.3.13)

'2

2*3 4

1MC

nBUA

=

π (2.3.14)

MCnBUA

−=

π1*

5 (2.3.15)

DCnBUP

−=

π1*

1 (2.3.16)

'*2 2

1DC

nBUP

=

π (2.3.17)

'2

2*

3 41

DCnBUP

=

π (2.3.18)

'*5 2

1DC

nBUP

=

π (2.3.19)

where ''' ,, MDL CCC are corresponding first derivatives of force coefficients with

respect to angle of attack α at 0=α ; n is structural frequency.

Like the researchers in the airfoil aeroelasticity, civil engineering researchers are

trying to expand the time-frequency domain hybrid format model to time domain. A

more general understanding of the unsteady aeroelastic force is found by

recognizing that the indicial function expression can be seen as a modification of

quasi-static nominal form of wind force under turbulent condition. The wind lift


18

force is given by

[ ]

+′+=

UuCCBUL LL

21)()(21

002 αααρ (2.3.20)

or

uCUBuUBCCBUBCUL LLLL ααραρααραρ )()()(21)(

21

0002

02 ′++′+= (2.3.21)

This is recognized to be only a nominal form that may hold for very slow changes

in the angle of attack and wind speed, but is strictly incorrect due to the known lag

of interaction force behind their angle of attack or wind velocity changes (Scanlan,

1993). Hence modification is needed:

∫∫∫

∫

−Φ′−Φ′′+

−Φ′+

−Φ′′+

Φ=

s

Lu

s

LL

s

LuL

s

LL

LL

dsudsCUB

udsuUBC

dsCBU

sBCUsL

0 2220 1110

00

002

02

)()()()()(

)()()(

)()()(21

)()(21)(

σσσσσσααρ

σσσαρ

σσσααρ

αρ

α

α

α

(2.3.22)

The first term represents an initial transient, arriving for ∞→s at the steady state

lift, and can be considered as constant. The second term represents self-excited lift,

the third horizontal impulse lift, and the last, interaction between the two lift forces.

It should be very small since the correlation between the fluctuation part of the

oncoming flow and the bridge motion is very weak. An expression is also available

for vertical movement related wind load.

In the time domain formulation of unsteady self-exited forces on a bridge deck,

indicial functions still remain the most important tool when the structure is

subjected to arbitrary motion. Scanlan et al. (1974) studied the aeroelastic moment

on a bluff bridge deck due to indicial angular movement. The characteristic of

indicial function corresponding to *2A of a bridge, according to their experiment, is


19

strongly different from those of the corresponding functions of airfoils. They

showed the relationship between the flutter derivatives and the indicial function by

recognizing that for a sinusoidal motion, the Duhamel integral is of the nature of a

Fourier transform and the inverse transform of frequency domain expression should

then produce the indicial function.

The direct measurement of indicial function is neither easy, nor conventional in the

sense of modern dynamic experiment techniques. Yoshimura and Nakamura (1979)

suggested, in their study on the measurement of the indicial aerodynamic moment

response of moving bluff prismatic sections in still air, that since the aeroelastic

moment arises from the relative motion between the fluid and body, it might be

expressed more conveniently by the time derivatives of the state variables. By

assuming the superposition of small disturbances to a linear aerodynamic system,

the moment due to the angular motion is decomposed into two parts, namely the

moment due to the angle of attack )(sα and the angular velocity of the body axis

relative to the fixed coordinated )(sq :

constconstq sqMsMsqsM == += ααα ))(())(())(),(( (2.3.23)

and the indicial dynamic moment response is also decomposed into two terms:

dsdss q /)()( Φ+Φ=Φ αθ . (2.3.24)

The first term is the indicial aerodynamic moment response for the angle of attack

motion and the second term is the indicial aerodynamic moment response for the

angular velocity motion. Three types of indicial motion were used. In the reported

study, it was found that the contribution of the angle of attack motion dominates,

while the second term contribution to the overall indicial function is small and

negligible.


20

2.3.2 Buffeting Forces

Buffeting force on bridge decks is also an important topic. The quasi-static

buffeting forces due to turbulence are (Scanlan 1988):

( )

+′+

−=

UwCC

UuCBUL DLLb

221 2ρ (2.3.25)

′+

=

UwC

UuCBUM MMb

221 22ρ (2.3.26)

′+

=

UwC

UuCBUD DDb

221 2ρ (2.3.27)

where MDL CCC ,, are mean lift, drag and moment coefficients, and the primes

denotes their first derivative with respect to angle of attack ( )α at 0=α . For an

accurate description, these expressions must be modified by aerodynamic

admittance factors (Davenport, 1962; Kumarasena, 1989):

)(KLL Lbb χ= (2.3.28)

)(KMM Mbb χ= (2.3.29)

)(KDD Dbb χ= (2.3.30)

DML χχχ ,, are aerodynamic admittance functions of lift, moment and drag

buffeting forces.

These functions are characteristic of the bridge deck shape and in fact are

approximately related to the flutter derivatives (Kumarasena, 1989) in the following


21

way when the body has long after-body geometry such that the flow reattachment

might occur:

( )*2

*3

0

2

)( iHHC

ddC

KKD

L

L ++

=

=αα

χ (2.3.31)

and

( )*2

*3

0

2

)( iAA

ddC

KKM

M +=

=αα

χ . (2.3.32)

Scanlan (2000) showed how important central characteristics of admittances can be

seen to be inherent in the measured flutter derivatives, and the buffeting forces can

be formulated by flutter derivatives:

+−=

UuKKH

UwKKHBULb )()(

21 *

5*1

2ρ ; (2.3.33)

+−=

UuKKA

UwKKABUM b )()(

21 *

5*1

2ρ and (2.3.34)

+−=

UuKKP

UwKKPBUDb )()(

21 *

1*

52ρ . (2.3.35)

In these expressions, the following replacements have been affected for the quasi-

static buffeting force terms:

)(2 *5 KKHCL −= (2.3.36)

)()( *1 KKHCC L

DL −=+′ χ (2.3.37)

)(2 *1 KKPC D

D −=χ (2.3.38)

)(*5 KKPC D

D −=′ χ (2.3.39)


22

)(2 *5 KKAC M

M −=χ (2.3.40)

)(*1 KKAC M

M −=′ χ (2.3.41)

All coefficients on the left are associated with zero angle of attack for a horizontal

wind, or to any other desired reference position. Because these terms are seen to be

functions of K rather than being, in general, simple constants, they reflect

frequency dependency and thus incorporate aerodynamic admittance effects. In

other words, aerodynamic admittance is inherently expressible in this context as a

function of the flutter derivatives.

2.4 Analytical Method in Frequency Domain

In the last several decades, the most significant advances have been made in

understanding aeroelastic phenomena. Most of current efforts are concentrated on

developing methods to alleviate the flutter instability, vortex-induced vibration and

buffeting. Modern approaches to address these issues are based on a combination of

state-of-the-art analytical, numerical and experimental techniques.

2.4.1 Analytical Method For Flutter Analysis

Techniques predicting flutter boundary of cable-supported bridges have been

developing in two parallel ways: one in frequency domain (Davenport 1962;

Scanlan 1978; Jain et al, 1996), the other in time domain (Matsumoto et al. 1994;

Chen et al. 2000). All these methods are developed to solve negative damping

driven flutter, due to the fact that for civil engineering structures, the wind speed is

rare to reach such a high value to bring about coalescence flutter.

Frequency domain analysis has dominated in the past due to the efficiency of

computation, especially when handling the unsteady aeroelastic forces that are

functions of reduced frequency. The nature of flutter analysis is generally a


23

complex eigenvalue problem, while buffeting analysis is conducted mainly by

mode-by-mode approach ignoring the aerodynamic coupling among modes.

As mentioned in the proceeding paragraph, the flutter instability of cable-supported

bridges is defined with respect to a negative damping threshold. It is reasonable to

postulate that a single mode will approximate the total response. This assumption is

justifiable from observation of the fact that typically just one predominant mode

will become unstable and dominate the flutter response of a three-dimensional

bridge model in the wind tunnel.

The so-called two-degree-of-freedom flutter analysis method supposes that there is

coherence between bending and torsional mode shapes along the span, and does not

consider the possible influence of transverse displacements. The small lack of

coherence between the bending and torsional mode shape in conventional

suspension bridges may have a non-negligible influence on critical wind velocity, as

has already been notice by other authors (Irwin 1979; Scanlan 1987; Lin and Yang

1983; Miyata et al. 1992). This effect is usually more important for shorter bridges

especially for cable-stayed bridges.

Other authors proposed three-dimensional flutter analysis on the basis of flutter

derivatives (Scanlan and Tomko 1971; Scanlan 1989; 1993). The main point in

studying the fully 3D stability consists in taking into consideration the degrees of

freedom in the lateral direction. Then the equations are more difficult to solve, since

the relations between the vertical, rotational and lateral displacement and the

aeroelastic forces become quite complex, as they depend on the deformation

patterns of the full bridge. In any case it will be supposed that there is no

aerodynamic coupling between these forces along the deck, so that the sectional

description will be integrated along the full bridge length to get total forces.

It has been common to use the combination of a set of mechanical modes, namely

the modes of the bridge structure under non-wind condition, as the flutter mode to

perform the flutter analysis. It is clear, however, due to the aeroelastic effects, the


24

combination of a limited number of the mechanical modes is only an approximation

of what happens in wind. Direct FEM flutter analysis by Miyata and Yamada

(1988), Miyata et al. (1995), and later development of the mode tracing method by

Dung et al. (1996, 1998) could serve as a better representation of the dynamic

behavior of the long-pan bridges in terms of complex flutter mode. Complex

eigenanalysis is made for an integrated system consisting of the 3-D FEM model of

a bridge and the aeroelastic force caused by the wind flow. To solve the complex

mode is an iterative procedure, tracing down the evolution of each aerodynamic

complex mode with step-by-step increment of wind speed. Finite element method is

the most common choice in this circumstance (Miyata, and Yamada 1988; Agar

1988; 1989; Namini 1991; Namini et al. 1992; Starossek 1993). The deck is usually

modeled by beam elements located along the bridge axis. Plate elements could also

be used provided that aeroelastic force is applied along the elastic axis of the deck.

By assuming harmonic oscillation, the self-exited force on a unit length of bridge

deck is incorporated into the element matrix. With the usual FE procedure, the

governing equations of the aeroelastically-influenced structure can be established.

Eignvalues and eigenvectors need to be found by iterative method since the

governing equation is reduced frequency dependent. In one step, a set of natural

frequencies of the aeroelastically-influenced structure is obtained with fixed wind

speed. The procedure repeats with a different wind speed covering the speed range

of interest.

More recently, an efficient scheme for coupled multimode flutter analysis has been

proposed introducing the unsteady self-exited aerodynamic forces in terms of

rational function approximations (Matsumoto et al. 1994; Chen et al. 2000). This

has led to a convenient transformation of the equation into a state space format

independent of reduced frequency. A significant feature of this approach is that an

iterative solution for determining flutter boundary is unnecessary because the

equations are independent of reduced frequency UBK /ω= where ω is the

circular frequency, B is the deck width and U is the wind velocity. In general,

frequency domain methods are restricted to linear structures excited by the

stationary wind load without aerodynamic nonlinearities. To include nonlinearities


25

of structural and aerodynamic origins, the time domain approach is more

appropriate. Time domain methods, however, involve the transformation of flutter

derivatives into indicial functions, which have inherent deficit, as will be shown in

the thesis. The effectiveness of time domain analysis in calculating buffeting

response depends on the establishment of an effective time domain model for the

self-excited wind force.

In this thesis, the traditional frequency domain analysis will be used for the flutter

instability analysis in chapter 6. Therefore, it will be reviewed in detail in the

following part.

2.4.2 Governing Equations of Flutter

(Jain et al, 1996)

Deck deflection components can be expressed by generalized mode coordinates

)(tiξ . If B is the bridge deck width, ),( txh is vertical displacement, ),( txα is

torsion displacement and ),( txp is lateral displacement. Deck deflections are

expressed in the following forms:

∑=i

ii tBxhtxh )()(),( ξ (2.4.1)

∑=i

ii txtx )()(),( ξαα (2.4.2)

∑=i

ii tBxptxp )()(),( ξ (2.4.3)

in which, )(xhi , )(xiα and )(xpi are dimensionless representations of thi mode in

each direction respectively.

The governing equation for the bridge deck motion can then be deduced as:

)()2( 2 tqI iiiiiiii =++ ξωξωζξ &&& (2.4.4)


26

The generalized force is defined as

[ ]dxMBDpBLhtphql

iiii ∫ ++=0

),,,( αα , (2.4.5)

where l is the deck span length; the generalized inertia is

( ) ( )∫= zyxdmzyxI ii ,,,,2η (2.4.6)

where iη is the full bridge mode, iω is the circular frequency and iζ is damping

ratio-to-critical and dm is infinite small mass.

The lift force L, drag force D and moment M in the governing equations are

decomposed to motion dependent force and motion independent force: aeroelastic

forces (ae) and buffeting force (b) as in Equation (2.3.1~2.3.3),

Substituting (2.4.5), (2.4.6) into (2.4.4), we have the dimensionless time domain

governing equation for the bridge deck motion:

),( sxQBAI b=+′+′′ ξξξ (2.4.7)

where BUts /= is the dimensionless time; ξ is the generalized coordinate vector; a

prime denotes the derivative with respect to dimensionless time s ; I is the identity

matrix and the general terms of matrix A , B and bQ are

]

[2

2)(

*5

*2

*1

*5

*2

*1

*5

*2

*1

4

jijijijiji

jijijiji

phhpp

ppphhhhi

ijiiij

GAGAGAGPGP

GPGHGHGHI

lKBKkA

ααααα

αρδζ

+++++

+++−= (2.4.8)


27

]

[2

)(

*6

*4

*3

*6

*4

*3

*6

*4

*3

242

jijijijiji

jijijiji

phhppp

pphhhhi

ijiij

GAGAGAGPGP

GPGHGHGHIlKBKkB

αααα

ααρδ

+++++

+++−= (2.4.9)

∫ ++=l

ibibibi

b ldxsxMpsxDhsxL

IlBsxQ

i 0

4

),(),(),(2

),( αρ (2.4.10)

where UBK /ω= is the reduced frequency and UBK ii /ω= is the reduced

frequency of mode i , *** ,, mmm PAH , )6,,1( L=m are flutter derivatives and ijδ is the

Kronecker delta function defined as:

≠=

=jiji

ij 01

δ . (2.4.11)

The modal integrals ji srG are obtained by integration over the deck, which is the

primarily aerodynamic load source

∫=l

jisr ldxxsxrG

ji 0)()( (2.4.12)

where iii phr ,= or iα ; jjj phs ,= or jα .

Note that the off-diagonal terms in equation (2.4.7) represent the aeroelastic

coupling through the flutter derivatives and mechanical coupling through the cross

mode integrals among different modes.

The new equation is Fourier transformed in to reduced frequency ( K ) domain

(Scanlan and Jones 1990) by

∫∞ −=

0)()( dsesfKf iks (2.4.13)

and is represented as


28

bQE =ξ (2.4.14)

where ξ and bQ are Fourier-transformed vectorsξ and bQ , respectively.

The general term of the impedance matrix is

)()(2 KBKiKAKE ijijijij ++−= δ (2.4.15)

where 1−=i .

The flutter condition is then defined as the aeroelasctically influenced eigenvalue

problem:

0=E . (2.4.16)

It is clear that the buffeting terms do not affect flutter stability. However it is

reasonable to argue that the turbulence effect on flutter stability can be take into

consideration by measuring the flutter derivatives in turbulent flow. These effects,

as will be shown in the thesis, will need further studies to include “relative

amplitude” effect.

The flutter mode is determined by the vector ξ in the homogeneous equation

0=ξE . (2.4.17)

The nontrivial solution of equation (2.4.16) and (2.4.17) yields the reduced

frequency at which flutter instability occurs and a non-zero vector ξ . This vector

indicates the relative magnitudes of participation structural modes at flutter. It can

be used as a tool to determine the flutter mode.


29

2.4.3 Buffeting Analysis

(Jain, 1997)

The vector of buffeting forces on the right hand side of Equation (2.4.14) is

=

∫

∫

∫

l

bnn

l

b

l

b

b

ldxF

I

ldxF

I

ldxF

I

lBQ

0

0 22

0 11

4

1

1

1

2M

ρ (2.4.18)

where the integrands in the vector are:

)(),()(),()(),(),( xKxMxpKxDxhKxLKxF ibibibbi α++= (2.4.19)

The buffeting force may include admittance functions. The power spectral density

(PSD) matrix is obtained by multiplying bQ and its complex conjugate transpose

vector *bQ :

=

∫ ∫∫ ∫

∫ ∫∫ ∫

l l BAbnbn

nn

l l BAbbn

n

l l BAbnb

n

l l BAbb

bb

ldx

ldxFF

IIldx

ldxFF

II

ldx

ldxFF

IIldx

ldxFF

II

UlBQQ

0 0

*

0 0

*1

1

0 0

*1

10 0

*11

1124

*

11

11

2L

MOM

L

ρ , (2.4.20)

where a ‘ * ’ denotes complex conjugate transpose.

Expressed in generalized displacement ξ (2.4.7), the PSD matrix becomes:

1**1 ][)( −−= EQQEKS bbξξ . (2.4.21)

Hence, physical displacement PSD is obtained.


30

Take vertical displacement as an example:

∑∑=i j

jiBjAiBAhh KSxhxhBKxxS )()()(),,( 2ξξ (2.4.22)

By integrating respective PSDs the mean-square value of the displacement can be

found:

∫∞

=0

2 ),,(),( dnnxxSxx BAhhBAhhσ . (2.4.23)

A covariance matrix for vertical displacement is thus obtained. Torsion and lateral

displacement can be treated in the same manner. The statistics of the displacements

can be calculated.

Summary

Besides the historical work in airfoil aerodynamics, selected research was obtained

from the literature on bridge aerodynamics for presentation in this chapter. Flutter

derivative models and analytical methods for flutter prediction and buffeting

response estimation are reviewed. Frequency domain methods are paid attention to.


31

CHAPTER THREE

Wind Tunnel Experiment to Extract

Flutter Derivatives

Abstract

The wind tunnel experiment remains the practical means to study the aeroelastic

behavior of bridges. In this part, methods of wind tunnel experiment to measure

flutter derivatives are reviewed; experimental design for the research is presented;

the experimental procedure is introduced and limitations of the experiment

discussed. Preliminary results are also presented.


32

3.1 Introduction

Wind tunnel experiments are usually used to evaluate the aerodynamic behavior of

bridge decks, as the reliable prediction based on purely theoretical methods has

proven hard to obtain. The confidence of wind tunnel testing has grown not only

because the principles of dynamic similarity lead one to expect good predictions

from wind tunnel models but also because good correspondence has been observed

between model and full-scale in a number of specific cases. While wind tunnel

models are frequently used as direct physical analog of full-scale structures, they

are now often employed more as a tool for obtaining values of empirical parameters

needed by various theories. An example of such theory is the semi-theoretical

flutter derivative model for the prediction of flutter boundary of cable-supported

bridges (Scanlan 1971, Scanlan & Jones 1990, Jain et al. 1996). The model does not

attempt to describe the detailed flow patterns around the bridge deck, and the

corresponding aerodynamic forces, but rather to provide a theoretical framework,

which contains a number of empirical parameters, flutter derivatives, describing the

wind forces. Although they are semi-empirical in nature, they can be helpful in

understanding the bridge behavior especially in more complex cases where the

bridge modes of vibration are complex. Wind tunnel experiments, primarily

sectional model testing, are used to quantify the flutter derivatives.

A sectional model is a span-wise representative segment of a full-scale structure.

The model itself is rigid, but is elastically suspended between end plates. The

model-to-full-scale ratio is typically in the range 30:1=Lλ to 200:1 . A sectional

model can be used in airflow as

1 An almost immobile object to measure time-average load for the whole

model or transducer forces and pressure data including everything except for

motion dependency;

2 Forced oscillating model to measure time series of driving forces and input

motion, plus possible pressure or

3 Free vibrating model to measure the displacement time series and/or

transducer forces.


33

The state-of-the-art use of sectional model is different from that used earlier

(Farquharson 1949). In the past, full-scale response was estimated from sectional

model testing directly. This estimation was at times of questionable reliability.

3.1.1 Similitude in the Experiment (Tanaka 1992)

The wind tunnel experiment is to correctly model the behavior of wind flow in a

certain space or area and its interaction between the geometrical and/or mechanical

characteristics of the boundaries of the field of concern. Based on Buckingham’s Pi-

theorem, it is required that a set of dimensionless parameters be invariant in the

model and prototype and with them, the governing equations also be dimensionless.

These parameters consist of suitable combinations of the reference quantities.

Various boundary conditions have also to be maintained in a dimensionless form.

Theoretically speaking, all of the dimensionless parameters in the prototype must be

duplicated in the model. However, almost inevitably, complete duplication of these

parameters is impracticable. As a mater of fact, all the requirements can be satisfied

exactly only when model and prototype are identical. Hence, decision must be made

as to which parameters can be relaxed or distorted to what extent for each testing

based on the understanding of the phenomenon and knowledge of dominant

parameters. Only a part of the process can be simulated to clarify the unknown

mechanism. The deficit part has to rely on analytical means for its solution.

The similitude requirements for testing have been well established and practiced for

a long time. The tradition of the aeroelastic model testing requires the similarities of

geometry and reduced frequencies (Cauchy Number) to be met (Hjorth-Hansen

1992). It is worthwhile, however, to have a review on all the dimensionless

numbers.

Cauchy Number is defined as the ratio of elastic force to the fluid inertia force:


34

2/ UECa ρ= . In aeroelastic testing of a sectional model, the Cauchy number is

usually rewritten to fBU / , where U is a short time average of wind speed; f is an

eigen-frequency of un-damped motion in some reference condition; B is the a

reference length. In most wind tunnel tests, rather than the original Cauchy number,

reduced velocity ( fBU / ) is much easier to use.

Reynolds number can be defined as the ratio of the fluid inertia force to the fluid

viscous force. In most wind tunnel tests, it is impractical to satisfy the Reynolds

number similitude. Indeed, the viscous force is usually of smaller magnitude and

relatively unimportant compared with the fluid inertia force for a large part of the

flow domain, but can be large for the parts close to the boundary. To have a correct

interpretation of the results, the consequences of neglecting the Reynolds number

similitude should be examined carefully, particularly in the case of vortex shedding

about structures with curved surface. Since vortex-shedding formation around such

structures is sensitive to Reynolds number. There is a shift of separation point with

the change of Reynolds number. The critical Reynolds number also dependents on

the surface roughness of the solid boundary and the turbulence level of the

oncoming flow. Civil engineering structures usually have sharp corners, their flow

separation points do not shift and the flow pattern is believed to be less sensitive to

the change of Reynolds number. However, a broad wake after separation from the

upstream corner may reattach to the body surface, depending on the aspect ratio of

the structure. The reattachment results in the reduction of drag force and increase of

Strouhal number in general. The critical body aspect ratio at which this change

occurs depends on Reynolds number as well as the corner radius and the air stream

turbulence level. This factor is also affected by the wind tunnel blockage ratio.

As the turbulence effect on the flutter boundary is an important issue, it is essential

to simulate the velocity spectra correctly if the experiment is done with turbulence

effect on the flutter derivatives being taken into consideration. A sectional model

may be exposed to flow of any mean speed profile and to different types of wind

turbulence. A sheared mean speed profile may not really be important for full-scales

with good clearance from water or land, but wind turbulence is presumably very


35

important. The target values for turbulence intensity may be encircled in the

satisfactory manner, but the flow inevitably contains too much fine-grained

fluctuation and too little at the low-frequency end, corresponding to long wave

eddies approaching the model. Therefore, while some results may hold their value

as reasonably good predictions for full-scale, the buffeting response of the model

will not be linked to that of the full-scale (Hjorth-Hansen 1992). Lack of similarity

of integral length-scales of the turbulent flow is not a unique feature of sectional

models, but rather a common error source for any stand-in for the full-scale.

For a Reynolds number that is large enough to allow turbulent flow, the flow

structures are almost the same for all Reynolds numbers. This is very important

since achieving the similitude of Reynolds number is in any case impractical.

However, Reynolds numbers does play a role in the existence of the inertia sub-

range of energy spectra.

Froude number represents the ratio of fluid inertia forces and vertical gravity or

buoyant forces. Consequently the Froude similitude becomes important when the

gravity predominates the dissipation of air born particles or wind induced response

of cable-supported structures. Although Froude similitude has been widely accepted

and employed for many aeroelastic studies in the past, it is not an essential

requirement unless the gravity or buoyancy plays an important role. In the

experiment of sectional model testing, the restoring force is provided mainly by

elastic force in the spring; the aeroelastic response does not require the Froude

similitude.

Density ratio is referred to the ratio of structural material density to air. In the case

where the model is only an equivalent model, which simply maintains the

geometrical and dynamical characteristics of the prototype, this ratio is thus of no

consequence.

The magnitude of structural damping in the system is obviously an important

parameter for the predication of structural response. The problem, however, is that


36

its magnitude is not known precisely until the structure has been constructed. As a

mater of fact, even for the existing structure, the magnitude of structural damping is

uncertain because of the difficulty in measuring its value and its dependency on

amplitude.

For the identification of the flutter derivatives, structural damping is not important.

Singh (1997) used additional dampers to increase the stability of very bluff model

when it became unstable under low wind velocity. For the study of vortex-shedding

effects, system damping and mass cannot be overlooked. Peak bending or rotational

amplitudes of beams under vortex-shedding excitations are clearly related to the

damping and mass of the system.

Structural details have significant effects on the flutter derivatives measured in wind

tunnel experiments. This was manifested by Ehsan et al (1993). Therefore, the

design of structure details for the bridge deck model is very important. The rail, for

example, need to be duplicated in such a way that the static force and moment

generated by it in the full scale and the model satisfy the requirement of similarity

in terms of geometric scale factor. The difficulty is that the dimension of the model

is very small and the Reynolds number in the wind tunnel corresponding to its

dimension is rather small, the drag or lift coefficients may have a significant change

if the rail was faithfully duplicated. In practice, instead of a replica of the full-scale

rail, wire mesh is usually used so that similarity requirement can be met. In such a

small dimension, Reynolds number effect on the drag coefficient of the circular

sectioned wire is not large.

End plates are usually used to reduce the end effect. For sectional model, it is

necessary to ensure the flow around it is a two-dimensional flow. However

according to the work of some researchers e.g. Hjorth-Hansen (1992), the end effect

is, hopefully, not significant.


37

3.1.2 Other Model Types

Besides sectional model, partial and full models are also used for the study on

aeroelastic behavior of cable-supported bridges.

Full models (Irwin, 1992) can simulate turbulent effects, topographic effects, mode

effects and responses during different construction stages. However, they are very

expensive and time consuming to build. Besides, a full model requires that

similarities pertaining to mass distribution, reduced frequency, mechanical damping

and mode shapes be met. Full models are not suitable for the investigation of flutter

mechanisms since the addition of tower and cable has influences on the bridge deck

behavior, making it even difficult to understand the mechanisms of bridge deck

vibrations.

Taut strip models or partial bridge model (Davenport, 1992) is a three dimensional

but simplified representation of the prototype. With properly designed sectional

model mounted on the taut wires or taut tubes stretched between two anchors, it can

match the lowest vertical, horizontal and torsional modes shape and frequency ratio

of the model to those of the prototype.

Because full and partial bridge models are built at too small a scale to present the

small details, they are not good at responding to vortex shedding. Also, vortex-

shedding excitations can happen at quite moderate wind speeds at full-scale which

scale down to very low values on the aeroelastic model in wind tunnel. This can

introduce the unwanted Reynolds number effects on the mean flow and turbulence

(Irwin, 1998).

3.2 Extraction of Flutter Derivatives

Figure (3.1) shows the sectional model and coordinates system for the bridge

section. In the experiment, we will denote the vertical, lateral and rotational


38

displacement with h , p andα , respectively. eR is the distance between the center of

mass and center of elasticity.

The motion governing equations for the bridge deck vibration is

)()()( tpxKKxCCxM aestraestr =++++ &&& (3.2.1)

e.g.

)(tpxKxCxM effeff =++ &&& (3.2.2)

in which [ ]Tphx α,,= is the displacement of the rigid model, M is the mass matrix

of the structure, aestreff CCC += is the effective damping matrix, the sum of

structural damping strC and aerodynamic damping aeC , aestreff KKK += is

effective stiffness matrix, including structural stiffness strK and aerodynamic

stiffness aeK . )(tp is buffeting force.

B

Re

p

h

U

Center of Mass

Center of Elasticity

Figure 3.1 Convention

The effective matrices of stiffness and damping can be measured with the system

identification method described in the next chapter. Subtracting the effective


39

matrices by structural matrices, aeroelastic derivatives can be obtained. Flutter

derivatives are as follows:

)(2)( 11112

*1 streff CC

BKH −−=

ωρ; (3.2.3)

)(2)( 12123

*2 streff CC

BKH −−=

ωρ; (3.2.4)

)(2)( 121223

*3 streff KK

BKH −−=

ωρ; (3.2.5)

)(2)( 111122

*4 streff KK

BKH −−=

ωρ; (3.2.6)

)(2)( 13132

*5 streff CC

BKH −−=

ωρ; (3.2.7)

)(2)( 131322

*6 streff KK

BKH −−=

ωρ; (3.2.8)

)(2)( 21213

*1 streff CC

BKA −−=

ωρ; (3.2.9)

)(2)( 22224

*2 streff CC

BKA −−=

ωρ; (3.2.10)

)(2)( 222224

*3 streff KK

BKA −−=

ωρ; (3.2.11)

)(2)( 212123

*4 streff KK

BKA −−=

ωρ; (3.2.12)

)(2)( 23233

*5 streff CC

BKA −−=

ωρ; (3.2.13)

)(2)( 232323

*6 streff KK

BKA −−=

ωρ; (3.2.14)

)(2)( 31312

*1 streff CC

BKP −−=

ωρ; (3.2.15)

)(2)( 32323

*2 streff CC

BKP −−=

ωρ; (3.2.16)


40

)(2)( 323223

*3 streff KK

BKP −−=

ωρ; (3.2.17)

)(2)( 313122

*4 streff KK

BKP −−=

ωρ; (3.2.18)

)(2)( 33332

*5 streff CC

BKP −−=

ωρ; (3.2.19)

)(2)( 333322

*6 streff KK

BKP −−=

ωρ (3.2.20)

3.3 The Experiment In this section, the experimental design for the research is presented.

3.3.1 The Wind Tunnel

The experiments were carried out in the Wind Tunnel Laboratory of School of Civil

and Environmental Engineering, Nanyang Technological University. This wind

tunnel is a 24-metre long open circuit environmental wind tunnel with a cross

section of 3-metre wide by 2-metre high. The variable speed controlled fan, the

remote controlled turntable and 3-dimensional traversing rig enable the versatility

of the tunnel for different applications. The maximum wind speed provided by it is

20m/s with a low level of turbulence over the testing length. No active device to

generate the turbulence are used. The models are installed in the middle of the wind

tunnel experimental section, therefore, the wind velocity distribution around it is

considered even. Information about the lateral turbulence in the wind tunnel is

shown in Table 3.1 and Figure 3.2. Vertical turbulence properties were not

measured.

Table 3.1 Intensity of Lateral Turbulence

U m/s %uI 2.15 1.54 5.69 2.04 10.36 1.76 15.59 1.73 19.01 1.71


41

10-1 100 101 102-18

-16

-14

-12

-10

-8

-6

-4

Frequency (Hz)

Pow

er S

pect

ral D

ensi

ty (d

B/H

z)

Figure 3.2 Power Spectral Density of Lateral Turbulence U=17.4m/s

3.3.2 Sectional Models

There are two bridge deck sectional models used in the experiment: a bluff twin

deck model and a partially streamline box girder model. They are chosen as

representatives of different types of bridge decks.

1. Model A (Figure 3.3) is a bluff twin deck model with an opening between

the two decks. Configuration of the model follows the design of Ting Kau

Bridge1. The scale ratio for the model is 1:80 to allow for adequate detailing

1 The Ting Kau Bridge (King et al, 1997) with its 1177m length is one of the longest cable-stayed bridges in the world. It consists of two main span of 448m and 475m and two side spans and provides a vital link in Hong Kong’s new Route 3, connecting Hong Kong Island, Kowloon and the new airport on Lantau to the New Territories and the border to Mainland. The deck has a varying chord with a minimum of 42.28m excluding fairings, and is separated into two carriageways. Each carriage way has a varying a chamber of approximately 2.5% with two longitudinal L-shaped edge girders, and I-shaped cross beams every 4.5m. Every 13.5m cross beams extend to connect with the other carriageway across at 5.26m void. The deck is very slender with a very high chord to depth ratio of approximately 25, making it potentially susceptible to aerodynamic actions. Since the bridge is situated in a typhoon area it will during its lifetime be subjected to very strong wind. It is thus important to establish the relationship between wind actions and bridge response. Furthermore, because of its slender section and bluff edges the aerodynamic stability is of great concern.


42

in the construction. The length of the model is 1.527m representing 121.6m

of deck with 10 cross beams at the ends and at equal intervals. The width is

0.574m and weight is 9.11kg. Some details of the full scale bridge were

duplicated, and the model was made for a previous experiment (Brownjohn

and Choi 2001). Due to its limitations, this model can only be tested in

2DOF or 1DOF.

Because the prototype is very thin, the model was unavoidably made

flexible. This affects the quality of the measured static wind load on the

model since the model vibrates itself when the wind is applied. However,

during the experiment, the model was observed having elastic bending

amplitudes much smaller than the rigid body motion amplitudes. According

to an earlier experiment on the model (Brownjohn and Xia 1999), the first

bending frequency is about 30Hz, considerably higher than the frequency of

the vertical rigid body motion, which is about 3.5Hz. Furthermore, the force

relating to “elastic vibration” of the model itself is with zero mean, so, the

measured static coefficients are still usable, but the frequency component is

not correct, therefore, cannot be used. For the identification of flutter

derivatives, the time history of displacement instead of the acceleration was

measured. There is no doubt that the two displacement components due to

rigid body movement and elastic bending are not of the same order. The

former should be much larger than the latter. Therefore, the stiffness

problem does not affect the flutter derivative identification very much.

Figure 3.3 Model A: Twin Deck Bluff Model (Dimension in mm)


43

2. Model B (Figure3.4) is a partially streamlined box girder deck model with a

wing on each side. Its prototype is the Humber Bridge2. More information of

the bridge can be found in Chapter 6. Length scale is 1:80. Its length is

1.08m; width is 0.35m; weight is 7.7 kg. This model was specifically made

for this study, so it can be tested in 3DOF experiment. The sectional model

is also strong since it has truss structure with a Perspex coat. The frequency

component of the wind load on the stationary model was also not correct,

since the rig was vibrating, which was all unfortunately transmitted to the

load cell.

227355

91

53

92

40 2464

20

92

402464

perspex coat

Figure 3.4 Model B: Streamlined Box Girder Model (Dimension mm)

2 The Humber Bridge connects the towns of Barton and Hessle across the Humber estuary upstream from Hull. Its main span of 1410m was the longest at the time it was built, with side span of 530 and 280m. The argument is not symmetric. It was opened to traffic in July 1981. All three spans at Humber are supported by cables. In the main span and Barton side span, the deck shapes parts of the circles in the vertical plane, tangential to each other at the tower. The Hessel side span is straight. The steel box-sections are 22m wide and 4.5m deep, carrying two traffic lanes each way. The cables consist of 37 strands each of which contains 404 galvanized drawn wires of 5 mm diameter. In each cable above the Hessle side span there are additional 800 wires divided into 4 strands fixed to strand shoes at both the main anchorage and tower saddles. This increased cable area is required to carry the greater tension generated in the steeper slop of this span. For 325.8 m either side of the mid-point of the main span, the hangers are articulated in both lateral and longitudinal directions at deck level; elsewhere, the articulation is in the longitudinal (i.e. about the X-axis) direction only. The towers at Humber are slip-formed concrete box section, 6 by 6 m at the base and 4.5 by 4.5 at the tower tops. The foundation for the Hessle tower is sited on the high-water line, being essentially a reinforced concrete slab 44 by 16 in plan and 11.5 deep. On the Barton side, the tower is supported on a concrete pier, 16 m deep, resting on twin hollow circular caissons each 24 m in diameter.


44

3.3.3 The Experimental Setup

1 Free Vibration Test:

A three DOF suspension system was developed by modifying the existing

two DOF suspension rig. The original 2DOF rig was made for the

experiment with model A in a previous study. The amendment was to

support the model using an arrangement of springs hung from the rig such

that the model would have vertical, rotational and lateral degrees of freedom

and to measure the time history of displacement associated with the degrees

of freedom.

The details of the setup for free vibration tests are shown in Figure 3.5. The

suspension system consists of a pair of steel frames with one end plate (not

shown) on each of them. The frames can be fixed to the floor and ceiling of

the wind tunnel. Eight vertical springs were hung from the frames to hold

the sectional model in position. Another four lateral springs were used to

provide lateral stiffness. The stiffness of the springs is chosen to make the

model vibrate at a suitable frequency, so that a suitable range of reduced

wind speeds can be covered by the experiment. The separations between the

springs can be adjusted according to the required rotational frequency. The

vertical tension in the springs can be adjusted by the height of the four bars

on the rig. Suitable lateral tension is obtained by relocation of the threaded

bars connecting the lateral springs. The four threaded bars are fixed to the

frame by screw nuts, so that their position is adjustable.

Four laser displacement sensors were used to measure the displacement

history of the model, with one below the front edge and another one below

the rear edge to record the vertical and rotational displacement, still another

two at two ends of the model for lateral displacement measurement. They

were all fixed on the frames. A single hot-wire anemometer was mounted in

front of the sectional model between two end plates to record the wind


45

speed. The anemometer has its own data acquisition system but is also

linked together with the four displacement sensors to an 8-channel data

acquisition system, which translated the voltage/current signals and

transmitted them to the connected notebook.

2 Wind Force Coefficient Measurement

The setup for static wind load coefficient is shown in Figure 3.6. The frames

are the same. The four lower springs and two lateral springs were replaced

by components with adjustable length. By adjusting the length of these

components, the attack angle could be changed. Springs were also pre-

stressed to make sure the model would not vibrate extensively in the wind.

Six load cells were installed: four of them to measure the vertical wind

forces, two to measure the lateral wind load. Therefore, the lift, moment and

drag forces could be calculated.

A single hot-wire anemometer was also mounted in front of the sectional

model between two end plates to record the wind speed. The Single hot-wire

anemometer and six load cells were linked to an eight channel data logger,

by which the input signal was translated and transmitted to a computer.

Figure 3.5 Setup for Free Vibration Test (One End)

Spring

Light of the Displacement Sensor

Spring

Model

Light of the Displacement SensorSpring


46

Figure 3.6 Setup for Static Force Coefficient Measurement (One End)

3.3.4 The Experimental Procedure

The experiment procedure for free vibration tests is as follows:

1) The suspension system was set up (Figure 3.5).

2) One of the bridge deck models, model A or model B, was installed.

Lateral DOF was constrained with thin wire for 2DOF experiment

3) Measurement of the system properties was made with free decaying

vibration of the model by giving it an initial displacement when there

was no wind. This test was repeated several times to allow a reliable

identification of the system parameters.

4) Five minute ambient response records of the model under the action of

wind were recorded.

5) Three to five free decay tests were conducted under the same condition

as in item 4) except for the initial conditions.

6) Wind speed was increased. Experiments in steps 4) and 5) were

repeated. The wind velocities varied from minimum to maximum wind

speed at a reasonable interval.

7) For model B, lateral constraints were released. Experiments from 3) to

6) were repeated for 3 DOF experiment.

Spring

Load Cell

Adjustable Section

Adjustable Section

Load Cell

ModelSpring


47

8) After the wind speed reached the upper limit (20m/s), the other model

replaced the first one. Experiments from 2) to 7) were repeated.

The experiment procedure to measure the static wind force is as follows:

1) The suspension system was setup (Figure 3.6).

2) One model was selected, and installed.

3) The mean attack angle was adjusted to 00.

4) The wind was applied. The loads in the six load cells were recorded

5) The wind speed was increased by the interval used in dynamic

experiments.

6) Steps 4) and 5) were repeated to cover the range of wind speed.

7) After the upper limit of the wind speed was reached, the mean attack

angle was increased or decreased by 1~20

8) Experiments in 4)-7) were repeated until the maximum attack angle 50

(or -50) was reached.

9) The other model was installed. Experiments from 3) to 8) were repeated.

3.3.5 Calibration

Before being used in the testing, all displacement sensors and load cells were

calibrated by measuring a series of known displacements and weight, respectively.

The measured relationships between voltage output and the input were compared

with those given in manuals. No remarkable difference was noticed. The data given

in manuals were adopted.

The stiffness, mass and moment of inertia are measured by adding a weight at each

end (front and rear) of the model. The system parameters in still air are identified by

the change of the frequencies. Suppose the added mass is m∆ , creating added

moment of inertia I∆ . The stiffness, mass and moment of inertia of the original

system is ),,( αphiKi = , ),( phjM j = and I . There exists the equation:


48

phiKm

KM

ii

i

i

,12 =

∆+=

ω (3.3.1)

or

αααω KI

KI ∆

+=2

1 . (3.3.2)

The mass, stiffness and moment of inertia can thus be identified by least squares

method.

3.4 Basic Measurements

The results of one-dimensional experiments are not to be presented here because the

models were found to be not vibrating well due to additional restraints and the

aeroelastic coupling effect. The aeroelastic coupling couples the free and restrained

DOFs together, demanding a feedback from the restrained DOFs. But the additional

restraints in those directions may cause the feedback to be “unnatural” and further

affect the free DOF vibration. For example, in the experiment with the partially

streamlined box girder model (model B), *3H was found to be large in value,

indicating a strong coupling between the rotational and vertical DOFs. If the

rotational DOF is free, force in the vertical direction will be generated, but in one

DOF experiments, the vertical DOF is restrained when the rotational DOF is free.

The restraint in the vertical direction may become a restraint to the rotational DOF

as well via the aeroelastic coupling. Under this condition, the vibration in the

rotational direction is affected.

Also, the design of the experimental devices did not provide an effective trigger

mechanism to trigger the model to vibrate in one dimension.

Table 3.2 summarizes the experiments conducted in the study. Due to the

limitations of model A, P derivatives were not measured for it. Therefore, the


49

experiments were 2DOF. For model B, both 2DOF experiments and 3DOF

experiment were carried out.

Table 3.2 Experimental Information

Items to measure Model

No.

M

I

Remax DML CCC ,, DML CCC ′′′ ,, *** ,, iii PAH

A 9.27 0.23 3*104 Y/Y/Y Y/Y/Y Y/Y/-

B 8.5 0.029 7*104 Y/Y/Y Y/Y/Y Y/Y/Y

Y=Measured, - =Not Measured

The maximum Reynolds numbers of the experiment is of the order of 104 based on

the depth of the deck. They are almost in the same order of Reynolds numbers in

other researches, for example, by Ehsan et al (1993). The Reynolds numbers in their

experiments were between 5*104 and 1.1*105 based on the section width.

Considering the reference length in the experiments of this thesis does not include

the height of the railings, the range of Reynolds numbers covered is not small.

The following figures show the measured static lift, moment and drag force

coefficients of model A and B vs. angle of attack. The reference length B was the

width of the bridge deck model. Due to the design and availability of experiment

apparatus, the angle of attack was not easy to adjust while keeping suitable pre-

tension in the load cell. Only a limited number of angles were tested, especially for

model B. The sampling rate was 20Hz, the record length was 300 seconds.

CL of Model A

-0.30-0.20-0.100.000.100.200.300.40

-8 -6 -4 -2 0 2 4 6 8

deg

Figure 3.7CL of Model A


50

Cm of Model A

-0.08

-0.06

-0.04-0.02

0.00

0.02

0.04

-8 -6 -4 -2 0 2 4 6 8

deg

Figure 3.8 Cm of Model A

Cd of Model A

0.00

0.05

0.10

0.15

0.20

0.25

-8 -6 -4 -2 0 2 4 6 8

deg

Figure 3.9 Cd of Model A

CL of Model B

-0.50-0.40-0.30-0.20-0.100.000.100.20

-6 -4 -2 0 2 4 6

deg

Figure 3.10 CL of Model B

Cm of Model B

-0.10

-0.05

0.00

0.05

0.10

0.15

-6 -4 -2 0 2 4 6

deg

Figure 3.11 Cm of Model B


51

Cd of Model B

0.00

0.05

0.10

0.15

-6 -4 -2 0 2 4 6

deg

Figure 3.12 Cd of Model B

Table 3.3 shows the corresponding derivative of the static force coefficient with

respect to the angle of attack: 0=ααd

dCi , ),,( PMLi = . The angle is in rad. These

derivatives are obtained by curve fitting the measured static force coefficient then

differentiating the equation when 0=α .

Table 3.3 Derivatives of Respective Static Force Coefficients

LC ′ MC ′ DC ′

Model A 2.6 1.0 0.2

Model B 4.0 1.0 -0.17

Summary In this section, issues relating to the wind tunnel experiments with sectional models

were reviewed. Design of the experiments for static force coefficient and flutter

derivatives was outlined. Some basic measurements were performed. The static

wind force coefficients of lift, moment and drag force were measured. Respective

derivatives of these coefficients with respect to the angle of attack were also

measured. The identification of flutter derivatives is to be discussed in the following

chapter.


52

CHAPTER FOUR

Method Used To Identify Flutter Derivatives

Abstract

Many identification methods have been used effectively to extract flutter derivatives.

In this section, the method of ERA (eigensystem realization algorithm) is reviewed

and selected as the tool to do the identification. The flutter derivative model is a

linear equation with reduced frequency dependent parameters. In the experiment, at

each wind speed level, it reduces to a linear equation with constant coefficients

corresponding to specified reduced frequency. It suffices to identify it as a time-

invariant linear dynamic system.


53

4.1 Introduction

Many methods have been tried for the identification of flutter derivatives.

Shinozuka (1982) used ARMA models to identify flutter derivatives. Yamada et al.

(1992) used the extended Kalman filter method to retrieve the aforementioned

parameters from coupled vibration time histories. Bogunovic-Jakobsen (1995)

followed Hoen (1991, 1993) to employ ambient response to recover covariance

function estimates for the identification of aeroelastic derivatives. There are many

other exercises not mentioned. In this work, the last mentioned method is followed.

The essence of this method is to employ the classic idea in stochastic identification

of using output covariance as Markov parameters, which was further used in

Eigensystem Realization Algorithm (ERA).

ERA is a minimum order realization method. Although for experiments with

sectional model, the number of degrees of freedom is known beforehand and there

is no need to determine the system order in the sense of minimum state-space

dimensions, this method offers us a good framework to discuss the errors in the

identification through singular-values.

4.2 Basics of ERA

In this part, the method of ERA is reviewed after a short introduction of the

historical development of the method.

4.2.1 The History of ERA

It is common to construct state-space representation of linear systems. Gilbert

(1963) and Kalman (1963) introduce the important principles of realization theory

in terms of the concepts of controllability and observability. Ho and Kalman (1966)

showed that the minimum realization problem is equivalent to a representation


54

problem involving a sequence of real matrices known as Markov Parameters (pulse

response function). By minimum realization is meant a model with the smallest

state-space dimension among systems realized that have the same input-output

relations within a specified degree of accuracy.

A common weakness of the schemes of that time is that the effects of noise on the

data analysis were not evaluated. Zeiger and McEwen (1974) proposed a

combination of the Ho-Kalman algorithm with the singular–value decomposition

technique for the treatment of noisy data. However, no theoretical or numerical

studies were reported by them. The singular-value decomposition technique has

been widely recognized as being very effective and numerically stable. It was used

by Juang and Pappa (1984) to form ERA for modal parameter identification of

flexible structures.

ERA consists of two major parts, namely, basic formulation of minimum-order

realization and modal parameter realization. In the first part, the Hankel matrix,

which represents the data structure for the Ho-Kalman algorithm, is generalized to

allow random distribution of Markov parameters generated by free decay response.

Then Ho-Kalman algorithm is combined with singular-value decomposition to

identify the dynamic system. In modal space, modal parameters are identified.

4.2.2 The Method of ERA

(Juang and Pappa, 1984, 1986; Juang 1994)

Linear dynamic systems can be expressed in the form of state equations. The

collection of state variables in state equations is the state of the system. The set of

all possible states is the state space of the system.

Rewriting the system equation of a bridge deck

)(tpxKxCxM effeff =++ &&& (4.2.1)


55

in state equation form, with input )(tp , output Tphx α,,= , we have,

)(tpBXAX cc +=& (4.2.2)

)(tDvCXY += (4.2.3)

with state matrix

−−

= −−effeff

c CMKMI

A 11

0, (4.2.4)

input matrix

= −1

0M

Bc , (4.2.5)

output matrix C , feed through matrix D and state variable TTT xxX &,= .

For bridge deck experiments, C= [ ]0I and D is zero.

The discrete time state space equation of the same systems above is in the form:

)()()1( iBpiAXiX +=+ (4.2.6)

)()( iCXiY = (4.2.7)

Where tAceA ∆= (4.2.8)

and

∫∆

=t

cA dBeB c

0ττ (4.2.9)


56

Let 1)0( =p and ),,2,1(,0)( L== iip giving rise to a sequence of pulse-response

matrices kY :

,0 DY =

,1 CBY =

,2 CABY =

,L

BCAY kk

1−= (4.2.10)

The constant matrices in the sequence are known as Markov parameters. Since the

Markov parameter sequence is the pulse response of the system, they must be

unique for a given system. It may be shown that Markov parameters are invariant

when subjected to coordinate transformation of the state vector, say )()( iTXiz = ,

yielding the same expression:

,,2,1;)( 1111 L=== −−−− kBCATBTATCTY kkk (4.2.11)

The realization starts from forming the generalized Hankel matrix composed of

Markov parameters.

( )

=−

−+++−+

+++

−++

21

21

11

1

βααα

β

β

kkk

kkk

kkk

YYY

YYYYYY

kH

L

MOMM

L

L

(4.2.12)

For the case when 1=k ,

( )

=

−++

+

11

132

21

0

βααα

β

β

YYY

YYYYYY

H

L

MOMM

L

L

(4.2.13)

DY =0 is not in )0(H .


57

If nn >> βα , ( n is the order of the state-space model), the matrix )(kH is of rank

n . To confirm this we can decompose )(kH into three matrices

βα QAPkH k=)( (4.2.14)

Where block matrix αP is the observability matrix:

=

−1

2

α

α

CA

CACAC

PM

(4.2.15)

and βQ is the controllability matrix:

[ ]BABAABBQ 12 −= ββ L (4.2.16)

If the system is controllable and observable, αP and βQ are of rank n . Therefore

nkHr =))(( . (4.2.17)

If the first Block Hankel matrix )0(H is factored by singular value decomposition

(SVD): TSRH Σ=)0( (4.2.18)

where the columns of R and S are orthonormal and Σ is a rectangular matrix

Σ=Σ

000n (4.2.19)

with


58

[ ]nn diag σσσ L21=Σ (4.2.20)

having diagonal elements monotonically non-increasing, i.e.

nσσσ ≥≥≥ L21 . (4.2.21)

Hence the number of non zero singular values n equals the rank of )0(H .

If R and TS are partitioned as

][ 0RRR n= and ][ 0TT

nT SSS = , (4.2.22)

observation on the expression suggests that matrices αP and βQ are related to R

and TS , respectively. Indeed,

2/1

nnRP Σ=α (4.2.23)

T

n SQ 2/1Σ=β (4.2.24)

In reality, the Block Hankel Matrix will always be full rank due to noise

components in the signal, non-linearity or computation round-off. However the

singular-value decomposition technique has been widely recognized as being very

effective and numerically stable in that small perturbations in the matrix generate

small perturbations in Σ . Some of the singular-values, inn σσσ ,,, 21 L++ may be

small and negligible in the sense that they contain more noise information than

system information. It is equivalent to saying that the directions determined by them

have less significant degrees of controllability and observability relative to the

noise. In this case, we should assume that:

βα QAPkH k≈)( (4.2.25)

Specifically,


59

βαQPH ≈)0( (4.2.26)

If the noise component in the Hankel Matrix is not too big, the solution should be

still acceptable. Therefore we have

T

nnn SARAQPH 2/12/1)1( ΣΣ== βα . (4.2.27)

Therefore 2/12/1* )1( −− ΣΣ= n

Tn

Tnn SHRA (4.2.28)

Tnn SB 2/1* Σ= (4.2.29)

2/1*

nnRC Σ= (4.2.30)

Where ' * ' denotes similar transformation of respective matrix, since it is known

that the identified result is the same with respect to similar transformation. In the

experiments to extract aeroelastic derivatives, all the degrees of freedom are

measured; it is possible to transform the identified result back to observability

canonical form.

Because it is possible to ensure in the experiment that the output matrix is

]0[IC = (4.2.31)

for the observability canonical realization, the observability matrix is equal to the

identity matrix, i.e.

ICAC

Q =

= (4.2.32)

Under the nonsingular transform,


60

*TXX = (4.2.33)

where T is a nonsingular transform matrix. Applying this in the state-space

equation,

)()()1( 1*1* iBpTiATXTiX −− +=+ (4.2.34)

)()( * iCTXiY = (4.2.35)

the identified observability matrix can be expressed as

TQTATCTT

CTAC

CQ ==

=

= −1**

** (4.2.36)

So, we have the original discrete time state matrix, output and input matrices as:

1* −= TTAA (4.2.37)

*TBB = (4.2.38)

1* −= TCC (4.2.39)

The method of ERA is completed.

Summary

In this section, system identification methods for the identification of flutter

derivatives were reviewed. ERA is the method to be used in the study.


61

CHAPTER FIVE

Effect of Relative Amplitude and Lateral Aerodynamic Derivatives on Bridge Deck Flutter I:

Experimental Detection of Nonlinearity in Self-Excited Forces

Abstract:

The self-excited wind forces on a bridge deck can be nonlinear even when the

vibration amplitude of the body is small. In this chapter, experiments detecting the

nonlinearity are described with the concept of “relative amplitude”, i.e. the

amplitude of the externally triggered free vibration relative to the envelope of the

ambient response of an elastically supported rigid sectional model. The effect of

relative amplitude on flutter derivatives and on the flutter boundary reveals, from

the structural point of view, a complex relationship between the self-excited forces

and the “structural vibration noise” due to signature turbulence related buffeting

forces. Although the aeroelastic forces are linear when the body motion due to an

external trigger is not affected significantly by the signature turbulence, they are

nonlinear when the noise component in the vibration due to the signature

turbulence related buffeting forces cannot be neglected. Two and three-dimensional

wind tunnel experiments with two types of sectional models were performed to

detect the existence of nonlinearity in the interactive wind forces.


62

5.1 Introduction:

This research studies the characteristics of interactive wind forces on bridge decks

at flutter.

Under the action of net wind forces on the bridge, the structure may deflect, causing

it as a whole to respond in its natural vibration modes. Some prominent modes may

attract interactive aerodynamic forces that lead to modal auto-excitation. In this

situation, the strong linear tendencies of a large elastic structure actually

predominate, permitting a simplified linear appreciation of what is indeed a highly

complex nonlinear phenomenon at least insofar as detailed loading is concerned. By

employing a flutter derivative model, a linearized representation of the self-excited

wind loads is obtained.

Previous researches (Scanlan 1997, Falco et al 1992) have indicated amplitude

dependency of the measured flutter derivatives. From this point of view,

nonlinearity does not exist for the small amplitude analysis. As a matter of fact, the

self-excited wind force due to a pulse input can be nonlinear even if the response

amplitude is small.

To illustrate this, it is necessary to include another issue: turbulence effects on

flutter derivatives and, further, on the flutter instability. As criticized by Cai et al

(1999), many theoretical and experimental studies of flutter were conducted in

smooth flow (Scanlan 1978a, 1978b; Namini 1989) for the reason of avoiding both

mathematical complexity and technical difficulty. Scanlan and Jones (1990) and

Jain et al. (1996) argued the effect of turbulence could be reflected by testing the

flutter derivatives under suitable conditions of turbulence and incorporating the

span-wise coherence effect of flutter derivatives into the formulation. By

recognition of a span-wise diminution of coherence in the associated flutter

derivatives, a higher critical wind speed for flutter is usually expected. The

argument is supported by experimental observations in wind tunnel. Typically, as

has been observed repeatedly, an extended span bridge model in the wind tunnel


63

becomes unstable at a certain crosswind velocity under smooth approaching flow

but at a distinctly higher one under turbulent flow (Scanlan 1997).

Lin and Li (1993, 1995) followed by Cai et al. (1999a) introduced the stochastic

method for predicting instability of wind-excited structures, making it possible to

consider the self-excited forces as random values to model the random nature of

turbulent wind. Flutter derivatives in smooth flow are used. The self-excited forces

are presented as a function of fluctuating wind speed.

The major problem of the methods, however, is that they do not carry the

information that the existence of buffeting response may change the feature of the

interactive forces and flutter derivatives, as will be shown in this chapter. A set of

flutter derivatives identified for flutter instability analysis may not be suitable for

the analysis when effects of buffeting forces are being considered. The argument of

a span-wise diminution of coherence in the associated flutter derivatives (Scanlan

1997) is a supplement for the free decay vibration testing in turbulent flow. Studies

(Sarkar et al 1994) have shown that turbulence effect on the measured flutter

derivatives is not noticeable. However, the tests were based on free decay vibration

testing. The tests with free decay or forced sinusoidal motion usually deal with a

clear harmonic motion, even if the turbulence is included. This situation, however,

exaggerates the effect of structural motion by forcing the surrounding fluid to

behave in phase with the body, increasing the span-wise coherence of the flutter

derivatives. If the magnitude of the pulse response of the rigid model is reduced to a

smaller level, i.e. within the ambient response vibration envelope, so that the

ambient response property of the aeroelastic system can be preserved, the effect of

turbulence on flutter derivatives will be manifested. This is a condition under which

more information from the fluid may be obtained.

5.2 The Relative Amplitude Effect

To investigate the nonlinearity in the self-excited wind forces, we shall limit our


64

discussions to an elastically supported rigid sectional bridge model subjected to a

pulse input to trigger the model to vibrate. The interactive forces under

investigation are the self-excited wind loads generated by the pulse response of the

rigid body.

The relative amplitude effect is defined as the triggered vibration amplitude of the

model relative to “structural noise” in the vibration due to the ambient wind

excitation. To quantify it, we define the relative amplitude ∆= /ARa (Figure 5.1),

where A is the mechanically triggered vibration amplitude and ∆ is a characteristic

measurement representing the ambient vibration magnitude.

The effect of relative amplitude on flutter derivatives and on the flutter boundary

reveals, from the structural point of view, a relationship between the self-excited

forces and the “structural vibration noise” due to buffeting forces. If ∞→aR , the

triggered vibration is totally smooth and the effect of turbulence is negligible. If

0→aR , the triggered vibration is severely affected by the ambient dynamic wind

load. However, in the experiments presented in this chapter, the turbulence in the

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1

1.5

∆

A

s

Figure 5.1 Definition of Relative Amplitude


65

oncoming flow was not generated artificially due to the limitations of equipment.

The buffeting response in the following experiment is mainly due to the signature

turbulence and turbulence in the approaching flow (turbulence intensity was less the

2% during the experiment, Table 3.1).

5.3 Physical Significance of Flutter Derivatives with Different

Relative Amplitudes

Discussions in the previous section may be summarized from another point of view.

We also limit our discussions to the interactive force on an elastically supported

rigid sectional bridge model due to a pulse response. In the wind tunnel experiment,

the sectional model is always responding to the ambient wind excitation, namely the

turbulence in the oncoming flow, and the signature turbulence generated by the

bluff structure itself when wind is passing by. This ambient response forms the

“environment” for the impulse response to exist. It could be a different story if the

pulse response of the sectional model is strong and destroys the ambient response,

or the pulse response is small with all the “ambient properties” of the aeroelastic

system unchanged.

The difference will be told by the various flutter derivatives identified with different

relative amplitudes. Experiments will show whether or not the turbulence effect

will manifest itself when the relative amplitude is small and causes nonlinearity in

the self-excited force.

For a flexible bridge in the wind, if there is no relative amplitude effect considered,

usually, the flutter analysis predicts the critical wind speed for a smooth sustained

sinusoidal motion of the deck, which indicates the bridge is going to lose its

stability due to negative damping. However, a “smooth” sinusoidal motion is rarely

the real case when there are disturbances in the aerodynamic forces. The ambient

vibration exists. An initially small amplitude sinusoidal motion can be only


66

considered as a “noisy” sinusoidal motion. It needs to be answered in the first place

whether or not the noisy sinusoidal motion will grow in the “environmental noise1”

and become larger in amplitude making itself smoother (the ambient vibration then

becomes relatively unimportant) before a flutter prediction is applied.

Based on this understanding, in reference to the frequency domain method to

predict the flutter boundary (Jain et al. 1996), there should be two thresholds AmbU

and TranU of the wind speed. The former is for a signal to diverge in the context of

ambient response, the latter is for a large amplitude signal to diverge after going

outside “ambient vibration envelope”. If a signal cannot diverge in the former case,

i.e. AmbUU < , it would not go to the stage of latter case. Theoretically, it is stable to

wind excitation. If the signal diverges from ambient response to larger amplitude

but subsequently decays, i.e. TranAmb UUU << , the response is bounded. If the

signal diverges in both cases, i.e. UU Amb < & UUTran < , the system is not stable.

It is interesting to consider the case where AmbTran UU < . Under this condition, the

judgment of the onset of flutter based on AmbU is not safe, since there could be other

sources of excitation contributing to the dynamic response. A strong gust, for

example, could push the bridge deck outside the normal ambient vibration

envelope. After the gust, the bridge deck may decay back from outside into the

ambient vibration envelope. Before the gust, the critical wind speed for flutter is

ambU , after the gust, however, the critical wind speed may change to tranU , which

may be lower than ambU . This is shown in Figure 5.2. If flutter happens in this way,

a transient rather than steady critical flutter wind speed exists. It will be

conservative to use TranU as the design flutter wind speed in this case.

1 This response is termed “noise” instead of buffeting response because the buffeting response of the flexible prototype carries many modes of different frequencies, which may be very difficult, if not impossible, to be reflected by sectional model testing. The discussion in this research is limited to the effect of “noise”. No special spectrum of it will be considered.


67

5.4 Use Of Output Covariance As Markov Parameters

It is difficult, if not impossible, to identify the flutter derivatives when the pulse

response is small in amplitude compared with the ambient vibration. However a

pulse response or Markov Parameters, if the discrete form is considered, can be

produced mathematically by using output covariance of ambient vibration. (Hoen et al

1993, Jakobsen and Hansen 1995). This corresponds to the extreme case where the

pulse response amplitude tends to zero.

The input term, buffeting force on the bridge deck model, is random and unknown

and assumed to be white noise. We assume the displacement measurement is zero

mean. If the deck has a static displacement due to the static wind load, it can be

treated as centered process, i.e.

[ ])()()( tYEtYtY −= (5.5.1)

where [ ]•E denotes expectation operation.

Therefore, we always assume we are dealing with a zero mean process.

As mentioned in Chapter 4, it is a classical result in stochastic identification that the

output covariance can be used as Markov parameters of deterministic linear time

invariant system when the measurement noise in the equipment and process noise

U

t

U amb Utran

Figure 5.2 Non-Stationary Flutter Boundary


68

(the small scale buffeting force due to signature turbulence and local variation of

wind speed in the oncoming flow) is white and zero mean. Markov parameters are

the discrete version of pulse response function; a proof is presented in the Appendix

I. The output covariance produces a new discrete state-space model of

)0(,,, iCCGA , where A is the state matrix; G is the input matrix;C is the output

matrix and )0(iC is the initial condition. Matrix G and )0(iC are defined in the

Appendix I.

One consideration is that within the time period of measurement of the ambient

vibration, the system is actually non-stationary due to uncontrollable experimental

conditions. Therefore, in calculating the output covariance, some numerical

consideration has been taken to overcome the problem, which will be addressed in

the following section.

5.5 Numerical Considerations for Computation of Output

Covariance

The structure function was introduced by Kolmogorov (1941) in his study of locally

isotropic and homogeneous stochastic turbulence. It is defined as:

[ ] ))()0((2)()(1lim)(

0

2 τττ tt

T

TCCdttYtY

TZ −=+−= ∫

∞→ (5.5.1)

where )(τtC is the auto-covariance function of the measurement after time τ+t .

For ambient vibration, the covariance function for zero lag time is constant, and the

structure function is the sum of the negative covariance function and a constant. If

the measurement shows slow fluctuation or has a time varying trend, i.e. shows

marked derivation from a stationary behavior, the method of structure function may

be advantageous because the structure function can tolerate more low frequency

noise than the correlation function (Solnes 1997).

Consider, for example, a simple harmonic function ttx ωsin)( = without random


69

amplitude and phase, the process can be shown to have an autocorrelation

function ttRx ωcos21)( = , by taking care to integrate over an integer number of half

circles πω nT = . Since this condition is almost impossible to realize in practice,

systematic error is introduced. Near the origin, 1<ωτ , the limit of error can be

shown as:

TTRx ω

ωτ2

)2sin(21)( −≈∆ . (5.5.2)

For the auto-structural function, the limit of error is

TTZ x ω

ωτωτ2

)2sin(21)( 22−≈∆ . (5.5.3)

This shows that the structural function is less distorted for 1<ωτ , hence is less

susceptible to low-frequency noise.

5.6 The Experiment

The two sectional models A & B described in Chapter 3 were tested. For the bluff

twin deck model A, two-dimensional ambient and transient vibration experiments

were carried out. For the partially streamlined box girder model B, both two and

three-dimensional tests were done with transient and ambient vibration. Transient

and ambient vibrations were recorded under the same condition. No turbulence in

the oncoming flow was artificially generated. The procedures were outlined in

previous chapters. In all of the experiments the signal outputs from the laser

displacement sensor and anemometer were recorded at 200Hz sampling rate.

System identification procedure was then applied. The method used was ERA

(eigensystem realization algorithm) as described in Chapter 4. After the state

matrices A and cA were recovered, the aeroelastic system parameters were

converted to flutter derivatives, the conversion is given in Chapter 3.


70

5.7 Results and Discussions

Figures 5.3-5.4 show typical experimental measurements of transient and ambient

vibration time histories and their Fourier transform at U=17.5m/s. Values in the y-

axis of FFT plots are just for comparison purpose. Figure 5.5 shows a typical

calculated output covariance )(kCi ( 1...2,1 −= Nk is the number of lags) from the

ambient response of the model B using sampling rate of 50 Hz. This sampling rate

is chosen to provide a longer covariance signal without increase the calculation

burden in the realization process.

Figures 5.6 show the flutter derivatives vs. reduced wind velocity (RU) of model A

identified with transient and ambient vibration. Figures 5.7 & 5.8 show the flutter

derivatives vs. RU of the partially streamlined box girder section identified with

transient and ambient vibration. Figures 5.7 are from 2 DOF experiments while

Figures 5.8 are from 3 DOF experiment. Great differences have been observed. The

features of some of the flutter derivatives are totally different, suggesting

nonlinearity in the self-excited wind load. Therefore, relative magnitude of the

response due to the pulse input has an effect on flutter derivatives.

In the flutter derivatives for the twin deck bluff model (model A), differences are

observed in *2H , *

3H and *4H . However there is no big difference in *

iA ,

)6,,1( L=i . Since this model was tested only in 2DOF experiments, no lateral

motion related flutter derivative was measured.


71

40 41 42 43 44 45 46 47 48 49 50-0.1

0

0.1

h(m

)

40 41 42 43 44 45 46 47 48 49 50-0.5

0

0.5

a(ra

d)

40 41 42 43 44 45 46 47 48 49 50-0.1

0

0.1

time (s)

p(m

)

40 41 42 43 44 45 46 47 48 49 5016

18

20

time (s)

U(m

/s)

Figure 5.3a. Transient Signal of Model B at u=17.5m/s

0 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

h

0 10 20 30 40 50 60 70 80 90 1000

0.02

0.04

a

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

p

Figure 5.3b. FFT of Transient Signal at U=17.5m/s


72

200 210 220 230 240 250 260 270 280 290 300-0.04

-0.02

0

h(m

)

200 210 220 230 240 250 260 270 280 290 300-0.1

0

0.1

a(ra

d)

200 210 220 230 240 250 260 270 280 290 3000.005

0.01

0.015

time (s)

p(m

)

200 210 220 230 240 250 260 270 280 290 30016

18

20

time (s)

U(m

/s)

Figure 5.4a. Ambient Vibration of Model B at U=17.5 m/s

0 10 20 30 40 50 60 70 80 90 1000

0.5

1x 10-3

h

0 10 20 30 40 50 60 70 80 90 1000

0.5

1x 10-3

a

0 10 20 30 40 50 60 70 80 90 1000

0.5

1x 10-3

p

(rad)

Figure 5.4b. FFT of Ambient Vibration at U=17.5 m/s


73

0 100 200-1

0

1H

0 100 200-0.5

0

0.5

0 100 200-0.2

0

0.2P

0 100 200-0.5

0

0.5

0 100 200-0.5

0

0.5

1

0 100 200-0.1

0

0.1

0 100 200-0.2

0

0.2

k0 100 200

-0.1

0

0.1

k0 100 200

-1

0

1

k

A

H

A

P

Figure 5.5. Output Covariance of Model B at U=17.5 m/s

0 5 10 15-20

-15

-10

-5

0H1

Amb Tran

0 5 10 15-30

-20

-10

0

10H2

0 5 10 15-10

0

10

20

30H3

0 5 10 15-6

-4

-2

0

2

4H4

Reduced Wind Velocity: RU

Figure 5.6a. 2DOF H (Model A)


74

0 5 10-1.5

-1

-0.5

0

0.5A1

0 5 10-1

-0.8

-0.6

-0.4

-0.2

0A2

0 5 10-0.5

0

0.5

1

1.5

2A3

0 5 10-0.4

-0.2

0

0.2

0.4A4


Amb Tran

Figure 5.6b. 2DOF A (Model A)

0 5 10 15-10

-5

0

5H1

0 5 10 15-60

-40

-20

0

20

40H2

0 5 10 15-50

0

50

100

150H3

0 5 10 15-20

-10

0

10

20H4


Amb Tran



75

0 5 10 15-1.5

-1

-0.5

0

0.5A1

0 5 10 15-2

-1.5

-1

-0.5

0

0.5A2

0 5 10 15-1

0

1

2A3

0 5 10 15-1

-0.5

0

0.5A4


Amb Tran


0 10 20-15

-10

-5

0

5H1

0 10 20-60

-40

-20

0

20

40H2

0 10 20-50

0

50

100

150

200H3

0 10 20-15

-10

-5

0

5

10H4

0 10 20-10

-5

0

5

10H5

0 10 20-20

-10

0

10

20H6


Amb Tran



76

0 10 20-1

-0.5

0

0.5A1

0 10 20-3

-2

-1

0

1A2

0 10 20-1

0

1

2

3

4A3

0 10 20-0.5

0

0.5

1A4

0 10 20-0.1

0

0.1

0.2

0.3

0.4A5

0 10 20-0.2

0

0.2

0.4

0.6A6


Amb Tran


0 10 20-4

-2

0

2

4P1

0 10 20-10

0

10

20

30

40P2

0 10 20-100

-50

0

50P3

0 10 20-5

0

5

10P4

0 10 20-10

-5

0

5

10P5

0 10 20-10

-5

0

5

10P6


Amb Tran

Figure 5.8c. 3DOF P (Model B)


77

For Model B, *2H and *

3H are among the most sensitive parameters to the vibration

type. In the transient vibration, the interaction between the vertical and rotational

DOF is strong within lower range of the reduced velocity both in terms of

aeroelastic stiffness and damping; while in the ambient response, this effect is

reduced greatly either in 2DOF or in 3DOF experiments. *4A , *

5A , *6A , *

1P , *2P and

*4P are also sensitive to the motion type as can be seen both in 2DOF and 3DOF

experiments.

A remarkable and interesting observation exists in *1P . According to quasi-static

theory shown by equation (2.3.16) in Chapter 2, *1P should be negative for all

reduced velocities and any type of section configuration. From the experimental

result, however, it is observed that the value of *1P is negative for ambient response,

but positive for transient response.

In case of quasi-static theory, a velocity of the bridge deck in the direction of the

wind speed will reduce the drag wind force. If there is only lateral motion p of the

body in the direction of the wind speed U , according to the quasi-static theory

aeroelastic force due to the lateral motion is

pUBCUpUBCD DDae && ρρ −≈−−= ])[(21 22 . (5.6.1)

where ρ is the air density, B is the deck width and DC is the wind drag force

coefficient. The aeroelastic drag force can also be expressed as

UpBKPUDae&*

12

21 ρ= . (5.6.2)

We have KCP D /2*1 −= , where K is reduced frequency. Because DC is positive,

*1P must be negative. The governing equation of the motion is thus in the form of

0)( =+++ pKpUBCCpM strDstr &&& ρ or (5.6.3)

0)21( *

1 =+−+ pKpBUKPCpM strstr &&& ρ . (5.6.4)

The effective damping is thus increased if the quasi-static theory is used or 0*1 <P .


78

However, the quasi-static theory is recognized to be only a nominal form that may

hold for very slow changes in the angle of attack and wind speed. It is, strictly

speaking, asymptotic case approached at high-reduced velocity e.g. the case when

vibration frequency 0→n and the reduced velocity ∞→= nBURU / .

For the triggered 3DOF experiment, however, the acceleration in the lateral

direction is the major reason for the generation of aerodynamic damping. Lin and

Yang (1983) suggested impulse response functions of the self-excited force due,

separately, to velocity and acceleration of the rigid body. By using Duhammel

integral, the acceleration related effect will become velocity related, i.e. added

damping effect. In this case, it is hard to say that the *1P must be negative because

the acceleration can be either positive or negative for a given velocity in the quasi-

static expressions. The time domain SEF model, which will be shown in Chapter 7

of this thesis, also points to the same conclusion: a velocity in the direction of wind

speed could increase the drag force. This may indicate that omitting the derivatives *

iP )6,,1( L=i may not be always safe for the prediction of the flutter boundary.

In the ambient response, the bridge deck model is driven by noise instead of a non-

zero initial condition as is in the transient vibration. The coherence of the interactive

fluid behavior along the axis of the model might be smaller in the former case than

in the later case. This is observed in the reduction in values of some of flutter

derivatives as *2H and *

3H , but not necessarily for others such as *1P and *

4P .

Transient vibration and ambient response represent the two cases where the

structural noise component accounts for a small part of the total signal energy and

100 percent of it, respectively. In the former case, the effect of initial condition is

the main factor of the experiment; while in the later case, the effect of the

“structural vibration noise” is exaggerated. The relative amplitude effect, i.e. the

significance of the consequence of the model being bluff in affecting the interactive

forces, is thus studied. The comparison of these two extreme conditions reveals that

the function of flutter derivatives of a particular section depends not only on

reduced velocity or reduced frequency but also relative amplitude of the vibration.


79

In transient vibration tests, the triggered vibration is clearly larger than the ambient

vibration when it starts, but decays very fast into the ambient vibration envelope. At

the beginning, the relative amplitude effect is negligible, but it is not at the end. The

identification of flutter derivatives corresponding to the free decay vibration

actually deals with a time-varying phenomenon if the vibration decays very fast. On

the contrary, in the experiment with ambient vibration, there is no change in the

relative amplitude because the pulse response is calculated numerically by output

covariance. The phenomenon to be identified is time-invariant under this condition.

As will be shown in the later chapter, the linear identification can approximate the

output covariance of the ambient response better than the triggered transient

vibration. This may support the idea that the relative amplitude effect on flutter

derivatives is not negligible.

Summary

In this study, nonlinearity in self-excited wind forces is detected through the

concept of relative amplitude effect. By comparing the flutter derivatives identified

from triggered free decay and ambient vibration of the elastically supported rigid

sectional bridge model, the relative amplitude effect on the interactive wind forces

is manifested. This effect, from the structural point of view, reveals a complex

relationship between the self-excited forces and the “structural vibration noise” due

to ambient wind excitations. Although the aeroelastic forces are linear when the

body motion due to an external trigger is not affected significantly by the

turbulence, they are nonlinear when the noise component in the vibration due to the

turbulence is not negligible. The major limitation in the research is that the

oncoming flow was considered smooth. The effect of signature turbulence on flutter

derivative was fully manifested but the effect of turbulence in the approaching flow

was not included. This effect needs to be considered in the same manner in a future

study.


80

CHAPTER SIX

Effect of Relative Amplitude and Lateral Aerodynamic Derivatives on Bridge Deck Flutter II:

Numerical Flutter Analysis

Abstract

Differences between flutter derivatives (FDs) identified from ambient and transient

vibration indicate, through the concept of relative amplitude, nonlinearity in self-

excited forces. In view of this, the relative amplitude effect on the flutter instability

of the cable-supported bridges is studied in this Chapter. The effect of P-related-

derivative is also studied by using flutter derivatives identified from 2 and 3 DOF

experiments. The flutter frequencies are obtained from Impedance Matrix Equation

(IME). To solve flutter modes, a numerically robust singular value decomposition

(SVD) method is used. This is because the approximate Impedance Matrix, due to

the unavoidable numerical errors, is not singular practically although it should be

singular theoretically. The direct solution of such equations for flutter modes is

sensitive to the numerical errors in the Impedance Matrix.


81

6.1 Introduction

As has been reviewed in Chapter 2, techniques predicting flutter boundary and

buffeting behavior of cable-supported bridges have been developing in two parallel

ways: one in frequency domain (Scanlan1978, Jain et al 1996), the other in time

domain (Xiang et al 1995, Diana et al 1998, Boonyapinyo 1999, Chen 2000). All

these methods are developed to solve negative damping driven flutter, considering

the change of the pure structural modes to aeroelastic ones in wind.

In this chapter, the frequency domain method developed by Jain et al (1996) is used

to perform the flutter instability analysis, as has been reviewed in Chapter 2. The

method is well established, but a systematic method is needed to solve for the flutter

modes.

The flutter condition is obtained at the reduced frequency satisfying equation

(2.4.16), i.e.

0)( =KE . (6.1.1)

As mentioned by Jain (1997), the nontrivial solution ξ of the aeroelastically

influenced eigenvalue problem stated in equation (2.4.17), i.e.

0=ξE (6.1.2)

indicates the relative participation of each structural mode under the flutter

condition. Solving ξ is useful for multi-mode flutter as it identifies the major

structural mode participating in the flutter state. To determine the vector ξ at flutter,

one of the elements must be preset at a certain value, usually unity for ease of

reference. Some care must be exercised in the assignment of the value due to the

numerical sensitivity of the system at flutter. However, multimode flutter is not

always initiated by the fundamental symmetric torsional mode (Miyata and Yamada

1988; Agar 1989); therefore, it is not always safe to assign the value corresponding


82

to the first torsional mode to be unity. In this chapter, a modification is made and

numerically robust singular value decomposition is used to overcome the

uncertainties in solving the flutter mode.

In the study in the previous chapter, through consideration of the relative amplitude

effect on the flutter derivatives, nonlinearity in self-excited forces was indicated by

comparing flutter derivatives from transient and ambient vibration. In Chapter 5, the

“relative amplitude” was defined as the triggered structural vibration amplitude

relative to “structural vibration noise” due to the ambient wind excitation. From this

point of view, the nonlinearity is related to both the structural vibration and

fluctuations in the flow. The effect of relative amplitude on flutter derivatives and

on the flutter boundary reveals, from the structural point of view, a complex

relationship between the self-excited forces and the “structural vibration noise” due

to signature turbulence related buffeting forces. The ambient response is the

extreme case where amplitude of bridge deck response due to an impulse input is

small comparing to the amplitude of ambient vibration. The properties of the

structural ambient vibration can be thus “preserved”. In the experiment with

transient vibration, on the other hand, the initial condition is larger than the ambient

vibration envelope; the structural behavior predominates by forcing the surrounding

fluid to behave more in phase with the structural vibration.

Differences have been found between the flutter derivatives identified from these

two cases, suggesting further study is needed on the effect of relative amplitude on

flutter prediction. In this section, a sample suspension bridge is used to manifest the

effect of relative amplitude on the flutter boundary.

The effect of the third dimension, the lateral direction, on flutter (Katsuchi 1999) is

also an interesting topic and will be studied at the same time by using the flutter

derivatives identified from the 2DOF and 3DOF experiment. The traditional

understanding based on the quasi-static theory indicates it will be safer to leave

aside the lateral motion related flutter derivatives because *1P should be negative all

the time, contributing to the stability of the bridge. However, the experimental


83

result in the previous section indicates quasi-static theory may fail to be a reliable

guidance and *1P can be positive. It is safer to include the lateral vibration in this

case.

The P derivative effect on flutter is discussed in three cases. The first case is to use

the two-dimensional test result; the second case corresponds to the use of flutter

derivatives from three-dimensional test but the P related flutter derivatives are

assigned zero and the last case corresponds to the use of all the flutter derivatives

identified from three-dimensional experiment.

6.2 The Suspension Bridge and Modeling

Figure 6.1 gives the three-dimensional view and finite element mesh of the Bridge.

An introduction of the bridge was presented in Chapter 3.

Figure 6.1 Plot of the Bridge


84

Table 6.1 summarizes the material properties and other dimensions required in

calculation. The main span is 1410m, with side span of 530 and 280m. The steel

box-sections are 22m wide and 4.5m deep and the shape is the same as the box

girder sectional model (model B in chapter 3). The towers are box section, 6 by 6 m

at the base and 4.5 by 4.5 at the tower tops.

Table 6.1 Material Properties of the Humber Bridge

Cable

Young’s modulus of cables 193 KN/mm2

Young’s modulus of hangers 140 KN/mm2

Area per hanger 0.0021 m2

Main Span 0.29 m2

Area of each cable Side Span 0.31 m2

Box Girder Deck

Young’s Modulus 200 KN/mm2

Axial area of steal 0.73 m2

Second moment of area for vertical bending 1.940 m4

Second moment of area for lateral bending 37.07 m4

Torsional rigidity 4.5 m4

Towers

Young’s modulus 20 KN/mm2

Average axial material area of each leg 20.37 m2

Average second moment of area of each tower leg for longitudinal

bending 66.8 m4

Average second moment of area of each tower leg for lateral bending 68.24 m4

Average torsional rigidity of each tower leg 113.1 m4

To facilitate the inclusion of an aeroelastic load model, 3-D beam deck formulation

was used to model the deck structure. Spar elements (having no flexural stiffness)

were used to present the main cable and hanger. They have the facility to


85

accommodate the initial strain value. The tower was analyzed using beam elements

with tension, compression, torsion and bending capabilities.

The modal analysis was conducted by using commercial software, ANSYS. Table

6.2 summarizes the modal analysis results. These resultant frequencies are generally

5-10% higher than the measured value on the prototype (Brownjohn et al. 1987).

Table 6.2 Dynamic Properties of the Bridge

Mode No. Mode Type Frequency

in (Hz)

Modal

damping

iς (%)

1 L, 1st S 0.0688 0

2 V, 1st S 0.1277 0

3 L, 1st AS 0.1591 0

4 V, 1st AS 0.1646 0

5 V, 2nd S 0.1897 0

13 V, 2nd AS 0.2498 0

14 LT, L, 2nd S; T, 1st S 0.2816 0

16 V, 3rd S 0.3246 0

26 V, 3rd AS 0.4022 0.5*

28 T, 1st AS 0.45853 0.5*

Note: S=symmetrical; AS=anti-symmetrical; L: Lateral; V=vertical; LT=lateral-torsion; T=torsion

and *=Assumed value

Structural modes used in flutter boundary computation are normalized with respect to mass matrix.

Figures 6.2.a-6.2.e show the structural mode shapes. These modes are normalized

by mass matrix. In the figures, H, P and A denote, respectively, vertical, lateral and

rotational components of the mode. In mode No. 14 & 28, the rotational component

is multiplied by half of the bridge width, so that its magnitude is comparable to

other components.


86

-1000 -500 0 500 1000 15000

2

4x 10-4 Mode No. 1 frequency: 0.068764(Hz)

Axial Position

P

-1000 -500 0 500 1000 1500-2

0

2


Axial Position

H

Figure 6.2.a Structural Modes of the Bridge Deck

-1000 -500 0 500 1000 1500-5

0


Axial Position

P

-1000 -500 0 500 1000 1500-5

0


Axial Position

H

Figure 6.2.b Structural Modes of the Bridge Deck


87

-1000 -500 0 500 1000 1500-5

0


Axial Position

H

-1000 -500 0 500 1000 1500-5

0


Axial Position

H

Figure 6.2.c Structural Modes of the Bridge Deck

-1000 -500 0 500 1000 1500-5

0


PA

-1000 -500 0 500 1000 1500-5

0


Axial Position

H

Figure 6.2.d Structural Modes of the Bridge Deck


88

-1000 -500 0 500 1000 1500-5

0


Axial Position

H

-1000 -500 0 500 1000 1500-1

0

1x 10-3

Axial Position

Mode No. 28 frequency: 0.45853(Hz)

A

Figure 6.2.e Structural Modes of the Bridge Deck

These mode shapes are similar to those modes experimentally identified in previous

study (Brownjohn et al. 1987).

After the modal parameters were obtained, they were exported to MATLAB (The

Mathworks 1998) environment to perform the flutter instability prediction.

6.3 Method to Solve the Aeroelastically Influenced Eigenvalue

Problem

Equation (6.1.1) is doubled up since both the real and imaginary parts of the

determinant have to be zero. Corresponding unknowns are reduced frequency K or

wind velocity U and vibration frequency ω . These equations are highly nonlinear

in both unknowns not only through the dependence that appears in the expression of

the elements in impedance matrix, but also through the flutter derivatives that are


89

implicit in these expressions. To solve numerically the flutter problem, different

schemes have been proposed (Agar 1988, Namini 1992). Alternatively, a somewhat

graphical method was proposed by Astiz (1998). In the method, it is necessary to

first compute E for an array of ω~K values: this is equivalent to defining two

surfaces, one for the real part and the other for the imaginary part of E . The

intersection of these surfaces with ω~K plane is obtained by linear interpolation.

Then the zero contour curves of the real surface and imaginary surface are obtained

with piecewise linear approximation and their intersections can be determined either

numerically or graphically. The intersection points define the flutter condition. This

method requires a certain computational effort since a full matrix of E values has

to be obtained. Also the computation task grows quickly with the number of modes

included in the analysis. Although not efficient, this method is graphically clear and

user friendly.

Figure 6.3 shows an example of contour line plot for impedance matrix of zero

determinant value in the plane of natural frequency (denoted by ‘freq’ in the plot)

vs. reduced wind velocity (denoted by ‘RU’ in the plot). The dash line is the zero

contour line for the imaginary part of determinant E ; the solid line is for the real

part zero contour line of impedance matrix determinant. The dotted line is the wind

speed contour line calculated from the frequency and reduced wind velocity at the

position where the dotted lines are drawn. Arrows indicate the intersection points.

As can be observed from the plot, there are three intersections. However, not all of

them are real solutions, since the first two are sensitive to the structural modal

damping and will disappear when the corresponding modal damping is increased

slightly. This will be shown later in the analysis. The third intersection point, which

is not sensitive to the structural modal damping, is the stable solution. The flutter

velocity in this plot, therefore, is around 48m/s, for a flutter frequency 0.33 Hz, with

reduced wind velocity is around 5.


90

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

RU

freq

30m/s

50m/s

70m/s

Real Part

Imag Part

Figure 6.3 Sensitivity of E-matrix to Damping Ratio

6.4 Approximating the Impedance Matrix

As mentioned above, the solution of equation (6.1.2) at flutter determines the

participation magnitude of each structural mode. Because the flutter frequency is

solved numerically or graphically, the determinant of impedance matrix obtained at

flutter is not strictly zero, i.e.

0≈fluuter

E (6.4.1)

Directly solving the equation

0=ξflutterE (6.4.2)


91

will not always give a reasonable result.

Jain (1997) mentioned that before solving the equation, one of the elements in

vector ξ should be preset at a certain value and some care should be exercised in

the assignment of the value, due to the numerical sensitivity of the system at flutter.

Typically for suspension bridges a single torsional mode is the most likely mode to

dominate flutter, while participation of other modes may not significantly alter the

outcome of the analysis. Therefore, Jain (1997) argued that for practical reason, the

preset value should be assigned to the entry corresponding to such flutter

dominating modes; otherwise, misleading result may be obtained.

Although practical, this method is not convincing. Some analytical results of

multimode flutter analysis of long-span bridges indicate that because of the closely

spaced natural frequencies and three-dimensional structural mode shapes, the

aerodynamic coupling mechanism among structural modes becomes complex.

Furthermore, the coupled multimode flutter is not always initiated by the

fundamental symmetric torsional mode (Miyata and Yamada 1988, Agar 1989).

These results seem to be sensitive to the structural and aerodynamic characteristics

of the system.

Therefore, it is needed to find an exact singular matrix E~ , so that the numerically

obtained impedance matrix can be approximated with the main structure of the

impedance matrix being maintained and the solution of equation 0~ =ξE produces

approximately the real eigenvector.

Singular value decomposition (SVD) of a matrix is a good tool to approximate a

matrix around a singular point. If the impedance matrix is decomposed with SVD

TUSVE = , (6.4.3)

where U and V are orthogonal singular vectors matrices, the diagonal singular

value matrix


92

=

0

1

00S

SS (6.4.4)

is affected by the perturbations in the numerically calculated impedance matrix and

tends to be full rank. The diagonal elements in S are nii ,,2,1, L=σ , and

nσσσ ≥≥≥ ,,21 L . n is the dimension of the E matrix.

By letting some of the last few singular values be zero, i.e.

[ ] 00 =S (6.4.5)

the rank of the matrix E is reduced:

)()( 1SRankSRank = . (6.4.6)

Multiplying Equation (6.4.2) by TU , since matrix U is orthogonal, we have:

00001 =

≈ ξξ TT V

SEU . (6.4.7)

It is easier to solve

[ ] 001 =ξTVS . (6.4.8)

Figure 6.4 shows a representative profile of singular values (S Value in the plot) of

the impedance matrix at flutter. It can be observed that the last point is much closer

to zero than others. It is safe to say this point is generated by the small perturbation

in the impedance matrix, and can be deleted.


93

0 5 100

2

4

6

8

10

12

S V

aluv

e

0 5 1010-5

10-4

10-3

10-2

10-1

100

101

102

Figure 6.4 Non-dimensional Singular Values at Flutter (2D FD Case, 1st Mode)

One problem that remains is how many singular values should be assigned zero. If

n singular values are assigned zero, that means there are n linearly independent

flutter modes at the given flutter frequency. It seems satisfactory to have one single

flutter mode corresponding to a given flutter frequency in current analysis.

However, there could be more than one flutter mode with the same frequency. This

question is open.

6.5 Description of the Analysis

The analysis adopts flutter derivatives identified from three and two-dimensional

experiments of both transient and ambient type. The relative amplitude effect is

studied by changing the FDs (flutter derivatives) identified by transient vibration to

FDs identified by ambient vibration. The flutter derivatives used are FDs of model


94

B presented in Chapter 5. Some of FDs are smoothed according to their trend to

avoid unexpected fluctuations in the calculation.

6.5.1 Analytical Cases

In the first step, the analysis is performed with 2D FDs, and the critical wind speeds

for flutter are obtained. In the second step, 2D FDs are obtained from 3D FDs by

setting the P related derivatives to zero for all the reduced velocities, i.e.

0~ *6

*1 =PP ; 0*

6*5 == AA ; 0*

6*5 == HH . In the third step, analysis is carried out

with 3D FDs. These three steps repeat for transient and ambient cases. The first step

indicates that there is no self-excited load relating to lateral structural vibration; the

second step assumes that there is self-excited load relating to lateral vibration but it

is neglected; the third step fully considers self-excited forced in 3DOF.

Note that the flutter derivatives ( *4

*1 ~ HH , *

4*1 ~ AA ) identified from 2D

experiment might not always be consistent with the results of 3D experiment. There

could be some effects of the restriction in the third dimension because in the

experimental setup of the first step, the structure is actually being restricted in the

lateral direction. If the aeroelastic coupling between the vertical and lateral

vibration, or between rotational and lateral vibration is not negligible, and must be

taken into consideration, the results could be different in step one and step two.

In total, 10 structural modes are included to be linearly combined to represent the

flutter mode in this analysis. Table 6.2 shows the structural modes included in the

flutter analysis. Modal damping of the first eight modes was assigned to be zero, so

that the analysis would be conservative. It was found that the flutter occurs at very

low wind speed if the last two modes have zero damping (as shown in Figure 6.3).

Therefore, accepting 0.5% damping is assigned to both modes. Considering that the

method of predicting flutter is a pure numerical procedure and the aeroelastic

parameters are experimentally measured, the sensitivity of the numerical procedure

does not necessarily mean that the structure’s flutter is sensitive to the structural


95

damping in the mechanical sense. The sensitivity, therefore, may reflect the “virtual

reality in the numerical world”. A physically meaningful solution will be robust to

the disturbances in the parameters, e.g. errors in the identified flutter derivatives. If

it is noticed that the geometrical features of the three intersection points in Figure

6.3 are different, it may be reasonable to argue that the tangential intersection points

(the first two intersection point in Figure 6.3) of the continuous lines and dashed

lines may be not reliable. The geometric feature is ill conditioned. The

perpendicular intersection point (the third intersection point in Figure 6.3) is more

reliable than the first two. If the first two intersection points are sensitive to

perturbations in the identified parameters (additional structural damping is

equivalent to a perturbation in the flutter derivatives), the solutions corresponding to

them may not be reliable.

Table 6.3 summarizes the analysis and the corresponding flutter wind speeds and

frequencies. Figures (6.5-6.7) show the plot of contour lines for impedance matrix

of zero determinant value. The intersection points of the solid line (zero value

contour line of real part of the determination of the impedance matrix) and the

dashed line (zero value contour line of imaginary part) define the flutter condition.

Table 6.3 Flutter Speeds & Frequencies in Different Combinations

Case Structural Modes Included Flutter Speed

flutterU (m/s)

Flutter

Frequency

flutterf (Hz)

1. Transient 1, 2, 3, 4, 5, 13, 14, 16, 26, 28 52.7 75.9 0.267 0.3922D FD

2. Ambient 1, 2, 3, 4, 5, 13, 14, 16, 26, 28 NA NA

3. Transient 1, 2, 3, 4, 5, 13, 14, 16, 26, 28 38.5 51.1 0.252 0.3412D from 3D

4. Ambient 1, 2, 3, 4, 5, 13, 14, 16, 26, 28 NA NA

5. Transient 1, 2, 3, 4, 5, 13, 14, 16, 26, 28 48.2 NA 0.332 NA 3D

6. Ambient 1, 2, 3, 4, 5, 13, 14, 16, 26, 28 NA NA

NA: Solution not found within the reduce velocity range covered by experiment


96

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

RU

freq

0

30m/s

50m/s

70m/s

Real Part

Imag Part

Figure 6.5 E Matrix of 2D FD

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

RU

freq

30m/s

50m/s

70m/s

Real Part

Imag Part

Figure 6.6 E Matrix From 2D FD By Deleting P Related FD


97

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

RU

freq

30m/s

50m/s

70m/s

Real Part

Imag Part

Figure 6.7 E Matrix From 3D FD

6.5.2 Effect of Relative Amplitude

The effects of relative amplitude are manifested by replacing the flutter derivatives

identified from transient vibration with those identified from ambient vibration. As

can be observed in Table 6.3, while the flutter derivatives identified from transient

vibration may produce flutter condition, the flutter derivatives from ambient

response are aeroelastically stable. No flutter occurs in all the study cases within the

range of reduced velocity covered by experiments. This indicates that when the

modal amplitude of a particular mode is smaller than the ambient vibration

amplitude it needs additional excitations, strong gust, for example, to put it outside

the “ambient vibration envelope” to be a flutter mode.

The weakness of the argument is that the ambient vibrations may have different

wind spectra in sectional model and full-scale studies. In view of this, in the

discussion of the relative modal amplitude effect on flutter, it is an open question


98

whether or not the experiment to identify the flutter derivatives based on sectional

model testing is applicable to the full bridge. However, the logic is that if there is no

such effect as relative amplitude effect on flutter, there should be no big difference

between the analytical prediction results by using either transient or ambient

vibration testing. Now that there is a difference between these two analyses, there

must be relative amplitude effect. Although the quantitative significance is

questionable, the qualitative significance stands.

A major conjecture relating to relative amplitude effect is that if turbulence exists in

the oncoming flow or is generated by the structure itself, the “ambient vibration

envelope” will grow, making it harder to put the modal amplitude outside the

envelope. Therefore the structure will become more stable. As has been indicated in

the previous study, the discussions on turbulence effect on flutter follow two ways,

one is to resort to stochastic differential equation, the other is to measure flutter

derivatives in turbulence with transient free or forced sinusoidal vibration. Neither

of these two ways, however, considers the relative amplitude effect on flutter

derivatives, i.e. the interaction between the buffeting force and interactive force, and

its further effect on flutter boundary.

6.5.3 Effect of Lateral Flutter Derivatives

As indicated above, there are three case studies to investigate the lateral flutter

derivative effect on flutter. The quasi-static theory is not included, since the

experimental result from transient vibration appears not to be consistent with it.

Although the ambient vibration testing gives rise to negative *1P , its resultant flutter

derivatives are aeroelastically stable, suggesting there is no need to include quasi-

static theory in this case.

The lowest wind speed for flutter occurs in the second step of the analysis, i.e. the

case which uses the 3D flutter derivatives with all P related flutter derivatives

being assigned zero. The flutter wind velocity in this case is 38.5m/s, and the flutter


99

frequency is 0.252Hz. 2D flutter derivatives give rise to the highest flutter wind

speed: 53m/s, and flutter frequency 0.267Hz. 3D flutter derivatives produce the

flutter wind velocity 48m/s and flutter frequency 0.332Hz.

It can be observed that there are two flutter modes found in analysis with 2D flutter

derivatives and 3D flutter derivatives with all P related flutter derivatives being

assigned zero, but only one flutter mode is found for the 3D flutter analysis. The P

derivatives push the lower one of the two unstable modes to higher reduced wind

velocity, which is outside the experimental range. The flutter mode shapes are

shown in Figure (6.8-6.12). Every mode vector x is normalized so that 12=x .

The flutter mode shape in the 3D flutter derivative case is close to the second flutter

mode in the other two cases. Table 6.4 summarizes the participation factors of the

structural modes at flutter. All the figures indicate that the vertical motion

predominates in the flutter mode.

Summary

Flutter derivatives of a partially streamlined box girder section identified from

transient and ambient vibration testing were used to manifest the relative amplitude

effect on flutter. The flutter derivatives from ambient vibration were found

aeroelastically stable. Therefore, the relative amplitude effect has a stabilizing

consequence. The relative amplitude effect may be used to study the turbulence

effect on flutter, but further studies are needed.

The analysis was carried out with 3DOF and 2DOF flutter derivatives, so that the

effect of lateral flutter derivatives can be investigated. In comparison with the two

study cases where flutter derivatives used are 2D or 3D, the analysis using 3D

flutter with P related flutter derivatives being assigned zero gives rise to the lowest

flutter wind speed. All these discussions are limited to the specific section type

under study.


100

Table 6.4 Participation Factors of Major Modes at Flutter

Case 1 Case 3 Case 5 Mode

1st Flutter Mode 2nd Flutter Mode 1st Flutter Mode 2nd Flutter Mode 1st Flutter Mode 2nd Flutter Mode

Real Imag Real Imag Real Imag Real Imag Real Imag

1 -0.15643 -0.08678 -0.00584 0.003522 0.10323 -0.36051 0.016181 0.009755 -3.3114 -0.14643 NA

2 -272.23 -236.09 -102.03 -35.351 215.24 -266.89 -101.27 -57.09 -122.65 -42.297 NA

3 -0.0112 0.022604 -0.3081 -0.01365 -0.1442 0.11976 -0.72114 -0.01898 -28.742 96.794 NA

4 68.786 99.641 -348.32 -143.39 -123.5 80.239 -340.84 -188.77 -326.58 -184.46 NA

5 -97.943 -101.14 204.65 99.371 125.69 -104.73 207.3 128.82 198.08 128.06 NA

13 9.7424 -39.41 3.3032 -3.1152 49.943 38.659 2.5506 -3.6734 -8.3657 7.4377 NA

14 -48.991 -15.955 0.5837 -0.25756 9.076 -48.741 -0.92603 -1.1227 -3.6219 1.9989 NA

16 26.78 1.6067 -0.21849 -3.7792 -5.6884 17.623 4.3437 -8.4397 0.17176 -13.89 NA

26 3.96 2.1752 -52.518 108.45 -2.6475 3.0993 -57.357 -1.8756 -45.981 -12.482 NA

28 0.99436 0 35.948 0 1.4402 0 34.814 0 35.436 0 NA

Note: Structural modes are normalized to mass matrix. The participation factors reflect the relative importance of mass-matrix normalized

structural modes in flutter.


101

-1000 -500 0 500 1000 1500-0.1

0

0.1

real

-1000 -500 0 500 1000 1500-0.1

0

0.1

imag

-1000 -500 0 500 1000 15000

0.1

0.2

abs

the rotational displacement(rad) is multiplied by B/2

HPA


-1000 -500 0 500 1000 1500-0.2

0

0.2

real

-1000 -500 0 500 1000 1500-0.1

0

0.1

imag

-1000 -500 0 500 1000 15000

0.1

0.2

abs


HPA

Figure 6.9 Second Flutter Mode From 2D FD


102

-1000 -500 0 500 1000 1500-0.1

0

0.1

real

-1000 -500 0 500 1000 1500-0.2

0

0.2

imag

-1000 -500 0 500 1000 15000

0.1

0.2

abs


HPA

Figure 6.10 the 1st flutter Mode (2D FD from 3D FD)

-1000 -500 0 500 1000 1500-0.2

0

0.2

real

-1000 -500 0 500 1000 1500-0.1

0

0.1

imag

-1000 -500 0 500 1000 15000

0.1

0.2

abs


HPA

Figure 6.11 The 2nd Flutter Mode (2D FD From 3D FD)


103

-1000 -500 0 500 1000 1500-0.2

0

0.2

real

-1000 -500 0 500 1000 1500-0.1

0

0.1

imag

-1000 -500 0 500 1000 15000

0.1

0.2

abs


HPA

Figure 6.12 First Flutter Mode From 3D FD


104

CHAPTER SEVEN

Time Domain Formulation of Self-Excited Forces on

Bridge Decks for Wind Tunnel Experiments

Abstract

Time domain formulation of the self-excited wind forces on bridge decks often

employs indicial functions. The usual practice in bridge aeroelasticity is to

transform the flutter derivative model to time domain. Studies in previous chapters

suggested, however, that the relative amplitude effect, i.e. the structural oscillation

amplitude relative to the ambient vibration amplitude on flutter derivatives, needs

to be considered. This effect indicates that although self-excited wind forces are

linear when the pulse response of an elastically supported body is smooth, they are

nonlinear when the motion is significantly affected by the buffeting forces. The

nonlinearity may cause unnoticed error in the transformation. An alternative is to

treat the self-excited forces (SEF) generation mechanism as a separate dynamic

system, so that the relative amplitude effect can be evaluated in more details. In this

part, SEF generation system coupled with the rigid bridge deck system is proposed

to overcome the difficulties in the measurement and derivation of the time domain

representation of self-excited forces on bridge decks. This expression can be linked


105

to the frequency-time-domain-hybrid flutter derivative model, and a transform

relationship between the two models is suggested. The SEF model enables the

treatment of the self-excited forces as an independent dynamic system and identify it

directly. This may change significantly the way of doing wind tunnel experiments to

formulate the self-excited wind force and make the experiment more in line with

system theory.

7.1 Introduction

The most fundamental task of bridge aeroelasticity lies in the formulation of the

self-excited forces, the wind load caused by the movement of the structure. Some of

the historical works were reviewed in Chapter 2. They are summarized in the next

several paragraphs.

Theodorsen (1935) derived the theoretical description of the unsteady aerodynamic

forces on the efficient airfoil under sinusoidal motion by employing the reduced

frequency dependent Theodorsen’s circulation function. Theodorsen and Garrick

(1943) further extended the work to characterize the non-stationary flow about a

wing-aileron-tab combination. Following Sears (1940) and Luke and Dengler

(1951), Edwards (1977) showed that these results could be generalized for arbitrary

motion. The unsteady aerodynamic force thus can be formulated by a reduced-

frequency dependent aerodynamic influence matrix. In time domain, Wagner

(1925) showed the lift evolution with dimensionless time acting on a theoretical flat

airfoil given a step change in angle of attack. Kussner (1936) considered the

problem of an airfoil with forward flight velocity penetrating a uniform vertical gust

of infinite downstream extent and vertical velocity. Sears (1940) derived the

corresponding oscillatory lift for a gust velocity distribution that is sinusoidal. Jones

(1941) introduced rational approximations of the unsteady loads.

However, the signature turbulence, in the case of efficient airfoils in smooth flow, is

intentionally reduced by careful streamlining with notable attention to introduction


106

of a sharp trailing edge. For bluff bodies, the situation is different. The use of

Theodorsen aerodynamics for bluff bodies is not guarantied to be correct. In view

of this, the formulation of self-excited forces on a civil engineering structures, such

as a bridge decks, is more experimental than theoretical. Scanlan and Tomko (1971)

suggested reduced frequency dependent flutter derivatives be used in the modelling

of self-excited wind load on bridge deck.

In the time domain formulation of self-excited forces on a bridge deck, indicial

functions are the most important tools when the structure is subjected to arbitrary

motion. Scanlan et al. (1974) studied the aeroelastic moment on a bluff bridge deck

due to indicial angular movement. The characteristic of corresponding indicial

function of a bridge, i.e. the rotational aerodynamic damping due to the rotational

motion, according to their experiment, is strongly different from those of the

corresponding functions of airfoils. It was showed that the relationship between the

flutter derivatives and the indicial function is obtained by recognizing that for a

sinusoidal motion, the Duhammel integral is of the nature of a Fourier transform

(Sabzevari, 1971) and the inverse transform of frequency domain expression should

then produce the indicial function. Other studies (Yoshimura and Nakamura 1979)

to measure indicial functions are available. Figure 7.1 shows the indicial functions

with different structural forms. The Jones approximation is for efficient airfoils. The

other two curves are experimental measurements from bluff bridge decks. Scanlan

et al (1974) used an exponential approximation form with two exponential terms to

curve fit the experimental data from a truss structure. The measurement of

Yoshimura and Nakamura (1979) is a direct measurement of indicial aerodynamic

moment response of moving bluff prismatic sections of H or T type in still air. This

curve is scaled to match the magnitude of other curves; it is shown here only

qualitatively. Striking differences can be observed from the exponential

approximation curves: an initial steep rise from a low negative value to a peak,

which “overshoots” the steady state value, then settles down asymptotically. The

oscillating component in the curve by Yoshimura and Nakamura is clear and cannot

be neglected.


107

However, the direct measurement of indicial functions is neither easy, nor

conventional. Bucher and Lin (1988), and Chen et al. (2000a, 2000b) treated the

surrounding airflow as a set of filter like devices in generating self-excited forces on

bridge by transforming the frequency domain flutter derivatives to time domain for

the flutter and buffeting analysis of cable supported bridges.

0 1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Dimensionless Time s

Val

ue o

f Ind

icia

l Fun

ctio

n

Scanlan et al

Yoshimura & Nakamura

Jones

Figure 7.1 Indicial Functions of Different Kinds

7.2 Relative Amplitude Effect on the Transformation of Flutter Derivative

Model to Time Domain

The relative amplitude effect is defined as the effect of triggered vibration

amplitude of the model relative to “structural noise” in the vibration due to the

ambient wind excitation. As have been observed in previous part of the study on the

identification of flutter derivatives from transient and ambient vibration, the relative


108

amplitude effect on flutter derivatives cannot be neglected. The effect of relative

amplitude on flutter derivatives and on the flutter boundary reveals, from the

structural point of view, a complex relationship between the self-excited forces and

the “structural vibration noise” due to signature turbulence related buffeting forces.

Although the aeroelastic forces are linear when the body motion due to an external

trigger is not affected significantly by the signature turbulence, they are nonlinear

when the noise component in the vibration due to the signature turbulence related

buffeting forces couldn’t be neglected.

In view of this, it may not be valid in some cases to transform the flutter derivative

model measured under sinusoidal or exponentially modified oscillation to time

domain for general-purpose analysis. Alternative models for the identification of

interactive forces from experiments may be favourable.

7.3 State Space Model for SEF Generation System

In this section, the SEF generation system is dealt with as a sperate dynamic

system. The effort is not to provide a nonlinear model, but to linearize the nonlinear

behaviour at different values of relative amplitude so that further experiment can be

done to study the interactive forces on a bridge deck at different relative amplitude

either in time domain format or conventional time-frequency-domain hybrid format.

It is speculated that the nonlinearity in the self-excited wind force on a bridge deck

is due to the change of the relative amplitude of the body vibration and can be

linearized if the relative amplitude is kept constant.

7.3.1 The Model

If the bridge deck motion is known, the nonlinear SEF generation mechanism can

be modeled approximately by a linear time invariant system, it is possible to

identify a linear state space model with a minimum number of states from the


109

experimental data to approximate the input-output relationship. The SEF system is

coupled with the bridge deck sectional model motion: the rigid body system gives

excitations to the SEF generation system and receives feedback from it.

The equation of the sectional model motion is formulated as:

[ ] [ ] [ ] buffsef ffxKxCxM +=++ &&& (7.3.1)

in which Tphx α= is the displacement vector comprising of vertical, lateral

and rotational motion, [ ]M is structural mass matrix, [ ]C is the structural damping;

[ ]K structural stiffness. bufff is the buffeting force vector due to the fluctuating

component in the oncoming flow and signature turbulence generated by the bluff

body itself. This term is considered independent of the structural motion. seff is the

self-excited force, i.e. the feedback from the SEF generation system to the structural

system.

Changing equation (7.3.1) to dimensionless time s domain via

tBUs = (7.3.2)

where s is the dimensionless time; U is the wind velocity, B is the width of the

bridge deck, one has

[ ] [ ] [ ] [ ]

[ ] ))((

)()()(

12

12

1

sffMUB

sxKMUBsxCM

UBsx

buffsef +

=

+′

+′′

−

−−

(7.3.3)

where ′ and ″ are the first and second derivatives of corresponding variable with

respect to dimensionless time s , respectively.


110

The state space form of equation (7.3.3) is:

)()(

))()(()()(

sXCsY

sfsfBsXAsX

s

buffsefss

=

++=′ (7.3.4)

with state vector

TsspshsspshsX )()()()()()()( αα ′′′= (7.3.5)

and the state matrix

[ ] [ ] [ ] [ ]

−

−= −− CM

UBKM

UB

IAs 11

20

. (7.3.6)

The input matrix for self-excited forces and buffeting forces vector is

[ ]

= −1

20

MUBBs ; (7.3.7)

and the output matrix is

=

000100000010000001

sC . (7.3.8)

Similarly, the state space formulation for the flutter derivative model is obtained by

using reduced frequency dependent matrices [ ]aeroC and [ ]aeroK :

[ ] [ ] )()()( txKtxCtf aeroaerosef += & . (7.3.9)


111

Therefore equation (7.3.4) can be rewritten as

)()(

)()()(

sXCsY

sfBsXAsX

s

buffsf

s

=

+=′ (7.3.10)

where

[ ] [ ] [ ] [ ]

−

−= −−

effeff

fs CM

UBKM

UB

IA 11

20

(7.3.11)

[ ] [ ] [ ]aeroeff KKK −= and (7.3.12)

[ ] [ ] [ ]aeroeff CCC −= (7.3.13)

SEF generation system is also formulated by state space equations in dimensionless

time domain. One question is how to determine the input of the SEF generation

system. It can be argued that the dimensionless time derivative of the rigid body

state vector can be considered as the input “force” of the SEF system.

A proof is due to Bisplinghoff and Ashley (1962). They have pointed out that

indicial response functions corresponding to lift and moment due, respectively, to

step changes in effective angle of attack and effective rate of change of angle of

attack, should be used to formulate the unsteady aeroelastic force on airfoil. Lin and

Yang (1983) held the same idea and suggested impulse response functions of the

self-excited forces due to velocity and acceleration of the rigid body, respectively. It

is justified to argue that the time derivative of rigid body state vector, i.e. )(sXdsd ,

can be the input of SEF system.

The SEF system may be modeled by a linear state space model, with order up to

experimental determination, in dimensionless time domain:


112

)()(

)()()(

sfCsf

sXdsdBsFfsf

fsef

f

=

+=′ (7.3.14)

in which )(sf is the 1×n state vector of SEF system,

)(sf ′ is the dimensionless time derivative of )(sf with respect to dimensionless

time,

F is the nn× square state matrix,

fB is the 6×n input matrix,

fC is the n×3 output matrix.

The value of n , i.e. the system order, is to be determined from experimental data.

Equation (7.3.14) reveals a deterministic relationship between the dimensionless

time derivative of rigid body state vector, )(sXdsd and the self-excited forces

)(sfsef . By no means will this imply that there is a similar relation between

)(sX and )(sfsef . The determinability of the relationship between )(sX and )(sfsef

depends on specific motion patterns of the rigid body. Using the input-output

relation )()( sfsXdsd

sef→ is physically different from using )()( sfsX sef→ when

the rigid body motion pattern is more general.

The coupled system governed by equation (7.3.4) and (7.3.14) can be expressed in

the form of the simulation diagram shown in Figure 7.2. In the diagram, the self-

excited forces system and the bridge deck system are referred to as SEF and BDS,

respectively. The SEF system takes the dimensionless time derivative of the BDS

state vector as its input and returns the self-excited forces as output to the BDS as a

part of its input.


113

Figure 7.2 Simulation Diagram of the SEF Model

7.3.2 Relation to Flutter Derivative Model

When there is exponentially modified or pure sinusoidal motion of the bridge deck

section, the self-excited forces can be described by a flutter derivative model (Jain,

et al 1996). For ambient response, we consider the self-excited wind forces due to a

small impulse response of the bridge deck in wind. As shown in the Appendix, this

is equivalent to considering self-excited wind forces due to the output covariance of

the ambient vibration of the body (Hoen et al 1993; Bogunovic-Jakobsen, 1995).

Due to the relative amplitude effect, the flutter derivatives in this case will be

quantitatively different from the former case, but the formulation of the self-excited

wind forces is the same.

Let the flutter derivative matrix be


114

[ ]

=Η

*2

*5

*1

*3

2*6

2*4

2

*2

*5

*1

*3

2*6

2*

4

2

*2

*5

*1

*3

2*6

2*4

2

)(

KAABKA

BKAKA

BKA

BK

KPPBKP

BKPKP

BKP

BK

KHHBKH

BKHKH

BKH

BK

K (7.3.15)

The self-excited forces model is rewritten in a matrix form:

[ ] )()(21)( 2 sXKBUsf sef Η= ρ (7.3.16)

in which Taeaeaesef MDLf = is the self-excited lift drag and moment forces, ρ

is the air density, UBK ω

= is the reduced frequency and 6,1,,, *** K=iAPH iii are

flutter derivatives.

It is possible to convert SEF system model to retrieve flutter derivatives. It seems

easier to be solved in a different format by using Duhammel integral to represent

the “fluid memory”. The “force” term is the time derivative of rigid body state

vector:

[ ]∫ ∞−−Φ=

s

sef dXddsBUf σσσ

σρ )()(21 2 (7.3.17)

in which

[ ] fFs

f BeCsBU =Φ )(21 2ρ (7.3.18)

is the pulse response function of SEF.

The relationship between the time-frequency domain hybrid model and the SEF

system model can be developed as follows. Substitute the rigid body state space

equation (7.3.10) into (7.3.17) and equate to (7.3.16):


115

[ ] [ ] σσσσ dfBXAssXKH sbuffs

fs∫ +−Φ= ∞− )()()()()( . (7.3.19)

Change the integration variable σ to στ −= s

[ ] [ ] [ ]∫∞∞ −Φ+∫ −Φ=

00 )()()()()()( ττττττ dsfBdsXAsXKH buffsf

s . (7.3.20)

Taking Laplace transform and using convolution property, we have

[ ] [ ] )()()()( KfBKXAKXKH buffsf

s +Φ= (7.3.21)

in which, an over bar denotes Laplace transform.

For the rigid body state space equation (7.3.10), because the buffeting force is

considered as process noise, which is white around the reduced natural frequency,

its magnitude should be much smaller that that of the state vector. Therefore

)()( KfBKXA buffsf

s >> . (7.3.22)

The covariance function equation (A.6) corresponds to the noise free case,

i.e. 0)( =Kfbuff .

Therefore, in both cases equation (7.3.21) yields

[ ] [ ] )()()( KXAKXKH fs⋅Φ≈ . (7.3.23)

There is no particular requirement for )(KX , therefore, the general relationship

between the flutter-derivative matrix and the transfer function of the SEF system

matrix is:

[ ] [ ]( ) 1)()( −Η≈Φ f

sAKK . (7.3.24)


116

7.3.3 The Transformation in Modal Coordinates

One observation is that the transformation depends on matrix ( ) 1−fsA , which

consists of aeroelastic coupling due to the self-excited forces in addition to the

structural properties. If the motion is decoupled, e.g. in sinusoidal rotational motion,

the system matrix must have the following format:

−

=0

02K

IA f

s , (7.3.25)

so that the aeroelastic matrix is

[ ] [ ]*2

*3

2 KAAK=Η , (7.3.26)

and the transfer function is

[ ] [ ][ ] [ ]*3

*2

-1s

fsA AKAHMM −==ΦΦ ′′′ αα . (7.3.27)

Because the state vector is

TiKsiKs eiKxexsX 00)( = , (7.3.28)

we define an impulse function for moment due to the rotational movement as

αα ′′′ Φ+Φ=Φ MaMM iK (7.3.29)

so that

][ *3

*2 iAAKM −=Φ α . (7.3.30)

This relation is valid for single DOF transfer function and flutter derivatives. For

multi-DOF in the α,, ph coordinates, the transformation of [ ])(KH into [ ])(kΦ

involves the state matrix of the aeroelastically-influenced system, which couples the


117

elements in matrix [ ])(KH to produce those in matrix [ ])(kΦ . For free vibrations,

the aeroelastic coupling is not always weak, the modal coordinate may be different

from the α,, ph coordinate. If modal coordinates [ ]ϕ are adopted,

[ ] )()( ssX ξϕ= (7.3.31)

it gives:

[ ][ ] [ ]ϕϕ Φ=Λ −1H (7.3.32)

in which

[ ] [ ][ ]ϕϕ HH = , (7.3.33)

[ ] [ ][ ]ϕϕ Φ=Φ , (7.3.34)

and

[ ] [ ] [ ][ ]

==Λ −

22

21

1

n

fs

K

KA Oϕϕ . (7.3.35)

nKK 21 ,..., is reduced natural frequency.

7.4 Suggestions for Future Experiments

Due to the relative amplitude effect, the self-excited wind forces are nonlinear. By

using the SEF model, it is possible to study the effect in more details. The practice

may need non-contact active drivers to force the model so that the model is not

constrained by the forcing device as it is in the case of conventional forced

vibration testing. Since the relative amplitude effect will vanish if the buffeting

response accounts for only a small percentage of the total signal energy, the non-

contact driving device is vital to the study.

If the driving force is sinusoidal, the experiment will be like the conventional forced

vibration testing. By choosing the magnitude of the driving force, the experiment


118

can be done with a set of controlled relative amplitude forced oscillations. If the

forced vibration amplitude is comparable to the buffeting response amplitude, the

identification of flutter derivatives will be a combined deterministic-stochastic

identification.

If a chirp signal is generated by the active device and a certain range of forcing

frequency is covered, it will be possible to identify the SEF model directly. The

relative amplitude also needs to be controlled.

Due to the limitation of experimental devices, in this study, the SEF model was not

identified directly. However, as a makeshift, flutter derivative model was

transformed to form the frequency response function of the SEF system.

Transforming flutter derivatives identified from transient vibration gives rise to the

SEF model for large relative amplitude, while transforming the flutter derivatives

from ambient response produces the SEF model for small relative amplitude. These

are only illustrative results, but the difference between respective large and small

relative amplitude FRF (frequency response function) can be roughly observed (If

the aforementioned equipment is available, better results can be obtained). The

model under test is a partially streamlined box girder sectional model B (Figure

3.3), flutter derivatives in two dimensions )4,1(, ** =iAH ii are shown respectively in

figure 5.5a and 5.5b. The resultant FRF matrices of the SEF system of respective

models are shown in figures 7.3 and 7.4. The FRF is presented in the figure in the

format as follows:

[ ]

=

′′′′′′

′′′′′′

αα

αα

MhMMhM

LhLLhL

RRRRRRRR

R (7.4.1)

in which ijR is the element in the FRF matrix, relating the self-excited forces in

ith direction due to the input in the jth dimension.

It is noted that these curves are obtained from limited number of flutter derivatives

and some of the details of the SEF system are not in good quality. Interpolating or


119

extrapolating the result will not improve the reliability of the identification,

therefore they are not suitable for the identification of the SEF model and direct

measurement mentioned above is needed.

Another issue regarding the FRF matrix so measured is that it is real. This is due to

the fact that the flutter derivative matrix [ ]Η and state matrix [ ]fsA are both real.

Identifying FRF of SEF from flutter derivative matrix is equivalent to finding a

matrix [ ]Φ , mapping the time history of rigid body state vector multiplied by the

flutter derivative matrix to weighted integration of its dimensionless time

derivative. The weighting function in this case is the impulse response function of

SEF system. Therefore, the calculated FRF is real. If the FRF of SEF is measured

directly, it is not necessary to be a real matrix.

Summary

Linear SEF dynamic system model for self-excited forces on bridge deck is

presented, making it possible to study the relative amplitude effect in more detail.

Direct experimental determination of the of SEF generation system in time domain

is possible if SEF model is adopted. To measure the FRF and further identify the

time domain representation of the SEF system, a new experimental method is

suggested but not exercised due to the limitation of equipment. SEF model may

serve as a more general formulation of self-excited forces.


120

0 0.5 1-3

-2

-1

0

1L-h

0 0.5 1-5

0

5L-a

0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6L-dh/ds

0 0.5 1-4

-3

-2

-1

0

1L-da/ds

0 0.5 1 1.5-0.3

-0.2

-0.1

0

0.1

0.2M-h

0 0.5 1 1.5-0.15

-0.1

-0.05

0

0.05M-a

0 0.5 1 1.5-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01M-dh/ds

0 0.5 1 1.5-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Reduced Frequency

M-da/ds

Figure 7.3 FRF Matrix via Flutter Derivatives (Transient Vibration)

0 0.2 0.4-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4L-h

0 0.2 0.4-2

-1

0

1

2

3

4

5L-a

0 0.2 0.4-0.8

-0.7

-0.6

-0.5

-0.4

-0.3L-dh/ds

0 0.2 0.4-0.8

-0.6

-0.4

-0.2

0L-da/ds

0 0.2 0.4-0.08

-0.07

-0.06

-0.05

-0.04

-0.03M-h

0 0.2 0.4-0.25

-0.2

-0.15

-0.1

-0.05

0M-a

0 0.2 0.4-0.01

0

0.01

0.02

0.03

0.04M-dh/ds

0 0.2 0.4-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

Reduced Frequency

M-da/ds

Figure 7.4 FRF Matrix via Flutter Derivatives (Ambient Vibration)


121

CHAPTER EIGHT

Errors in the Identification of

Aeroelastic Derivatives

Abstract:

Errors in the identification of flutter derivatives may be due to nonlinearity in the

aeroelastic system as indicated by the relative amplitude effect, possible strong

wake oscillator behind the body or poor experimental conditions such as the high

level noise in the identified Markov parameters. In this chapter, error indices are

used as a confidence indicator of the identification. These indices are built on the

instantaneous energy discrepancy between the identified and the measured system

in subspace, and the real values of flutter derivatives are not needed during the

evaluation. The discussion is within the framework of eigensystem realization

algorithm (ERA).


122

8.1 Introduction:

There are some error sources in the identification of flutter derivative model such as

the nonlinearity in the self-excited wind force as indicated by the relative amplitude

effect, noise in the measurement, non-whiteness of the buffeting force (Bogunovic-

Jacobsen 1995) and so on.

8.1.1 Errors due to Non-White Noise

The experiments with ambient response might be sensitive to the nature of the

external excitations since the model is totally “noise driven” in this case. An

identification method based on response data only cannot distinguish between the

properties belonging to the structure or the load. The nonwhite noise in the system

can be attributed to at least the two factors listed below.

8.1.1.1 Non-White Noise Due to the Fluid Oscillation

The time-frequency domain hybrid linear model using flutter derivatives has always

been dissociated from those bridge deck forms that exhibit strong vortex-shedding

effects and lock-in in certain velocity ranges. In other words, at an earlier stage

either some bluff deck forms prone to vortex excitation are deemed to be entirely

untreatable by flutter derivative methods or the application of these method was

provisionally suspended over strong vortex shedding lock in velocity regimes.

Bogunovic-Jakobsen (1995) discussed the effect of non-white noise effects of the

buffeting wind load on the identification of aeroelastic derivatives. The power

spectral density of wind load on a stationary sectional model is needed to create a

filter to correct the measured data. One possible disadvantage is that buffeting load

is measured in a separate experiment. Putting together two experiments possibly

under different conditions might not be an acceptable way to evaluate the result of


123

the identification. This method also leads to difficulties when the DOF of the

system is more than one.

8.1.1.2 Non-White Noise Due to Structural Vibration.

The other source of non-whiteness in the input is due to the additional vibration

source in the suspension system. If we expand the order of the state space model

for the aeroelastic system to include this part of vibration, the state vector is

expressed as

TTs

Tm tXtXtX )()()( = (8.1.1)

In the state vector, mX is the rigid body degree of freedom (DOF), if the

experiment is 3D, its dimension, mDim , is six, sX is DOF due to other vibration

sources, its dimension, sDim , is unknown. If the suspension system is not strong

enough the measurement could possibly contain other sources of vibration, such as

the ground vibration or the suspension system vibration and so on.

The corresponding state space equation will be

buffBftAXtX += )()(&

)()( tCXtY = (8.1.2)

in which A is the state matrix, having dimension

( ) ( )ssA DimDimDim +×+= 66 , (8.1.3)

Matrix B is the input matrix for random buffeting load and

Matrix C is the output matrix, having dimension


124

)6(3 sC DimDim +×= . (8.1.4)

The column vector )(tY is the output vector at time t . Since only the rigid body

motion is measured, the dimension of the output vector remains unchanged.

However, our goal is not to identify this model of unknown order, but rather, unlike

other situations in which system identification is used, to identify a system that has

a fixed order, i.e. the state matrix A , in 3 DOF experiment, must have order 6.

However, in the experiment, it is not necessarily true that the signal measured

carries nothing else but the desired information. Other sources of vibration can also

be represented. The vibration of the suspension system and other vibration sources

could affect the pre-assumption that the signal has an order of 6. By truncating the

system to order six, the originally internal forces between the truncated degrees of

freedom and the remaining ones become external, and may contribute to the non-

white component of the unknown buffeting force.

8.1.2 Errors Due To Nonlinearity in the Self-Excited Forces

It is difficult, however, to attribute the errors in the identification of flutter

derivatives solely to the nonwhite noise effect, since the argument above is as weak

as either the fluid states of the aeroelastic system or the additional structural states

of the suspension system. The major errors come from, at least in the experiment

described in the thesis, the non-stationary nature of the triggered free decay

vibration.

The study in previous chapters indicates that the relative amplitude has a notable

effect on the identified flutter derivatives. Flutter derivatives identified under the

conditions of small and large relative amplitude may be different. In transient

vibration tests, the triggered vibration is clearly larger than the ambient vibration at

the beginning, but decays very fast into the ambient vibration envelope. When the

vibration starts, the relative amplitude effect is negligible but it is not at the end.


125

The identification of flutter derivatives corresponding to the free decay vibration

actually deals with a time-varying phenomenon. On the contrary, in the experiment

with ambient vibration, there is no change in the relative amplitude because the

impulse response is calculated numerically by output covariance. The phenomenon

to be identified is time-invariant under this condition.

As in these two cases of the identification of free decay and ambient vibration, other

factors remain almost the same; it is predictable that if the identification errors in

the former case are considerably larger than those in the latter case, the transient

nature of the free decay vibration may be considered to have a contribution to errors

in the identification. The remaining part of the chapter will serve as a proof.

8.2 Evaluation Based on Block Hankel Matrix

Based on the analysis in the previous part, it is necessary to present an index of

confidence together with the identified flutter derivatives. It is straightforward to

create a ratio of the signal power of the rigid body to the signal power of the cut off

parts in the Block Hankel Matrix.

By the fundamental ideas of ERA (Juang, 1994), The measured Block Hankel

Matrix is decomposed with SVD

[ ] [ ]Tnn

n SSPPH 00

0 00

)0(ˆ

Σ

Σ= (8.2.1)

[ ] Nn PPP ≡0 and [ ] Nn SSS ≡0 are orthogonal, and eigenvector matrices of

)0(ˆ)0(ˆ HH T and )0(ˆ)0(ˆ THH , respectively. nΣ and 0Σ are diagonal matrices, their

elements are positive singular values. Their magnitudes reflect the contribution of

the signal in that dimension to the overall measurements. The dimensions of

nΣ equal the order of the state-space model. Elements in 0Σ are small and

considered negligible. If the measured Hankel matrix is exactly generated by a


126

linear time invariant system, elements in 0Σ are exactly zero. Due to the

incompleteness of the flutter derivative model and other error sources 0Σ might not

be small, but deemed as zero in the identification.

Because the ERA interprets singular vectors as directions in the system, in the

evaluation of errors in the identification, it is needed to find a common coordinate,

onto which both realized and unrealized parts are projected. By such projection, the

“realized signal ellipsoid” and “unrealized signal ellipsoid” can be constructed. The

length of each semi-axis of the ellipsoids can quantify the contribution of the signal

to the overall Block Hankel Matrix in each direction defined by the projection

matrix. The power corresponding to each direction can be compared without

confusing the dimensions the signal powers correspond to.

By observation, the matrix [ ]0PPn is a good candidate for the aforementioned

base matrix.

Projecting the Block Hankel Matrix to the subspace TnP we have )0(HPT

n . We

define the realized signal power matrix on the subspace TnP of the defined base

matrix as

nTT

n PHHP )0()0(=Θ . (8.2.3)

It is clear that the signal power matrix so defined is symmetric and positive semi-

definite. The singular values of the signal power matrix Θ equal to its eigenvalues,

i.e.

TQQD 2=Θ (8.2.4)

where ),,,( 21 nddddiagD L= , 2id is the eigenvalue as well as singular values of

signal power matrix Θ . Hence the singular values define the length of semi-axes of


127

the signal ellipsoid, while the column vectors of the orthogonal matrix Q define the

directions of these axes.

The error signal matrix is defined as

−=−

=

)0(ˆˆ

)0(ˆ0

)0()(

0 HPNPHP

HPnE

T

TnT

N

Tn

r (8.2.5)

in which ][ 0PPP nN = is the orthogonal projection matrix; )0(ˆ)0(ˆ HHN −= is

the error matrix between the identified block Hankel matrix )0(H and the

originally measured block Hankel matrix )0(H in the original coordinate.

The power matrix of the error signal is defined as:

==

Ξ

Ξ

002

1

)0(ˆ)0(ˆ00ˆˆ

)()(0

0PHHP

PNNPnEnETT

nTT

nTrr . (8.2.6)

The first part is the projection of the difference between the measured and identified

Block Hankel Matrix in the direction of the identified singular vectors TnP . The

second part is projection of the measured Block Hankel Matrix on the unidentified

part of the singular vector TP0 , the orthogonal complement of TnP . Because of the

truncation of singular values, and singular vectors, the SVD of the identified Block

Hankel Matrix does not produce TP0 and the subspace formed by TP0 is not a part of

)0(H . The projection of )0(H onto these directions are zero, so only )0(H is

considered in projecting onto TP0 .

The 1Ξ part is the error due to the computation, which will be shown, in the

following result, is small.


128

It is clear that energy of 2Ξ belongs to the directions that are not a part of the

aeroelastically influenced rigid body motion. Within the structure of ERA, it is

excluded from the system and cannot be incorporated into the rigid body motion

direction, unless additional states are included. If this part of energy is unacceptably

large, the identification may cease to be effective to represent the behavior of the

deck.

Define the sum of all semi-axes of the signal ellipsoid as the overall power in the

subspace defined by the semi-axes. The trace of the matrix Θ or Ξ produces the

sum of the eigenvalues and, in this case, the sum of singular values as well. Hence

we have the summation of realized power in the rigid body motion direction of the

state space

∑=

=Θn

ii

TTi pHHpTr

1)0()0()( . (8.2.7)

The summary of error power in the rigid body motion direction of the state space is

∑=

=Ξn

ii

TTi pNNpTr

11

ˆˆ)( , (8.2.8)

in other directions

∑+=

=ΞN

njj

TTj pHHpTr

12 )0(ˆ)0(ˆ)( . (8.2.9)

We have two ratio values: the ratio of error to signal power in the rigid body motion

direction:

)()( 1

1 ΘΞ

=ΩTrTr

(8.2.10)

and in other directions


129

)(( )2

2 Θ

Ξ=Ω

TrTr

. (8.2.11)

The first one reflects the signal to noise energy ratio in the subspace of TnP . The

second one reflects the energy ratio of signal in subspace of TnP to signal in the

subspace of TP0 , reflecting how much signal energy has been left in subspace TP0

unidentified.

8.3 Data from Experiment

In this section, results from the two models are under evaluation. The experiments

are transient and ambient vibration testing. The first model is the one-meter long

partially streamlined box girder deck, which is shown in Figure (3.3). The second

model is the twin deck bluff body model shown in Figure (3.2). The same

experimental condition exists in both ambient and transient testing.

Figure 8.1 shows the first 20 singular values from the SVD during a representative

ERA procedure for the 3DOF experiments with partially streamline box girder

model. The wind speed is 14m/s.

0 2 4 6 8 10 12 14 16 18 2010

-1

100

101

102

Singular Values

Ambient Transient

Figure 8.1 Singular Value at U=14m/s


130

It can be observed that in the identification of free decay vibration the first singular

value is about 90, and the sixth singular value is round 50, a little higher than the

seventh singular value that is around 25. Because the experiment is three

dimensional, the cutoff criteria allows only six singular values to be nonzero, others

must be forced to be zero, regardless of the structure of the singular value matrix.

This mandatory cutoff criterion can cause severely large errors in the identification

result. As shown in figure 8.2, the error index 2Ω is as high as 25%.

The bad profile of the singular values of the free decay vibration can hardly be

explained by the argument that the additional states are too strong to be cut off. It

may be due to the transient nature of the free decay vibration. The argument is

supported by the good profile of singular values in the ambient vibration

identification, since the free decay and ambient vibration differ only in the values of

relative amplitude.

In the ambient vibration identification, the first singular value is around 20, the

sixth is around 7 and the seventh is around 0.7. There is a sudden drop between the

sixth and seventh singular values. Under this circumstance, it is safe to say that the

system has an order of six, and the cutoff of other directions will not change the

property of the signal. Correspondingly, the 2Ω value is as low as 4 percent.

The evaluation above might show that the output covariance of the ambient

vibration gives rise to a better linear dynamics than the decay response, indicating

that the ambient vibration, by averaging over some length of time, can be

represented well by a linear dynamic system, whose parameters are however

unfortunately somewhat different from the transient vibration dynamics. Since in

the decay response, the “noise level” due to the signature turbulence is changing,

while it is not in the ambient vibration, the discussion above may imply that the

self-excited force can be linearized with respect to the “noise level”, i.e. relative

amplitude, of the system.


131

2 4 6 8 10 12 14 16 180

2

4

6x 10-29

2 4 6 8 10 12 14 16 180

10

20

303D Tran3D Amb

Omega 2 HB

Omega 1 HB

(%)

Wind Speed (m/s)

Figure 8.2 Error Index for Transient and Ambient Vibration Testing (Model B)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5x 10-28

0 2 4 6 8 10 12 14 16 18 200

20

40

602D Tran2D Amb

Omega 2 HB

Omega 1 HB

Wind Speed (m/s)

(%)

Figure 8.3 Error Index for Transient and Ambient Vibration Testing (Model B)


132

2 4 6 8 10 12 14 16 18 20 220

1

2

3x 10-29

2 4 6 8 10 12 14 16 18 20 220

10

20

30

402D Tran2D Amb

Omega 2 TK

Omega 1 TK

Wind Speed (m/s)

(%)

Figure 8.4 Error Index for Transient and Ambient Vibration Testing (Model A)

Figure 8.3 shows the evaluation of the 2DOF experiment of the streamlined model.

Figure 8.4 show the error index of 2DOF experiments of the twin deck bluff

sectional models. In all the cases above, ambient vibration output covariance gives a

better fitting with a linear dynamic system than the decay response.

Summary

In this chapter, error index is introduced, to serve as a quantitative tool to evaluate

the quality of the experiments to extract flutter derivatives. The flutter derivative

model may formulate the output covariance of ambient vibration better than the free

decay vibration.


133

CHAPTER NINE

Conclusions and Future Work

9.1 Conclusions

The research described herein consists of studies on properties of interactive wind

forces on elastically supported bluff bridge sectional models. Two and three-

dimensional sectional model tests in a wind tunnel were carried out to detect the

nonlinearity in the self-excited wind forces. This work is needed because the

transformation of frequency-time domain hybrid flutter derivative model to either

time or frequency domain usually requires the linearity assumption of the self-

excited wind forces, which has not been investigated thoroughly.

In this study,

1: Nonlinearity in self-excited wind forces was detected through the concept of

relative amplitude of the aeroelastically-influenced system.


134

This method is different from the usual practice of examining the amplitude

effect on flutter derivative. Because aeroelastic analysis is by and large

meant to predict the structural behavior when the structural vibration

amplitude and the angle of attack of the oncoming wind are both small, it is

important to detect the existence of nonlinearity in the self-excited force

under the same condition.

Relative amplitude is defined as the body oscillatory amplitude relative to

the ambient vibration envelope. It reflects the relation between the

mechanically triggered body oscillations and aerodynamically induced

random vibration due to turbulence, indicating the relative importance of the

two factors. Physically, the larger the buffeting component in the total

response, the more the information about the fluid feature than about the

structure the system carries, and vice versa. These two conditions are

different in that the body oscillation may change the behavior of the

surrounding fluid if its amplitude exceeds the ambient vibration amplitude,

and the identified flutter derivatives may also be changed.

Ambient vibration and triggered free decay vibration are two extreme cases

of the tests on the relative amplitude effect. By comparing resultant flutter

derivatives of these two extreme cases, the nonlinearity is detected.

The effect of relative amplitude on flutter derivatives and on the flutter

boundary reveals, from the structural point of view, a complex relationship

between the self-excited forces and the “structural vibration noise” due to

signature turbulence related buffeting forces. Although the aeroelastic forces

are linear when the body motion due to an external trigger is not affected

significantly by the signature turbulence, they are nonlinear when the noise

component in the vibration due to the signature turbulence related buffeting

forces could not be neglected.


135

2: By using flutter derivatives of a partially streamlined box girder section

identified form transient and ambient vibration testing, the relative

amplitude effect on flutter instability is studied.

The flutter derivatives from ambient vibration were found aeroelastically

stable. Therefore, the relative amplitude effect has a stabilizing

consequence. The relative amplitude effect may also be used to study the

turbulence effect on flutter, but further studies are needed.

3: The effect of lateral flutter derivatives was investigated by using 3DOF and

2DOF flutter derivatives.

In comparison with the two cases where flutter derivatives used are 2D or

3D, the analysis with 3D flutter derivatives with P related flutter derivatives

being assigned zero gives rise to the lowest flutter wind speed. The 2D

flutter derivatives produce the highest wind speed for instability. Both in the

two cases of 2D and 3D with P derivatives assigned zero, two flutter

conditions were found within the reduced velocity range covered by

experiments producing two flutter frequencies and modes in both cases.

Only one flutter condition was found in the 3D case. It corresponds to the

higher second flutter mode in the other analytical cases. The lower flutter

mode is “pushed”, by the P derivatives, outside the reduced velocity range

covered by experiment

All these discussions are limited to the specific section type under study.

Because the experimental result is not in consistence with that from quasi-

static theory, the conclusion of the investigation is somewhat different from

the usual idea based on the quasi-static theory.

4: A more general formulation of self-excited wind forces is discussed.

The formulation in time domain of the self-exited force on bridge decks

often employs indicial function by transforming the flutter derivative model.


136

However the relative amplitude effect may make the transformation

questionable if the flutter derivatives identified from transient and ambient

vibration are different.

As it is suspected in this study that the self-excited wind forces might be

linearized with respect to the relative amplitude, an alternative model

treating the self-exited force generation mechanism as a separate dynamic

system may be favorable to offer more flexibilities in manipulating the

sectional model testing.

In this paper, SEF generation system coupled with the rigid bridge deck

system is proposed. This expression was linked to the frequency-time

domain hybrid form of flutter derivative model and transform relationship

between the two models was also suggested. The SEF model enables the

treatment of the self-exited forces as an independent dynamic system and

direct identification in time domain.

5: One of the error sources in extracting flutter derivatives was identified.

The ineffectiveness of the identification of flutter derivatives may be due to

nonlinearity in the aeroelastic system as has been indicated by the relative

amplitude effect, possible strong wake oscillator behind the body or poor

experimental conditions such as the high level noise in the identified

Markov parameters.

As all factors remain almost the same in the two cases of the identification

of free decay and ambient vibration, it is predictable that if the identification

errors in the former case are considerably larger than those in the latter case,

the non-stationary nature of the free decay vibration will contribute to errors

in the identification.

The evaluation might suggest that the output covariance of the ambient

vibration gives rise to a better linear dynamics than the decay response does.


137

This indicates that the ambient vibration, by averaging over some length of

time, can be represented by a linear dynamic system, whose parameters are

however unfortunately somewhat different from the transient vibration

dynamics. It is noticed that in the decay response, the “noise” level due to

the turbulence is changing, while it is not in the ambient vibration. This

might imply that the self-excited force can be linearized with respect to the

noise level in the system.

9.2 Suggestions for Future Work

Studies on the formulation of interactive wind forces on bridge decks are far from

ending. It must be recognized that the frequency-time domain hybrid flutter

derivative model works under specific conditions. Any transformation of the model

will only extend it nominally. The surplus information so generated may be

factious. To formulate the self-excited force for more general purpose, conceptual

models for experiments that can work in time domain are needed. The following

subjects are among the most important:

1: Criteria evaluating the quality of transforming the flutter derivative model to

time domain.

2: Nonlinear model in time domain for the self-excited wind force.

3: Nonlinear identification method to identify the aeroelastically influenced

model directly.

Therefore, in order to have better approaches for flutter and buffeting analysis, it is

important to have a better understanding of the self-excited forces.

138

Appendix I Output Covariance of Discrete-Time State Space System (Overschee and Moor 1996)

When the measurement noise in the equipment and process noise (the small scale

buffeting force due to signature turbulence and local variation of wind speed in the

oncoming flow) is white and zero mean, then

0][ =⋅ TqXE and 0][ =⋅ TwXE (5.A.1)

where )()( iBpiq = and )()( iDviw = .

The Lyapunov equation for the state covariance matrix is

QAAiXiXE Tsi

Tsi +Σ=+⋅+=Σ + )]1()1([1 (5.A.2)

where )]()([ iqiqEQ T⋅= .

If output covariance is defined as

)]()([)( iYkiYEkC Ti ⋅+= (5.A.3)

then

RCCiYiYEC Tsi

Ti +Σ=⋅= )]()([)0( (5.A.4)

where )]()([ iwiwER T⋅= .

If we define

SCAiYiXEG Tsi

T +Σ=⋅+= )]()1([ , (5.A.5)

where )]()([ iwiqES ⋅= , we have for L,2,1=k

GCAkC ki

1)( −= . (5.A.6)

This produces a new state-space model of )0(,,, iCCGA .

139

Appendix II: The Correlation Function of Continuous-Time State Space System

The correlation function of the output signal is:

+=+ ∫

+ −+δ τδδ ττδs

s

TTbuffs

sAA CsXdfBesXeCEsYsYCovfs

fs )()()())(),(( )(

+= ∫

+ −+δ τδδ ττs

s

TTbuffs

sATs

A CsXdfBeCECCovCefs

fs )()()(

(A.1)

in which

[ ])()( sXsXECov Ts = (A.2)

and [ ]•E is the mathematical expectation.

If the condition is met that the system is subjected to white noise, and the

fluctuating buffeting force contains no memory of the bridge deck vibration history,

considering this is a centered process, [ ] [ ] 0)()( == sXCEsYE , the second term in

equation (A.1) vanishes, i.e.,

[ ] 0)()()( =

∫

+ −+ TTs

s buffssA CsXEdfBeCE

fs

δ τδ ττ . (A.3)

Hence

covA BCesYsYCovfs δδ =+ ))(),(( (A.4)

where T

scov CCovB = . (A.5)

This gives rise to a linear dynamic system:

)()()()(

sCXsYBsXAsX cov

fs

=+=′ δ

(A.6)

140

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