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공학석사학위논문

Estimation of Damping Ratio from

Operational Monitoring of Cable-Stayed

Bridge

진도대교의 운용계측을 통한 감쇠계수 추정

2013년 2월

서울대학교 대학원

건설환경공학부

Radiance Calmer

i

ABSTRACT

Estimation of Damping Ratio from Operational Monitoring of

Cable-Stayed Bridge

Radiance Calmer

Department of Civil and Environmental Engineering

College of Engineering

Seoul National University

Interference Vortex-Induced Vibration (VIV) has been observed for Second Jindo

Bridge. A previous study with wind tunnel tests suggests that VIV was due to low

structural damping ratio (0.2%). Multiple Tuned Mass Damper (MTMD) has been

installed in Second Jindo deck to mitigate amplitude of vibration. This study verifies the

assumption of low structural damping ratio for the first vertical mode of vibration with

field measurements. Based on monitored data, modal parameters are obtained using

NExT-ERA time domain method. Results of damping ratio are in good agreement with

the wind tunnel tests (average 0.3%). However, this structural damping ratio is lower than

the recommended value by Korean Design Guidelines. Estimation of modal parameters

for First Jindo Bridge shows a higher damping ratio than second bridge (average 0.5%).

As expected, the acceleration monitored with the MTMD operating in Second Jindo

ii

Bridge provides a higher damping ratio. When wind conditions create VIV, larger RMS

accelerations can be observed, while MTMD is kept under fixed conditions. On the other

hand, after releasing the MTMD, the amplitude of vibration is within the usual range and

damping ratio increases with an average of 3.4%. In the present investigation, it has been

successfully demonstrated that MTMD design is an efficient method to mitigate VIV.

Keywords: Cable-stayed bridge, damping ratio, structural health monitoring, Vortex

Induced Vibration

Student Number: 2011-24090

iii

CONTENTS

ABSTRACT ........................................................................................................................ i

CONTENTS ..................................................................................................................... iii

List of Tables .................................................................................................................... vi

List of Figures .................................................................................................................. vii

CHAPTER 1 INTRODUCTION ..................................................................................... 1

1.1 BACKGROUND AND MOTIVATION ............................................................................. 1

1.2 REVIEWS ON PREVIOUS WORKS ................................................................................. 4

CHAPTER 2 THEORETICAL BACKGROUND ......................................................... 8

2.1. NATURAL EXCITATION TECHNIQUE NEXT .............................................................. 8

2.2 EIGENSYSTEM REALIZATION ALGORITHM .............................................................. 15

CHAPTER 3: NUMERICAL EXAMPLE TWO-STORY-SHEAR BUILDING ...... 26

3.1 CALCULATION OF TWO-STORY-SHEAR BUILDING RESPONSE .................................. 26

3.1.1 Damping matrix ............................................................................................... 26

3.1.2 Normally distributed random load ................................................................... 28

3.1.3 Newmark’s method ........................................................................................... 29

iv

3.2 NUMERICAL APPLICATION....................................................................................... 31

3.2.1 Parametric study on Nfft .................................................................................. 33

3.2.2 Parametric study on modal participation ........................................................ 39

CHAPTER 4 ANALYSIS OF FIELD MEASUREMENT DATA .............................. 42

4.1. JINDO BRIDGES DESCRIPTION ................................................................................. 42

4.2 FIELD MEASUREMENT DATA SETS ........................................................................... 44

Without MTMD operating: 2012/10/15 10:00 to 2012/10/17 0:00 .......................... 45

With MTMD operating: 2012/11/24 10:00 to 2012/11/28 10:00 ............................. 47

4.3 PARAMETERS FOR FIELD MEASUREMENTS ANALYSIS ............................................. 48

4.4 SECOND JINDO BRIDGE WITHOUT MTMD OPERATING ........................................... 51

4.5 FIRST JINDO BRIDGE WITHOUT MTMD OPERATING ............................................... 56

4.6 SECOND JINDO BRIDGE WITH MTMD OPERATING ................................................. 59

4.7 COMPARISON OF RESPONSE FROM SECOND JINDO BRIDGE WITH AND WITHOUT

MTMD OPERATING ....................................................................................................... 63

4.8 WIND VELOCITY / AMPLITUDE OF ACCELERATION RELATIONSHIP ......................... 64

4.9 INVESTIGATION ON VIV .......................................................................................... 66

Traffic load induced vibration .................................................................................. 66

Vortex Induced Vibration .......................................................................................... 68

Response due to combined wind and traffic loads .................................................... 70

Buffeting response ..................................................................................................... 72

v

4.10 EFFICIENCY OF MTMD ......................................................................................... 75

CHAPTER 5 CONCLUSION ........................................................................................ 81

BIBLIOGRAPHY: .......................................................................................................... 82

ABTRACT (IN KOREAN) ............................................................................................ 86

vi

List of Tables

Table 1.1 Characteristics of MTMD installed in Second Jindo deck ................................ 7

Table 3.1 Parametric study on Nfft ................................................................................... 34

Table 3.2 Theoretical modal parameters for first mode of vibration ................................ 40

Table 3.3 Damping ratio, frequency and modal participation for mode 1, Nfft = 216

....... 40

Table 3.4 Damping ratio, frequency and modal participation for mode 1, Nfft = 215

....... 40

Table 3.5 Damping ratio, frequency and modal participation for mode 1, Nfft = 214

....... 41

Table 4.1 Natural frequencies for first vertical mode of vibration [2] .............................. 43

Table 4.4 Damping ratio Second Jindo Bridge without MTMD ...................................... 51

Table 4.5 Damping ratio First Jindo Bridge ..................................................................... 56

Table 4.6 Damping ratio Second Jindo Bridge with MTMD operating ........................... 59

Table 4.9.1 Vehicle crossing time characteristics ............................................................. 66

Table 4.9.2 Amplitude of acceleration and peak factor .................................................... 69

Table 4.9.3 Amplitude of acceleration and peak factor .................................................... 71

Table 4.9.4 Amplitude of acceleration and peak factor .................................................... 74

vii

List of Figures

Figure 1.1 First (right) and Second (left) Jindo Bridges ..................................................... 2

Figure 1.2 Mitigating the interference VIV of Second Jindo Bridge by increasing the

damping ratio (wind tunnel tests) [4] ............................................................................... 3

Figure 1.3 Geographic localization of Jindo Bridges ......................................................... 6

Figure 1.4 Tuned Mass Damper (TMD) installed in Second Jindo Bridge ........................ 7

Figure 3.1 Two-story-shear building and its modes of vibration ...................................... 26

Figure 3.2 Normally distributed random load over 1200 sec............................................ 28

Figure 3.3 a) displacement b) velocity c) acceleration of the two-story-shear building ... 31

Figure 3.4 PSD of a) first floor acceleration b) second floor acceleration........................ 32

Figure 3.5 Auto-correlation function (blue) and theoretical IRF (red) for a) first floor b)

second floor .................................................................................................................... 33

Figure 3.6 Auto-correlation function for the first floor and its corresponding PSD for

a) Nfft =216

b) Nfft = 215

c) Nfft = 214

............................................................................ 35

Figure 3.7 Zoom of auto-correlation function (0~20s) and its corresponding PSD (first

floor) a) Nfft =216

b) Nfft = 215

c) Nfft = 214

.................................................................. 36

Figure 3.8 Auto-correlation function for the second floor and its corresponding PSD for

a) Nfft =216 b) Nfft = 215 c) Nfft = 214 ........................................................................ 37

Figure 3.9 Zoom of auto-correlation function (0~20s) and its corresponding PSD (Second

floor) a) Nfft =216

b) Nfft = 215

c) Nfft = 214

.................................................................. 38

viii

Figure 4.1.1 Elevation of Second Jindo Bridge [2] ........................................................... 42

Figure 4.1.2 Cross sections (m) a) Second Jindo Bridge b) First Jindo Bridge [4] .......... 43

Figure 4.1.3 Distance between Second and First Jindo Bridges [4] ................................. 43

Figure 4.2.1 Measurement of vertical acceleration a) Second Jindo Bridge b) First Jindo

Bridge [26] ..................................................................................................................... 44

Figure 4.2.2 Wind time history without MTMD a) wind velocity 20 min average b) wind

direction 20 min average c) Turbulence intensity Iu d) wind rose ................................. 46

Figure 4.2.3 Wind time history with MTMD a) wind velocity 20 min average b) wind

direction 20 min average c) turbulence intensity Iu d) wind rose .................................. 48

Figure 4.3.1 2012/10/15 11:20 to 11:40 a) raw acceleration b) PSD of raw acceleration c)

Auto-correlation function d) PSD of auto-correlation function ..................................... 49

Figure 4.3.2 2012/10/15 11:20 to 11:40, depending on Hankel matrix size variation of a)

frequency b) damping c) modal participation d) Frequency vs. modal participation ..... 50

Figure 4.4.1 Time history of Second Jindo Bridge without MTMD a) damping ratio b)

RMS acceleration c) MAX acceleration d) peak factor ................................................. 53

Figure 4.4.2 Wind velocity history of Second Jindo Bridge without MTMD a) damping

ratio b) RMS acceleration c) peak factor ........................................................................ 54

Figure 4.4.3 RMS acceleration vs. damping ratio Second Jindo Bridge without MTMD 55

Figure 4.5.1 Time history of First Jindo Bridge without MTMD a) damping ratio b) RMS

acceleration c) MAX acceleration d) peak factor ........................................................... 57

ix

Figure 4.5.2 Wind velocity history of First Jindo Bridge without MTMD a) damping ratio

b) RMS acceleration c) peak factor ................................................................................ 58

Figure 4.5.3 RMS acceleration vs. damping ratio First Jindo Bridge without MTMD .... 59

Figure 4.6.1 Time history Second Jindo Bridge with MTMD operating, between red bars

wind velocity >10m/s a) damping ratio b) RMS acceleration c) MAX acceleration d)

peak factor ...................................................................................................................... 61

Figure 4.6.2 Wind velocity history of Second Jindo Bridge with MTMD a) damping ratio

b) RMS acceleration c) peak factor ................................................................................ 62

Figure 4.6.3 RMS acceleration vs. damping ratio Second Jindo Bridge with MTMD ..... 63

Figure 4.7.1 Mean wind velocity vs. damping ratio, Second Jindo Bridge ...................... 64

Figure 4.7.2 Mean wind velocity vs. RMS acceleration, Second Jindo Bridge ................ 64

Figure 4.8.1 mean wind velocity vs. RMS of acceleration, First Jindo Bridge ................ 65

Figure 4.8.2 mean wind velocity vs. RMS of acceleration, Second Jindo Bridge ............ 65

Figure 4.9.1 Raw acceleration and its broad-banded PSD 2012/10/15 13:40................... 67

Figure 4.9.2 Zoom on amplitude of acceleration assumed to be due to vehicle load ....... 67

Figure 4.9.3 Raw acceleration and its PSD 2011-01-16 09:07:35 .................................... 68

Figure 4.9.4 a) Filter acceleration (0.3~0.55 Hz) and b) wind rose .................................. 69

Figure 4.9.5 Wind velocity and wind direction (40 sec) ................................................... 69

Figure 4.9.6 Acceleration in blue raw and in red filtered (0.3~0.55Hz) and its PSD

2012/10/17 13:20 ............................................................................................................ 71

Figure 4.9.7 Wind rose...................................................................................................... 71

x

Figure 4.9.8 Wind direction .............................................................................................. 71

Figure 4.9.9 Acceleration, in red raw, in blue filtered acceleration and its PSD

2012/08/28 3:40.............................................................................................................. 73

Figure 4.9.10 Wind rose.................................................................................................... 73

Figure 4.9.11 a) Horizontal wind velocity and b) wind direction ..................................... 73

Figure 4.10.1 Wind history a) mean wind velocity b) Turbulence intensity Iu c) wind rose

for 2012/10/17 9:20 to 18:00 .......................................................................................... 76

Figure 4.10.2 Second Jindo Bridge response without MTMD Between blue bars VIV

occurs ............................................................................................................................. 77

Figure 4.10.3 Wind history a) mean wind velocity b) Turbulence intensity Iu c) wind rose

for 2012/11/26 10:00 to 0:00 .......................................................................................... 78

Figure 4.10.4 Second Jindo Bridge response with MTMD Between blue bars response of

the bridge to VIV conditions .......................................................................................... 80

1

Chapter 1 Introduction

1.1 Background and Motivation

Jindo Bridges consist of two cable-stayed bridges connecting the Korean peninsula and

Jindo Island. The First Jindo Bridge opened in 1984, but twenty years later, to meet the

increase in traffic demands, the Second Jindo Bridge was opened in 2005 (Fig.1.1).

However amplitudes of vibrations were far beyond the recommended limits as specified

by the Korean Guidelines [1] leading to serviceability issues. A wireless monitoring

system was also implemented to demonstrate the efficiency of a wireless smart sensor

framework, and to facilitate the monitoring response parameters of the bridge. The test-

bed study led to collaboration between the University of Illinois at Urbana-Champaign,

University of Tokyo, KAIST and Seoul National University. This international

collaboration resulted in a large number of publications (for example see references [2],

[3]). At Seoul National University, wind tunnel tests have been performed on Jindo

Bridges to understand sources of Vortex Induced Vibration (VIV) [4]. It has been

suggested by the experimental model that for the first vertical mode, vibrations on

Second Jindo Bridge are due to a low structural damping ratio (~0.2 %). According to

wind tunnel tests, a slight variation in the damping ratio could create a large displacement

at the center of the main span (Fig.1.2). These results are analogous to wind tunnel tests

for Great Belt East Bridge [5]. Interference VIV would also be amplified by closely

2

spaced decks. Different aerodynamic additives were experimented to decrease the

vibration phenomena, but the only efficient way was to increase the structural damping

ratio. In Marsh 2012, Hyundai Engineering & Construction Co. adopted Multiple Tuned

Mass Damper (MTMD) to mitigate VIV of Second Jindo Bridge.

Figure 1.1 First (right) and Second (left) Jindo Bridges

The purpose of the present study is to investigate assumptions of low structural

damping ratio for the first vertical mode of vibration with field measurements. Two sets

of data are analyzed in the present work. Initially, the acceleration at the middle of the

center span with stationary MTMD in Second Jindo deck was considered. Secondly, data

was also monitored with the MTMD operating in the released state . Results of damping

from the First and Second Jindo Bridges are also compared. Furthermore, for the Second

Jindo Bridge, wind and traffic conditions are investigated to identify the response of the

bridge, particularly for VIV and buffeting.

3

The time domain analysis NExT-ERA (Natural Excitation Technique – Eigensystem

Realization Algorithm) is employed to estimate damping ratio for Jindo Bridges.

Figure 1.2 Mitigating the interference VIV of Second Jindo Bridge

by increasing the damping ratio (wind tunnel tests) [4]

0

5

10

15

20

25

30

0 1 2 3

ξh(%)

Dis

pla

cem

ent

at

cente

r of

the d

eck(c

m)

MAX

RMS

Allowable amplitude

4

1.2 Reviews on previous works

Analysis methods to calculate modal parameters have been developed since the 1970s.

Earlier methods used to work in the frequency domain. Because of some problems

associated with frequency resolution and leakage, researchers started looking at time

domain methods. Some years ago, frequency domain methods were readopted to improve

algorithms by increasing the model order for example [6]. Through a comparative study

of modal analysis methods, two techniques were highlighted to calculate dynamic

characteristics of a bridge [7]. Rational Fraction Polynomial method using data from

shaker excitement of the bridge provided the most complete and consistent modal

parameters. Subspace method or ERA also led to complete and consistent results

obtained from ambient vibrations. The ERA has been applied to identify modal

parameters on different types of structures, particularly on bridges. Qin & Al. (2001) used

an improved ERA technique to obtain modal parameters of Tsing Ma Bridge [8].

Siringoringo (2008) demonstrated successfully the efficiency of ERA method combined

with NExT to calculate dynamic characteristics from output-only monitored data [9].

Damping ratio is always a difficult parameter to determine as the group of Magalhaes

(2010) emphasized in their studies dedicated to comparison between numerical

simulations and analysis from full-scaled measurements [10]. Values of damping ratio are

very scattered whatever analysis method is employed. Damping ratio is also decomposed

into two parts, the first one is related to structural characteristics of damping and the

5

second concerns aerodynamic properties. Macdonald (2005) was the first researcher to

separate contributions of aerodynamic and structural damping for a long-span bridge

[11]. According to the quasi-steady theory, aerodynamic damping is proportional to wind

speed [12], but linearity is not always observed in reality. To estimate structural damping

ratio, data monitored under a wind velocity lower than 10 m/s and not associated with

significant VIV should be considered [5]. In case of Vortex-Induced oscillations,

damping ratio is expected to increase with amplitude of vibration. Estimation of damping

is a key parameter when evaluating cross-wind VIV even if it is associated with high

uncertainty.

Vortex-Induced oscillations have been observed on several bridges, among them

Trans-Tokyo Bay Bridge [13], Rio-Niteroi Bridge [14], Osteroy suspension bridge [15]

and the Great Belt East suspension bridge [5]. These four bridges have geographic

similarities with Jindo Bridges. Wind from the sea blows perpendicular to the bridge due

to the corridor created by surrounding lands (Fig.1.3). This localization shapes a laminar

flow. In case of both Norwegian suspension bridges (Osteroy and Great Belt East

bridges), aerodynamic additives were sufficient to reduce VIV. For the other two bridges,

Tuned Mass Dampers have been adopted to reduce oscillations. Damping ratio of TMD

should be selected at its maximum value reached when the natural frequency of the deck

for the mode subjected to VIV is nearly equal to that of TMD. But the drawback of TMD

concerns its sensitivity problem due to the fluctuation in tuning the TMD frequency [16].

6

A multiple tuned mass damper is shown to be more effective in the mitigation of

oscillations of structures compared to TMD. In conventional practice, all the TMDs are

designed to have the same mass, same spring and same dashpot [17]. Their natural

frequencies are tuned in the critical bandwidth. Definition of bandwidth is as following:

1Bandwidth B N

Where N is the number of TMDs and is the non-dimensionalized frequency spacing.

In case of Second Jindo Bridge, MTMD are designed to mitigate vibrations of the first

vertical mode. Four TMDs have been installed in the deck at the center of the main span

(Fig.1.4). See Table 1.1 for MTMD characteristics.

Figure 1.3 Geographic localization of Jindo Bridges

7

Figure 1.4 Tuned Mass Damper (TMD) installed in Second Jindo Bridge

Table 1.1 Characteristics of MTMD installed in Second Jindo deck [18]

TMD 1 TMD 2 TMD 3 TMD 4

Mass (ton) 3.25 3.25 3.25 3.25

Natural frequency (Hz) 0.404 0.423 0.445 0.461

Damping ratio (%) 3.435 3.435 3.435 3.465

8

Chapter 2 Theoretical background

2.1. Natural Excitation Technique NExT

To use classical curve-fitting methods to extract modal parameters of a structure like

the ERA, signals from ambient excited structures should have the form of an impulse

response function. The Natural Excitation Technique gives the cross-correlation function

of responses from field measurement. This cross-correlation function is assumed to have

the same form as the system impulse response function. Therefore, time domain curve-

fitting algorithms can be applied to obtain modal parameters. The input signal, which is

not measurable, is assumed to be white noise [19].

The theoretical development of NExT mainly comes from references [20] and [21].

The equation of motion for an n-degree-of-freedom system can be written as:

( ) ( ) ( ) ( )M x t C x t K x t p t (1)

With

is the n x n mass matrix

is the n x n damping matrix

is the n x n stiffness matrix

is the n x 1 acceleration vector

is the n x 1 velocity vector

is the n x 1 displacement vector

is the n x 1 applied force vector

9

With the assumption:

0M x t M x t (2)

Combining equation (1) and equation (2) results the state-space model corresponding

to the dynamic equations:

A X t B X t F t (3)

where

0

0 0 0

x t f tC M KA B X t F t

M M x t

(4)

The modal transformation can be used:

X t q t q t

(5)

Where

is the 2n x 2n complex modal matrix

q t is the 2n x 1 vector of modal coordinates

is the n x n mode shape matrix

is the 2n x 2n complex eigenvalue matrix

Then

10

2

1

n

r r

r

x t q t q t

(6)

Where

r is the vector of the rth mode shape

rq is the rth component of vector q t

Using the orthogonality of mode shapes:

T T

A a B b (7)

With a

and b

diagonal matrices

Premultiplying equation (3) by T

gives:

T T T

A X t B X t F t (8)

Substituting equation (5) into (8):

T T T

A q t B q t F t (9)

A set of scalar equations in the modal coordinates can be given as:

T

r r ra q t b q t f t (10)

Where ,r ra b are the elements of diagonal matrices a and b .

11

The solution of equation (10) can be obtained from Duhamel integral assuming zero

initial conditions:

1r

tT t

r rr

q t f e da

(11)

Where r r rb a

Equations (6) and (11) provide the solution for x t

2

1

1r

tnT t

r rrr

x t f e da

(12)

The response ilx t at the ith DOF due to a single input force lf t at the lth DOF can

be derived as:

2

1

1r

tnt

il ir lr lrr

x t f t e da

(13)

Where nr is the ith component of the rth mode shape vector r .

The cross-correlation function iplR T can be defined [22]. The function links two

measured responses at location i and p caused by a white noise input at l, with E[]

representing the expectation operator:

12

[ ( ) ( )]ipl il plR T E x t T x t (14)

Subsisting equation (13) into (14), noticing lf t that is white noise, hence:

2 2

1 1

1( ) [ ( ) ( )]r r

t t Tn nt T t

ipl il lr ps ls l l

r s r s

R T e e E f f d da a

(15)

The autocorrelation can be rewritten as:

[ ( ) ( )] ( )l l lE f f (16)

Where is a constant and t is the Dirac delta function.

Substituting equation (16) into (15) and collapsing the integration results the

following:

2 2

1 1

( ) r

n nl il lr ps ls T

ipl

r s r s r s

R T ea a

(17)

Summing over all the inputs , 1,2,...,lf t l L which are assumed to be uncorrelated

with one another, the cross-correlation function ( )iplR T between the output i and the

output p is obtained:

2

1

( ) r

nT

ipl ir pr

r

R T Q e

(18)

Where prQ is a new constant defined by:

13

2

1 1

n Ll lr ps ls

pr

s l r s r s

Qa a

(19)

It shows that the cross-correlation function in equation (18) is a sum of complex

exponential functions of the same form as the impulse response of the original system in

the following:

2

1

r

nt

pr ir lr

r

h t W e

(20)

Where prh t is the impulse response at the point i due to the input force at the point l,

lrW is the modal participation factor.

The Fourier transform of the impulse response function prh t gives the frequency

response function ilH j :

2

1

nir lr

ilrr

WH j

j

(21)

The cross-power spectral density function npG j is the Fourier transform of the

cross-correlation ( )iplR T in equation (18):

2

1

nir pr

iprr

QG j

j

(22)

14

Note that in practice, it is easier to work in frequency domain. Cross-power spectral

density is first estimated and then, the cross-correlation function is obtained from the

Inverse Fourier Transform. The following study focuses on estimation of modal

parameters for the first mode of vibration. Signals are monitored on one point of the

structure. That is why only the auto-correlation function will be used in further parts.

15

2.2 Eigensystem Realization Algorithm

Once the impulse response is obtained, the ERA can be applied to obtain modal

parameters. ERA is based on the concept of minimum realization which identifies a

system model with the smallest state dimension. From the free-decay response of the

system, matrix A which provides frequencies and damping is defined. Identification of

modal parameters from ambient measurements is then possible with this output-only

method. References [6] and [9] provide the theoretical background.

1st step: from the general equation of motion to a discretized time response.

The equation of motion of an n degree of freedom system

( ) ( ) ( ) ( )M x t C x t K x t p t

(1)

Rewrite the equation with the mathematical form of displacement vector as:

( )( )

( )

x tu t

x t

(2)

With a vector with 2nx1 dimension

1 1

(2 2 )

0'

n n

I

M K M CA

(3)

16

1 1

( ) ( )n n q q

p t F t

(4)

1

2

0'

n qM F

B

(5)

Where is the input vector represented by the Dirac Delta function at q locations

and is a matrix of input coefficients. So the equation of motion can be rewritten as:

2 1 2 2 2 1 2 1

( ) ' ( ) ' ( )n n n n n q q

u t A u t B t

(6)

To go back to the initial displacement matrix where the response is measured at

p physical coordinates, the matrix is used:

1 2 2 1

( ) ( )p p n n

x t R u t

(7)

The response to the impulse function is given by the convolution integral:

0

0

' ( ) ' ( )

0 0( ) ( ) ' ( )

tA t t A t

t

u t e u t e B d

(8)

To give a discrete representation of (8), interval time is defined, it can be multiplied

by k to span the total duration of the excitation, then take and .

(8) becomes:

( 1)' ' (( 1) )

0(( 1) ) ( ) ' ( )k tA t A k t

k tu k t e u k t e B d

(9)

17

During the time interval it is assumed that the delta function is

constant. Variables are changed as :

' ' '

00

(( 1) ) ( ) ' ( )tA t A

u k t e u k t e B k t d

(10)

For simplicity, terms are rewritten as:

[ ']A tA e [ '] '

0[ ']

tAB e d B

(11)

( 1) (( 1)u k u k t ( ) ( )k k t (12)

Finally the response (10) becomes:

( 1) ( ) ( )u k A u k B k for (13)

And the transition to physical coordinates:

( ) ( )x k R u k (22)

Now, the response function is discretized. The 2nd

step concerns the construction of the

Hankel matrix.

2nd

step: arrangements for the construction of the Hankel matrix

The response to an impulse is considered at k=0, given and

for

18

(1) [ ] (0) [ ]u A u B and (1) [ ] (1)x R u (15)

Hence

(1) [ ][ ] (0) [ ][ ]x R A u R B (16)

Considering for simplicity, the initial condition , then

2 12 1

{ (1)} [ ]nn

u B

and 21 2 1

{ (1)} [ ]{ }p np n

x R B

(17)

For further time intervals,

{ (2)} [ ]{ (1)}u A u and { (2)} [ ][ ]{ }x R A B (18)

And so on…

2{ (3)} [ ]{ (3)} [ ][ ]{ (2)} [ ][ ] { }x R u R A u R A B (19)

For the general case:

1{ ( )} [ ][ ] { }kx k R A B (20)

If an input at all the q input locations is considered, it follows that:

1[ ( )] [ ][ ] [ ]kX k R A B (21)

These matrices are called the Markov parameters and they are used to form the

generalized Hankel matrix:

19

[ ( )] [ ( 1)] [ ( )]

[ ( 1)] [ ( 2)] [ ( 1)][ ( 1)]

[ ( )] [ ( 1)] [ ( )]

pr qs

X k X k X k j

X k X k X k jH k

X k i X k i X k i j

(22)

With and and

If there is an initial measurement, is simply replaced by . If the

Hankel matrix is rewritten according to the equation (21), then

[ ( )] [ ][ ] [ ]kH k Q A W for (23)

Where

2

1

[ ]

[ ][ ][ ]

[ ][ ]

pr n

r

R

R AQ

R A

(24.a)

1

2

[ ] [ ] [ ][ ] [ ] [ ]s

n qs

W B A B A B

(24.b)

The process to determine the matrices is called realization. There are an

infinite number of sets of these three matrices according to the equation (20), i.e. there

are an infinite number of realizations. The objective is to obtain a minimum realization,

i.e. the realization corresponding to the minimum order of state-space formulation that

can still represent the dynamic behavior of the structure.

20

Step 3: Calculation of the pseudo-inverse of

Let’s define a matrix as

2 2 22

[ ][ ]' [ ] [ ]n qs qs qr n nqr n

W H Q I

(25)

Multiplying by and ,

[ ][ ][ ]'[ ][ ] [ ][ ]Q W H Q W Q W (26)

According to the equation (23)

[ ][ ] [ (0)]Q W H (27)

Hence

[ (0)][ ]'[ (0)] [ (0)]H H H H (28)

Therefore is the pseudo-inverse of

[ ]' [ (0)]H H (29)

The Singular Value Decomposition allow to calculate the pseudo-inverse of

[ (0)] [ ] [ ] [ ]T

pr pr pr ps ps pspr qs

H U V

(30)

The matrix has 2n non-zero singular values equivalent to the state-space

system. So can be recomputed using only the first 2n columns of and

21

2 2 22 2 2 2

[ (0)] [ ][ ][ ]T

n n npr qs pr n n n n qs

U U V

(31)

With

2 2 2 2[ ] [ ] [ ][ ] [ ]T

n n n nU U V V I (32)

The pseudo-inverse is obtained as:

1

2 2 2[ ]' [ (0)] [ ][ ] [ ]T

n n nH H V U (33)

Step 4: rewritten of the Hankel matrix using the pseudo-inverse matrix

From the equation (21)

[ ( 1)] [ ][ ] [ ]kX k R A B (34)

Or, using the identity matrices

[ ( 1)] [ ] [ ( )][ ]T

p qp q pr qsp pr qs q

X k E H k E

(35)

[ ] [ ] [0] [0]T

pp p p p p pp pr

E I

and

[ ]

[0][ ]

[0]

q

qs q

I

E

(36)

Using equations (23), (25), (27), (31), (32) and (33), matrices can be reorganized to

obtain:

22

1/2

2 2[ ( 1)] [ ] [ ][ ]T

p n nX k E U

1/2 1/2

2 2 2 2[ ] [ ] [ ][ ] [ ] [ ][ ]T k

n n n nU Q A W V

1/2

2 2[ ] [ ] [ ]T

n n qV E

(37)

To obtain modifications in (37) are made.

1/2

2 2[ ( 1)] [ ] [ ][ ]T

p n nX k E U

1/2 1/2

2 2 2 2[ ] [ ] (1) [ ][ ]k

T

n n n nU H V

1/2

2 2[ ] [ ] [ ]T

n n qV E

(38)

The realization has been achieved, comparing equations (34) and (38), matrices can be

determined:

1/2

2 2[ ] [ ] [ ][ ]T

p n nR E U

1/2 1/2

2 2 2 2[ ] [ ] [ ] (1) [ ][ ]k

T

n n n nA U H V (39)

1/2

2 2[ ] [ ] [ ] [ ]T

n n qB V E

The modal parameters are obtained by solving the eigenproblem based on the

‘realized’ matrix :

[ ]{ } { }u uA (40)

The mode shapes in terms of physical coordinates of the system are determined

through the matrix :

23

1 2 2 1

x u

p p n n

R

(41)

From the solution of the eigenvalue problem, we obtain:

r r ri ,r r r n 2 2

, ,

12 4

2r r r n r r (42)

2 2 2 2 2 2 2 2, , , ,r r r r n r r n r n r n (43)

2 2,r n r r

2 2,

r rr

r n r r

(44)

With

,r n damped frequency at the rth mode,

r undamped frequency at the rth mode,

r damping ratio for the rth mode.

In practice, the number of columns and rows of the Hankel matrix is very important to

calculate the damping ratio. Practical guidelines to apply ERA [23] advise to select the

number of columns as four times the number of expected modes. The number of rows is

based on the number of points available in the cross-spectral density function. The aim is

to use as much data from cross-correlation function as possible without including noisy

signals found at the end. As selected in reference [9], a square Hankel matrix is

preferable to a rectangular matrix due to one processing signal.

24

Modal participation

Modal participation vectors are a result of multiple references from modal parameter

estimation algorithms and relate how well each modal vector is excited from each of the

reference locations included in the measured data. The combination of the modal

participation vector and the modal vector for a given mode provides the residue matrix

for that mode [23].

Modal participation indicator (MPI)

In the ambient/operational modal analysis, correlation or covariance function can be

measured as Markov parameter, and expressed via eigenvalue, modal vector (mode

shape) and modal participation factor:

1

1

nk T

k r r r

r

Y

(1)

With kY covariance function (Markov parameter), , ,r r r r-th modal vector,

eigenvalue and modal participation factor, respectively

Choosing all the measurement coordinates as references, the dimension of modal

partition vector is then equal to corresponding mode shape. We can therefore define

Modal Participation Scale r as:

r r (2)

25

The contribution of the r-th mode to the covariance matrix can be expressed as:

1r H kk r r r rY (3)

MPI represents a type of “kinetic energy” in time domain, and can be adopted as a

modal indicator to distinguish structural from computational modes. MPI can be

calculated via least square solution of two vectors as the following formula:

Hr r

r r Hr r

MPI

(4)

When implementing, r-th modal participation indicator MPI is normalized as the

percentage of the “total energy”.

26

Chapter 3: Numerical example two-story-shear building

A numerical simulation is useful to verify the reliability of programs. In this case,

acceleration obtained from numerical simulation of a two-story-shear building is used to

verify damping ratio calculated with NExT-ERA. The two-story-shear building idealized

as a 2DOF structure is defined (Fig.3.1). Mass and stiffness matrices and damping ratio ξ

are selected to represent a real structure. Damping matrix is calculated based on M, K and

ξ, assuming Rayleigh damping. Sampling frequency is defined as 100 Hz. Duration of

response from the structure is selected as 1200 s, 20 min.

Figure 3.1 Two-story-shear building and its modes of vibration

3.1 Calculation of two-story-shear building response

3.1.1 Damping matrix

27

The damping matrix C is built based on mass and stiffness matrices. The first step

consists of solving the eigen-system to obtain eigen-vectors and eigen-values. C matrix is

then calculated assuming Rayleigh damping. Rayleigh damping is defined as [25]:

0 1c a m a k (1)

The damping ratio for the nth mode of such a system is

0 11

2 2n n

n

a a

with n natural frequency (2)

The coefficient 0a and 1a can be determined from specified damping ratios i and j for

the ith and jth modes, respectively.

0

1

1/1

1/2

ii i

j j j

a

a

(3)

These two algebraic equations can be solved to determine the coefficients 0a and 1a . If

both modes are assumed to have the same damping ratio , which is reasonable based on

experimental data, then:

0

2 i j

i j

a

1

2

i j

a

(4)

0 1c a m a k (5)

28

3.1.2 Normally distributed random load

The second step of this numerical example consists of defining the load applied to the

numerical structure. To simulate ambient vibration, a normally distributed random load is

chosen. P is defined with mean value = 0 and standard deviation =5.

Figure 3.2 Normally distributed random load over 1200 sec

0 200 400 600 800 1000 1200-25

-20

-15

-10

-5

0

5

10

15

20

25

Time (sec)

Load (

N)

1st floor

2nd floor

29

3.1.3 Newmark’s method

To generate displacement, velocity and acceleration of the two-story-shear building,

Newmark’s method is employed. This time-stepping method is defined in reference [25].

The current numerical example uses the linear acceleration method, (parameters

1 1,

2 6 )

Initial calculations

0 00

P cu kuu

m

(1)

2

1k̂ k c m

t t

(2)

1a m c

t

and

11

2 2b m t c

(3)

Calculation for each time step i

ˆi i i ip p au bu

(4)

ˆ

ˆi

i

pu

k

(5)

12

i i i iu u u t ut

(6)

2

1 1 1

2i i i iu u u u

tt

(7)

30

1 1, ,i i i i i i i i iu u u u u u u u u (8)

And so on…

0 200 400 600 800 1000 1200-6

-4

-2

0

2

4

6x 10

-3

dis

pla

cem

ent

(m)

Time (sec)

1st floor

2nd floor

0 200 400 600 800 1000 1200-0.03

-0.02

-0.01

0

0.01

0.02

0.03

velo

city (

m/s

)

Time (sec)

1st floor

2nd floor

31

Figure 3.3 a) displacement b) velocity c) acceleration of the two-story-shear building

3.2 Numerical application

For the numerical example, mass of each floor is chosen as m = 100 ton and stiffness k

= 3.5 MN/m. Damping ratio is fixed at ξ = 3%. Matrices for analysis are as below:

100 000 0( )

0 100 000M kg

and

70 000 000 35 000 000( / )

35 000 000 35 000 000K N m

Damping matrix and natural frequencies are calculated:

47 624 15 875

. /15 875 31 749

C N s m

and

3.6563

/9.5724

rad s

or

0.5819

1.5235f Hz

0 200 400 600 800 1000 1200

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

accele

ration (

m/s

2)

Time (sec)

1st floor

2nd floor

32

Sampling frequency is defined as 100 Hz and the signal is 1200 s-long (20 min). Power

spectral density (PSD) of each floor is presented in Fig.3.4. In the following part, the

auto-correlation function is calculated with the acceleration of the first floor or the second

floor.

Figure 3.4 PSD of a) first floor acceleration b) second floor acceleration

The auto-correlation function is obtained from NExT. If the theoretical impulse

response is compared with the obtained auto-correlation function, a good match could be

observed for the beginning of the signal (Fig.3.5).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

33

Figure 3.5 Auto-correlation function (blue) and theoretical IRF (red) for

a) first floor b) second floor

3.2.1 Parametric study on Nfft

An important parameter in the calculation of auto-correlation function is the value Nfft.

Nfft represents FFT length which determines the frequencies at which the power spectral

density is estimated. Then, PSD is calculated by averaging the squared magnitude of the

spectral FFT [22].

2

2

kx k

cS

With k kth natural frequency, kc magnitude, frequency resolution

Use of the inverse Fourier transform provides the auto-correlation function. Nfft

controls frequency resolution and time length of auto-correlation function. In fact, as PSD

is symmetric, half of PSD is meaningful for this study, so time length of auto-correlation

0 2 4 6 8 10 12 14 16 18 20

-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Norm

aliz

ed a

ccele

ration

IRF and auto-correlation 1st floor

NExT

Theoretical

0 5 10 15 20 25 30-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

IRF and auto-correlation 2nd floor

NExT

Theoretical

34

function corresponds to Nfft/2. In this case of numerical application, three values of Nfft

are used to study their influence on NExT-ERA algorithm (see Table 3.1). The aim is to

obtain modal parameters of the first mode as close as possible to the theoretical values.

Table 3.1 Parametric study on Nfft

Nfft Frequency

resolution points Nfft/2

Corresponding

time

216

0.0015 65 536 32 768 ≈ 330 sec

215

0.0031 32 768 16 384 ≈ 160 sec

214

0.0061 16 384 8 192 ≈ 80 sec

35

Figure 3.6 Auto-correlation function for the first floor and its corresponding PSD for

a) Nfft =216

b) Nfft = 215

c) Nfft = 214

0 50 100 150 200 250 300-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 50 100 150 200 250 300-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 50 100 150 200 250 300-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

36

Figure 3.7 Zoom of auto-correlation function (0~20s) and its corresponding PSD (first floor)

a) Nfft =216

b) Nfft = 215

c) Nfft = 214

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

37

Figure 3.8 Auto-correlation function for the second floor and its corresponding PSD for

a) Nfft =216 b) Nfft = 215 c) Nfft = 214

0 50 100 150 200 250 300-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 50 100 150 200 250 300-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 50 100 150 200 250 300-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

38

Figure 3.9 Zoom of auto-correlation function (0~20s) and its corresponding PSD (Second floor)

a) Nfft =216

b) Nfft = 215

c) Nfft = 214

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time (sec)

Norm

aliz

ed a

ccele

ration

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

39

As it can be seen in Fig.3.6 to Fig.3.9, in time domain, Nfft influences time length and

shape of auto-correlation functions. In frequency domain, when Nfft decreases, the first

peak in PSD figures corresponding to the first mode appears smoother and deteriorated.

This phenomenon is due to the gap increase in frequency resolution. In practice, ERA

estimates modal parameters from the initial measurement of the auto-correlation function.

It can be seen in Fig.3.7 and Fig.3.9 that the influence of Nfft is the same during the first

20 s of the auto-correlation function. To obtain modal parameters of the first mode, Nfft=

216

appears to be the most suitable for NExT. This assumption will be verified from

damping and natural frequency calculations of the first mode.

3.2.2 Parametric study on modal participation

In ERA, key parameter is the size of the Hankel matrix. This size is based on the

number of points available in the auto-spectral density [23]. Considering the shape of the

auto-correlation function, the duration between 2 ~ 17 sec could be employed to calculate

modal parameters with ERA. These time lengths correspond to 200x200 to 1700x1700

for the size of the Hankel matrix, with a sampling frequency of 100 Hz. In case of field

measurement data, the exact value of damping ratio is unknown. The size of the Hankel

matrix will be selected with modal participation indicator. After obtaining auto-

correlation function for acceleration of the first and second floor of the two-story-shear

building, ERA is applied with the variation in the size of Hankel matrix. The modal

order is two modes as the structure has two degrees of freedom. Tables 3.2 to 3.5 present

40

comparison of results. In pink are the highlighted results with maximum modal

participation of the first mode and in blue, the minimum modal participation is defined.

Table 3.2 Theoretical modal parameters for first mode of vibration

Theoretical

values

Mode 1

Frequency (Hz) Damping ratio (%)

0.5819 3

Table 3.3 Damping ratio, frequency and modal participation for mode 1, Nfft = 216

Nfft = 216

1st

floor acceleration 2nd

floor acceleration

Hankel

matrix size

Damp.

ratio (%)

Freq.

(Hz)

Modal

part. (%)

Damp.

ratio (%) Freq. (Hz)

Modal

part. (%)

200 3.985 0.586 28.529 3.462 0.583 50.172

1200 3.036 0.584 27.638 2.962 0.583 49.104

1300 3.031 0.584 27.603 2.961 0.583 49.122

1700 3.064 0.587 27.788 2.969 0.584 49.268

Table 3.4 Damping ratio, frequency and modal participation for mode 1, Nfft = 215

Nfft = 215

1st

floor acceleration 2nd

floor acceleration

Hankel

matrix size

Damp.

ratio (%)

Freq.

(Hz)

Modal

part. (%)

Damp.

ratio (%) Freq. (Hz)

Modal

part. (%)

200 3.639 0.582 28.156 3.476 0.580 49.978

850 3.292 0.582 27.839 3.261 0.581 49.476

1300 3.291 0.584 27.989 3.227 0.583 49.591

1700 3.069 0.587 27.774 3.104 0.584 49.567

41

Table 3.5 Damping ratio, frequency and modal participation for mode 1, Nfft = 214

Nfft = 214

1st

floor acceleration 2nd

floor acceleration

Hankel

matrix size

Damp.

ratio (%)

Freq.

(Hz)

Modal

part. (%)

Damp.

ratio (%) Freq. (Hz)

Modal

part. (%)

200 3.571 0.582 27.910 3.505 0.580 49.761

500 3.604 0.580 27.759 3.498 0.580 49.486

1100 3.689 0.580 28.173 3.618 0.580 50.011

1700 3.580 0.588 27.690 3.691 0.580 50.671

As expected, auto-correlation function calculated with Nfft = 216

provides damping

ratios closest to the theoretical value. Assessing the size of the Hankel matrix, damping

ratio close to 3% corresponds to the minimum modal participation. NExT-ERA gives a

damping ratio almost equal to the theoretical value. Natural frequency is also reasonable.

This numerical application verifies the accuracy of NExT-ERA algorithm and helps to

visualize the influence of parameters as Nfft and Hankel matrix size.

42

Chapter 4 Analysis of field measurement data

4.1. Jindo Bridges description

Jindo Bridges are twin cable-stayed bridges with a 344 m main span and two 70 m side

span (Fig.4.1.1). Both bridges have streamlined steel box girder with diamond-shape

pylons (Fig.4.1.2). Decks are separated only by 10 m (Fig.4.1.3).

Figure 4.1.1 Elevation of Second Jindo Bridge [2]

43

Figure 4.1.2 Cross sections (m) a) Second Jindo Bridge b) First Jindo Bridge [4]

Figure 4.1.3 Distance between Second and First Jindo Bridges [4]

Table 4.1 Natural frequencies for first vertical mode of vibration [2]

Second Jindo Bridge First Jindo Bridge

0.434 Hz 0.511 Hz

44

4.2 Field measurement data sets

Bridges are subjected to ambient vibrations created by both traffic and wind load. The

combination of these loadings is considered as white noise [19]. Vertical acceleration of

bridges was monitored at the middle of center spans. To obtain acceleration at the center

of the deck, mean values of accelerometers at each edge is calculated. However, in case

of First Jindo Bridge, one accelerometer was broken (Fig.4.2.1).

Figure 4.2.1 Measurement of vertical acceleration

a) Second Jindo Bridge b) First Jindo Bridge [26]

Data were monitored during two different periods. During the first one, MTMD did not

operate in Second Jindo Bridge although they did during the second period. Korea

Infrastructure Safety Cooperation (KISTEC) provided measurement data. Acceleration

a

b

45

was measured in gal (cm/s2) with a sampling frequency of 100 Hz. It has been decided to

split the data into 1200 s-long files. In following parts, results are calculated with 1200 s-

long data.

Without MTMD operating: 2012/10/15 10:00 to 2012/10/17 0:00

During this period, mean wind velocity was less than 14 m/s. Mean wind direction

varied from North-West (NW) to South-East (SE) with stronger wind in NW direction

(Fig.4.2.2). The horizontal component of wind velocity was decomposed into

perpendicular and longitudinal wind velocity. It can be seen that the transverse

component is almost equal to the horizontal wind velocity. Turbulence intensity Iu is

calculated with transverse wind velocity. Iu is the ratio between standard deviation and

mean wind velocity. Its value varies from 10 to 80%. High turbulence intensity is due to

low wind speed. It is also influenced by local topographical variation.

0

2

4

6

8

10

12

14

10:00 22:00 10:00 22:00 10:00 22:00

me

an w

ind

ve

loci

ty (

m/s

)

time (hours)

mean wind velocity (horizontal)

mean wind velocity (transverse)

46

Figure 4.2.2 Wind time history without MTMD a) wind velocity 20 min average b) wind direction

20 min average c) Turbulence intensity Iu d) wind rose

0

50

100

150

200

250

300

350

10:00 22:00 10:00 22:00 10:00 22:00

me

an w

ind

dir

ect

ion

(d

eg)

time (hours)

mean wind direction

0

20

40

60

80

100

10:00 22:00 10:00 22:00 10:00 22:00

Iu t

urb

ule

nce

inte

nsi

ty (

%)

time (hours)

5%

10%

15%

20%

WEST 270° EAST 90°

SOUTH 180°

NORTH 0°

0

2

4

6

8

10

12

14wind velocity (m/s)

47

With MTMD operating: 2012/11/24 10:00 to 2012/11/28 10:00

During one of the four days, wind blew from NW direction with a velocity higher than

10 m/s, reaching almost 15 m/s occasionally. In consideration of the wind rose, it is clear

that the strongest wind originates from the NW direction, transverse to the bridges

(Fig.4.2.3). Turbulence intensity Iu is similar to previously measured wind conditions, of

low turbulence (Iu=10~15%) and higher scattered intensities (80~90%).

0

2

4

6

8

10

12

14

16

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

me

an w

ind

ve

loci

ty (

m/s

)

time (hours)

mean wind velocity (horizontal)

mean wind velocity (transverse)

0

50

100

150

200

250

300

350

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

me

an w

ind

dir

ect

ion

(d

eg)

time (hours)

mean wind direction

48

Figure 4.2.3 Wind time history with MTMD a) wind velocity 20 min average b) wind direction 20

min average c) turbulence intensity Iu d) wind rose

4.3 Parameters for field measurements analysis

Based on parametric studies from the numerical example, in NExT, Nfft equal to 216

has been selected. For ERA, modal order has been chosen based on the number of peaks

that could be visualized in PSD figures. In most cases, ten to fifteen peaks can be

observed (Fig.4.3.1). Modal order equivalent to fifteen modes is employed to obtain

damping ratio, frequency and modal participation. ERA provides modal parameters for

these fifteen modes however, only the first is under consideration in the study. Size of the

0

20

40

60

80

100

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

Iu T

urb

ule

nce

Inte

nsi

ty (

%)

time (hours)

5%

10%

15%

20%

WEST 270° EAST 90°

SOUTH 180°

NORTH 0°

0

2

4

6

8

10

12

14

16wind velocity (m/s)

49

Hankel matrix varies between 500x500 and 1700x1700, corresponding to 5 and 17 sec of

auto-correlation function. In the analysis of field measurement data, minimum modal

participation is related with a frequency value which is not consistent with the natural

frequency of the first mode. However, maximum modal participation provides a close

natural frequency and a damping ratio which seems reasonable. So damping ratio is

selected for the maximal modal participation when Hankel matrix size varies from

500x500 to 1700x1700 (Fig.4.3.2).

Figure 4.3.1 2012/10/15 11:20 to 11:40 a) raw acceleration b) PSD of raw acceleration c) Auto-

correlation function d) PSD of auto-correlation function

0 200 400 600 800 1000 1200-25

-20

-15

-10

-5

0

5

10

15

20

25

Time (sec)

Am

plit

ude a

ccele

ration (

gal)

raw acceleration 5

0 5 10 15 20 250

10

20

30

40

50

60

70

80

90

100

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

0 50 100 150 200 250 300 350-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time (sec)

Norm

aliz

ed a

ccele

ration

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

50

Figure 4.3.2 2012/10/15 11:20 to 11:40, depending on Hankel matrix size variation of

a) frequency b) damping c) modal participation d) Frequency vs. modal participation

0.43385

0.4339

0.43395

0.434

0.43405

0.4341

0.43415

0.4342

0.43425

500 700 900 1100 1300 1500 1700

Fre

qu

en

cy (H

z)

frequency

natural frequency

0

0.05

0.1

0.15

0.2

0.25

0.3

500 700 900 1100 1300 1500 1700

dam

pin

g ra

tio

(%

)

0

2

4

6

8

10

12

500 700 900 1100 1300 1500 1700

mo

dal

par

tici

pat

ion

(%

)

Hankel matrix size

0.43385

0.4339

0.43395

0.434

0.43405

0.4341

0.43415

0.4342

0.43425

0 2 4 6 8 10 12

Fre

qu

en

cy (H

z)

Modal participation (%)

51

4.4 Second Jindo Bridge without MTMD operating

From 2012/10/15 10:00 to 2012/10/17 0:00, acceleration of Jindo Bridges was

monitored with a sampling frequency of 100 Hz. MTMD did not operate in Second Jindo

Bridge. Using NExT-ERA methods, damping ratio is estimated. Each value is calculated

with 1200 s acceleration data. Results of damping ratio are presented in figures 4.4.1 to

4.4.3. It can be seen that values are quite scattered between 0.1 and 0.6 %with a mean

value of 0.3 %. As wind velocity is generally lower than 10m/s, these results correspond

to the structural damping ratio. However, the average is lower than the recommended

value of damping from Korean Design Guidelines. For the aerodynamic design of steel

deck cable-stayed bridge, damping ratio should be equal to 0.4 % [1]. Estimation of

damping ratio for Second Jindo Bridge is obviously beyond recommendation. Results of

field measurement data match with damping ratio obtained from wind tunnel tests,

estimated around 0.2 % [4]. No clear relationship can be drawn between damping ratio

and wind velocity or damping ratio and RMS acceleration.

Table 4.4 Damping ratio Second Jindo Bridge without MTMD

Mean Median Mode

Damping (%) 0.292 0.283 0.215

52

A non-linear relationship between RMS acceleration and wind velocity is discernable.

In Fig.4.4.2.b, RMS acceleration remains less than 4 gal when wind blows from 0 to 10

m/s. As soon as wind velocity exceeds 10 m/s, amplitude of acceleration becomes larger,

increasing until 7 gal. However maximum wind speed monitored during this period is

equal to 12 m/s, which is quite low. Further investigation to establish this relationship

will be conducted in part 4.8.

Concerning peak factor, ratio between maximum acceleration and RMS acceleration,

its value decreases with increasing wind speed. In case of bridge subjected to random

vibration, peak factor is expected around 3 and 4. However, if the bridge is controlled by

VIV, its motion would be close to a harmonic motion at lock-in and the related peak

factor around 2 [27]. In Fig.4.4.2.c, peak factor becomes lower when wind velocity

exceeds 10 m/s. More investigation will be discussed in part 4.9 to assess if the bridge is

subjected to VIV between 2012/10/17 9:00 and 21:00.

53

Figure 4.4.1 Time history of Second Jindo Bridge without MTMD

a) damping ratio b) RMS acceleration c) MAX acceleration d) peak factor

0

0.2

0.4

0.6

0.8

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

dam

pin

g ra

tio

(%

)

ERA

mean =0.292

median = 0.283

mode =0.215

0

2

4

6

8

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

RM

S ac

cele

rati

on

(ga

l)

0

10

20

30

40

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

MA

X a

cce

lera

tio

n (

gal)

0

5

10

15

20

25

30

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

pe

ak f

acto

r

time (hours)

54

Figure 4.4.2 Wind velocity history of Second Jindo Bridge without MTMD

a) damping ratio b) RMS acceleration c) peak factor

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14

dam

pin

g ra

tio

(%

)

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14

RM

S ac

cele

rati

on

(ga

l)

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14

pe

ak f

acto

r

mean wind velocity (m/s)

55

Figure 4.4.3 RMS acceleration vs. damping ratio

Second Jindo Bridge without MTMD

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 8

dam

pin

g ra

tio

(%

)

RMS acceleration (gal)

ERA mean =0.292 median = 0.283 mode =0.215

56

4.5 First Jindo Bridge without MTMD operating

From 2012/10/15 10:00 to 2012/10/17 0:00, acceleration was also monitored on First

Jindo Bridge. During this period, the bridge was closed to traffic and ambient vibration

was due only to wind load. As it can be seen in Table 4.5 and Fig.4.5.1, estimated

damping ratio is higher than damping from Second Jindo Bridge. Moreover, it has been

found in previous works [28] that damping ratio calculated only with wind load is lower

than damping from wind and traffic loads.

Table 4.5 Damping ratio First Jindo Bridge

Mean Median Mode

Damping (%) 0.479 0.431 0.243

Considering First Jindo Bridge data, it is clearer that damping ratio increases with wind

velocity (Fig.4.5.2). It was also observed that higher values of damping increased with

RMS acceleration (Fig.4.5.3). A clear trend is highlighted in Fig.4.5.2.b between wind

velocity and amplitude of acceleration, a parabolic curve is observed.

Peak factor follows the same decay for First and Second Jindo Bridge. In Fig.4.5.2.c,

its value decreases lower than 5 when wind speed exceeds 10 m/s.

57

Figure 4.5.1 Time history of First Jindo Bridge without MTMD

a) damping ratio b) RMS acceleration c) MAX acceleration d) peak factor

0

0.2

0.4

0.6

0.8

1

1.2

1.4

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

dam

pin

g ra

tio

(%

)

ERA mean = 0.479 mode = 0.243 median = 0.431

0

1

2

3

4

5

6

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

RM

S ac

cele

rati

on

(ga

l)

0

5

10

15

20

25

30

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

MA

X a

cce

lera

tio

n (

gal)

0

5

10

15

20

25

30

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

pe

ak f

acto

r

time (hours)

58

Figure 4.5.2 Wind velocity history of First Jindo Bridge without MTMD

a) damping ratio b) RMS acceleration c) peak factor

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14

dam

pin

g ra

tio

(%

)

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14

RM

S ac

cele

rati

on

(ga

l)

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14

Pe

ak f

acto

r

mean wind velocity (m/s)

59

Figure 4.5.3 RMS acceleration vs. damping ratio

First Jindo Bridge without MTMD

4.6 Second Jindo Bridge with MTMD operating

Acceleration of Second Jindo Bridge was monitored during four days, MTMD were

released in the deck. Results can be considered in three phases.

2012/11/24 10:00 to 2012/11/26 9:00 for a wind velocity lower than 10 m/s

2012/11/26 9:00 to 0:00 for wind velocity higher than 10 m/s

2012/11/27 0:00 to 2012/11/28 10:00 for a wind velocity lower than 10 m/s

Table 4.6 Damping ratio Second Jindo Bridge with MTMD operating

Damping ratio

Mean (%) Median (%) Mode (%)

Phase 1 (wind<10 m/s) 0.57 0.484 0.249

Phase 2 (wind>10 m/s) 3.379 3.62 4.05

Phase 3 (wind<10 m/s) 0.792 0.757 0.538

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6

dam

pin

g ra

tio

(%

)

RMS acceleration (gal)

ERA mean = 0.479 mode = 0.243 median = 0.431

60

In Table 4.6 and Fig.4.6.1, damping ratio is very different between Phase 1 and 3 and

Phase 2. Firstly, when wind velocity is lower than 10 m/s, the structural damping ratio is

higher with MTMD operating rather than when it is fixed. Secondly, damping ratio

reaches very high values with an average very close to damping ratio of MTMD (Table

1.1). In Fig.4.6.2, damping ratio obviously increases with wind speed. And high damping

ratio is in most cases associated with RMS acceleration higher than 2.5 gal (Fig.4.6.3).

Even if wind velocity exceeds 10 m/s in Phase 2, RMS and MAX acceleration remains

within the general range, without exceeding, 4 and 50 gal respectively. In case of the

Second Jindo Bridge without MTMD operating, RMS acceleration exceeded 6 gal when

wind speed was higher than 10 m/s.

61

Figure 4.6.1 Time history Second Jindo Bridge with MTMD operating, between red bars wind

velocity >10m/s a) damping ratio b) RMS acceleration c) MAX acceleration d) peak factor

0

1

2

3

4

5

6

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

dam

pin

g ra

tio

(%

)

0

1

2

3

4

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

RM

S ac

cele

rati

on

(ga

l)

0

10

20

30

40

50

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

MA

X a

cce

lera

tio

n (

gal)

0

5

10

15

20

25

30

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

Pe

ak f

acto

r

time (hours)

mean=3.379 mode=4.05 median=3.62

mean=0.792 mode=0.538 median=0.757

mean=0.57 mode=0.249 median=0.484

62

Figure 4.6.2 Wind velocity history of Second Jindo Bridge with MTMD

a) damping ratio b) RMS acceleration c) peak factor

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16

dam

pin

g ra

tio

(%

)

0

1

2

3

4

0 2 4 6 8 10 12 14 16

RM

S ac

cele

rati

on

(ga

l)

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16

pe

ak f

acto

r

wind velocity (m/s)

63

Figure 4.6.3 RMS acceleration vs. damping ratio

Second Jindo Bridge with MTMD

4.7 Comparison of response from Second Jindo Bridge with and

without MTMD operating

Damping ratio shows large differences with or without MTMD operating. Without

MTMD released in the deck, damping ratio is low and does not increase with wind

velocity. In contrast, when MTMD are operating, the damping ratio exceeds 1 % to reach

almost 7 % (Fig.4.7.1). Concerning RMS acceleration, amplitude of vibration is much

larger without MTMD, but is mitigated in the other case (Fig.4.7.2). These two figures

highlight the efficiency of MTMD. As expected, damper device increases the damping

ratio and mitigates amplitude of vibration of the bridge. MTMD have been designed to

operate in case of VIV. These two sets of data provide a good opportunity to identify

features of VIV in case of Second Jindo Bridge.

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

dam

pin

g ra

tio

(%

)

RMS acceleration (gal)

64

Figure 4.7.1 Mean wind velocity vs. damping ratio, Second Jindo Bridge

Figure 4.7.2 Mean wind velocity vs. RMS acceleration, Second Jindo Bridge

4.8 Wind velocity / amplitude of acceleration relationship

In previous work, a parabolic relationship between wind velocity and RMS

acceleration has been identified [29]. As Jindo bridges, Hakucho Bridge is a steel box

girder bridge and acceleration of the deck were monitored at the center of the main span

for an orthogonal wind. The same relationship can be identified from Jindo Bridges

response. Parabolic curve is well defined in case of First Jindo Bridge (Fig.4.8.1),

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16

dam

pin

g ra

tio

(%

)

wind velocity (m/s)

without MTMD

with MTMD

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16

RM

S ac

cele

rati

o (

gal)

wind velocity (m/s)

without MTMD

with MTMD

65

ambient vibration coming only from the wind. For Second Jindo Bridge, acceleration has

also been monitored during a typhoon in August 2012, 27th, 28

th. However, only the

beginning of this event can be visualized. A deficiency of sensors provided wrong

measurements when wind velocity reached 35m/s. In Fig.4.8.2 daily and typhoon

measurements show a good continuity.

Figure 4.8.1 mean wind velocity vs. RMS of acceleration, First Jindo Bridge

Figure 4.8.2 mean wind velocity vs. RMS of acceleration, Second Jindo Bridge

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14

RM

S ac

cele

rati

on

(ga

l)

mean wind velocity (m/s)

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35

RM

S ac

cele

rati

on

(ga

l)

mean wind velocity (m/s)

daily wind

typhoon wind

66

4.9 Investigation on VIV

Response of the bridge can come from traffic load or wind load or a combination of

both. The purpose of this particular study is to identify features of each source and

response. When wind velocity is low, acceleration distinctly shows vehicles passing on

the bridge. If wind velocity is higher than 10 m/s, depending on wind direction, the

bridge can be subjected to VIV or buffeting.

Traffic load induced vibration

Looking at monitored acceleration, amplitude of vibration exceeds 5 gal when a

vehicle crosses the bridge. Maximum amplitude can reach 30 gal depending on the

vehicle type. Time length of vehicles crossing the bridge lasts between 20 and 40 sec.

This time length corresponds to the necessary period to cross the main span, considering

vehicle velocity and length of the bridge (Table 4.9.1). PSD of acceleration is broad-

banded, showing excitation of several modes (Fig.4.9.1). Wind speed lower than 10 m/s

does not influence the response of the bridge. Wind can blow in any direction when

traffic load is dominant.

Table 4.9.1 Vehicle crossing time characteristics

Main span

length

Vehicle

velocity range

Crossing time of the

main span

Observed time length in

acceleration figures

344 m 40~60 km/h

11~17 m/s 30~20 sec 40~20 sec

67

Figure 4.9.1 Raw acceleration and its broad-banded PSD 2012/10/15 13:40

Figure 4.9.2 Zoom on amplitude of acceleration assumed to be due to vehicle load

From these data, features of vibration induced by traffic can be assumed as:

Acceleration

Amplitude exceeding 5 gal

Time length of crossing vehicle between 40~20 sec

PSD is broad-banded

Wind

Any directions

Velocity lower 10 m/s

0 200 400 600 800 1000 1200-30

-20

-10

0

10

20

30

Time (sec)

Am

plit

ude a

ccele

ration (

gal)

filtered acceleration 12

0 5 10 15 20 250

20

40

60

80

100

120

140

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

960 970 980 990 1000 1010

-15

-10

-5

0

5

10

15

Time (sec)

Am

plit

ude a

ccele

ration (

gal)

filtered acceleration 12

68

Vortex Induced Vibration

To identify response of VIV alone, different acceleration data have been employed.

Monitored data come from KAIST network [30] installed on Second Jindo Bridge.

MTMD were not settled in the deck in 2011. As records are very short corresponding to

only 40 s, it can be assumed that the sensor is not being affected by crossing vehicles

during this time. Acceleration comes from sensor at the center of the main span. As it can

be seen in Fig.4.9.3, amplitude of acceleration is almost harmonic. Filter function is used

to isolate the first vertical mode of vibration. But filtered acceleration is the same as the

raw acceleration and only this mode is excited. This assumption is confirmed by the

narrow-banded PSD. One peak is significant. The peak factor is very close to 2 , which

is the expected value at lock-in (Table 4.9.2). Looking at wind speed and direction, a

wind from North-West between 12 and 16 m/s can be identified. Turbulence intensity Iu

is very low and equal to 2.48%.

Figure 4.9.3 Raw acceleration and its PSD 2011-01-16 09:07:35

0 5 10 15 20 25 30 35 40-25

-20

-15

-10

-5

0

5

10

15

20

25

Time (sec)

Am

plit

ude o

f accele

ration (

gal)

0 1 2 3 4 5 6 70

50

100

150

200

250

300

350

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density

69

Figure 4.9.4 a) Filter acceleration (0.3~0.55 Hz) and b) wind rose

Figure 4.9.5 Wind velocity and wind direction (40 sec)

Table 4.9.2 Amplitude of acceleration and peak factor

2011/01/16

09:07:35

RMS

acceleration

MAX

acceleration Peak factor

Raw data 14.375 gal 23.304 gal 1.621

Filtered data 14.337 gal 22.665 gal 1.581

0 5 10 15 20 25 30 35 40-25

-20

-15

-10

-5

0

5

10

15

20

25

Time (sec)

Am

plit

ude o

f accele

ration (

gal)

10%

30%

50%

70%

90%

WEST 270° EAST 90°

SOUTH 180°

NORTH 0°

0

2

4

6

8

10

12

14

16

18

20wind velocity (m/s)

0 5 10 15 20 25 30 35 40

12.8

13

13.2

13.4

13.6

13.8

14

14.2

14.4

14.6

Time (sec)

win

d v

elo

city (

m/s

)

0 5 10 15 20 25 30 35 40325

330

335

340

345

350

355

Time (sec)

win

d d

irection (

deg)

70

From these data, features of VIV can be assumed as:

Acceleration

Amplitude of filter acceleration is the same as raw

acceleration

PSD is narrow-banded with one significant peak

for the first mode

Wind

Direction from North-West

Velocity exceeding 10 m/s

Low turbulence intensity

Response due to combined wind and traffic loads

As it can be seen in this example, response of the bridge comes from a combination of

wind and traffic load. In Fig.4.9.6, raw acceleration shows some amplitude due to traffic

occurring between 20 and 40 sec. However, filtered acceleration is almost the same as the

raw acceleration and the first peak in PSD figure is largely dominant compare to the other

modes. However, peak factor is higher than 2 due to traffic influence (Table 4.9.3).

Considering wind characteristics, the same situation is observed: wind blowing from NW

direction and exceeding 10 m/s (Fig.4.9.8). Considering turbulence intensity Iu, its value

is higher than Iu from VIV alone because of the increased time length of monitored data.

In this example Iu = 13.2%. However, this value of turbulence intensity is low compared

to turbulence from daily wind. More investigation about turbulence intensity history is

presented in part 4.10. Finally features of VIV can be identified.

.

71

Figure 4.9.6 Acceleration in blue raw and in red filtered (0.3~0.55Hz)

and its PSD 2012/10/17 13:20

Figure 4.9.7 Wind rose Figure 4.9.8 Wind direction

Table 4.9.3 Amplitude of acceleration and peak factor

2012/10/17 13:20 RMS

acceleration

MAX

acceleration Peak factor

Raw data 6.439 gal 29.363 gal 4.56

Filtered data 5.876 gal 16.065 gal 2.73

0 200 400 600 800 1000 1200-30

-20

-10

0

10

20

30

Time (sec)

Am

plit

ude a

ccele

ration (

gal)

Filtered acceleration 155

without filter

with filter

0 5 10 15 20 250

100

200

300

400

500

600

700

800

900

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density155

without filter

with filter

10%

30%

50%

70%

90%

WEST 270° EAST 90°

SOUTH 180°

NORTH 0°

0

2

4

6

8

10

12

14

16

18

20

20121017-132000

wind velocity (m/s)

0 200 400 600 800 1000 1200270

280

290

300

310

320

330

340

Time (sec)

Win

d d

irection (

deg)

20121017-132000

72

Buffeting response

When the wind originates from a different direction other than NW, the response of the

bridge is totally different even if the wind speed largely exceeds 10 m/s. An example can

be observed during Bolaven Typhoon. Filtered acceleration is different than the raw

acceleration (Fig.4.9.9). Several modes are expressed in PSD figure. Peak factor is far

from 2 . Wind velocity is high, between 20~30 m/s but wind direction is close to the

South direction and non transverse to the bridges. Looking at the turbulence intensity Iu,

it can be seen that its value at the beginning of the typhoon is much higher than in case of

VIV. Mean Iu is equal to 29% which is comparatively higher than daily turbulence

intensity. Wind conditions and behavior of the bridge are completely different from VIV

features. Response of the bridge can be identified as buffeting.

73

Figure 4.9.9 Acceleration, in red raw, in blue filtered acceleration and its PSD 2012/08/28 3:40

Figure 4.9.10 Wind rose

Figure 4.9.11 a) Horizontal wind velocity and b) wind direction

0 200 400 600 800 1000 1200-50

-40

-30

-20

-10

0

10

20

30

40

Time (sec)

Am

plit

ude a

ccele

ration (

gal)

Acceleration 21

0 200 400 600 800 1000 1200-50

-40

-30

-20

-10

0

10

20

30

40

Time (sec)

Am

plit

ude a

ccele

ration (

gal)

Acceleration 21

without filter

with filter

0 5 10 15 20 250

100

200

300

400

500

600

Frequency (Hz)

Fourier

Am

p (

cm

/s)

Power spectral density21

without filter

with filter

10%

30%

50%

70%

90%

WEST 270° EAST 90°

SOUTH 180°

NORTH 0°

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

wind velocity (m/s)

0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

40

Time (sec)

Horizonta

l w

ind v

elo

city (

m/s

)

0 200 400 600 800 1000 1200120

130

140

150

160

170

180

190

200

210

220

Time (sec)

Win

d d

irection (

deg)

74

Table 4.9.4 Amplitude of acceleration and peak factor

2012/08/28 3:40 RMS

acceleration

MAX

acceleration

Peak factor

Raw data 8.40 gal 42.13 gal 5.01

Filtered data 4.39 gal 17.37 gal 3.96

From these data, features of buffeting can be assumed as:

Acceleration

Amplitude of filter acceleration is different

than the raw acceleration

PSD is broad-banded with at least two

significant peaks

Wind

All directions

Velocity exceeding 10 m/s

High turbulence intensity

75

4.10 Efficiency of MTMD

After identification of response of the bridge to traffic and wind load, data with and

without MTMD could be re-analyzed further. From 2012/10/17 9:20 to 18:00, mean wind

velocity is higher than 10 m/s, wind blows from NW direction and turbulence intensity Iu

is the lowest (Figure 4.10.1).

0

2

4

6

8

10

12

14

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

me

an w

ind

ve

loci

ty (

m/s

)

time (hours)

0

20

40

60

80

100

10:00 22:00 10:00 22:00 10:00 22:00

Iu t

urb

ule

nce

inte

nsi

ty (

%)

time (hours)

76

Figure 4.10.1 Wind history a) mean wind velocity b) Turbulence intensity Iu

c) wind rose for 2012/10/17 9:20 to 18:00

Acceleration data show same features as the response from the bridge to VIV. Filtered

acceleration for the first mode of vibration is almost the same than the raw acceleration

with a narrow-banded PSD (See Fig.4.9.6 to 4.9.8). In Fig.4.10.2, RMS acceleration is

the highest with scattered values during this period. This tendency is another clue about

the influence of the wind. It can also be seen that peak factors are the lowest during this

period. Peak factors are not equal to 2 because wind load is combined with traffic load,

but RMS and MAX acceleration are closer in this situation compared to usual wind

conditions. However, it is quite difficult to define a trend looking at damping ratio

values. From previous studies [5], aerodynamic damping during VIV can be negative and

could decrease the value of the damping ratio. This phenomenon does not seem to occur.

Structural damping ratio is low and Second Jindo Bridge is easily subjected to large

vibrations as observed in Fig.4.10.2.b.

10%

30%

50%

70%

90%

WEST 270° EAST 90°

SOUTH 180°

NORTH 0°

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14wind velocity (m/s)

77

Figure 4.10.2 Second Jindo Bridge response without MTMD

Between blue bars VIV occurs

If monitored data with MTMD released are observed, wind conditions for VIV are

similar. From 2012/11/26 10:00 to 0:00, mean wind velocity exceeds 10 m/s, wind

originates from NW direction and turbulence intensity Iu is the lowest, around 15%

(Fig.4.10.3).

0

0.2

0.4

0.6

0.8

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

dam

pin

g ra

tio

(%

)

0

2

4

6

8

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

RM

S ac

cele

rati

on

(ga

l)

0

5

10

15

20

25

30

10:00 16:00 22:00 4:00 10:00 16:00 22:00 4:00 10:00 16:00 22:00

pe

ak f

acto

r

Time (hours)

78

Figure 4.10.3 Wind history a) mean wind velocity b) Turbulence intensity Iu

c) wind rose for 2012/11/26 10:00 to 0:00

0

2

4

6

8

10

12

14

16

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

win

d v

elo

city

(m

/s)

time (hours)

0

20

40

60

80

100

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

Iu T

urb

ule

nce

Inte

nsi

ty (

%)

time (hours)

10%

30%

50%

70%

90%

WEST 270° EAST 90°

SOUTH 180°

NORTH 0°

0

2

4

6

8

10

12

14

16wind velocity (m/s)

79

However, even if conditions for VIV occur, response of Second Jindo Bridge is

completely different because of operating MTMD (Fig.4.10.4). The observed increase of

damping ratio can be attributed to MTMD. RMS acceleration remains in the usual range

and is not scattered or higher as it was observed without MTMD. Finally, this response

from the bridge demonstrates the efficiency of MTMD to mitigate amplitude of vibration

when wind blows from NW with a velocity higher than 10 m/s. Vibration is controlled by

increasing damping ratio of the first vertical mode with MTMD.

80

Figure 4.10.4 Second Jindo Bridge response with MTMD

Between blue bars response of the bridge to VIV conditions

0

1

2

3

4

5

6

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

dam

pin

g ra

tio

(%

)

0

0.5

1

1.5

2

2.5

3

3.5

4

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

RM

S ac

cele

rati

on

(ga

l)

0

5

10

15

20

25

30

10:00 22:00 10:00 22:00 10:00 22:00 10:00 22:00 10:00

Pe

ak f

acto

r

time (hours)

81

Chapter 5 Conclusion

Damping ratio of the first vertical mode of vibration was estimated for Second and

First Jindo Bridges using NExT-ERA modal analysis method. Structural damping ratio of

Second Jindo Bridge was lower than damping of the first bridge and less than the

recommended value specified by Korean Design Guidelines. Estimated value of damping

from field measurements show good agreement with damping obtained with wind tunnel

tests. When MTMD are released in Second Jindo deck, structural damping ratio is

increased. A parabolic relationship between wind velocity and amplitude of vibration is

observed for both bridges when MTMD did not operate. This study also provided the

opportunity to characterize features of Vortex Induced Vibration for Second Jindo

Bridge. When wind is transverse to the bridge with an upward stream and a velocity

higher than 10 m/s, Second Jindo Bridge was subjected to VIV. Under the same wind

conditions and operating MTMD, the bridge did not experience VIV. Efficiency of

MTMD was demonstrated, leading to the mitigation of vibration amplitude by increasing

damping ratio. In fact, when wind conditions caused VIV conditions, structural damping

ratio of the bridge is increased from 0.5 % to 4 % by MTMD. However, the acceleration

did not exceed 4 gal and remained in the usual range.

82

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86

ABTRACT (in Korean)

초 록

진도대교의 운용계측을 통한 감쇠계수 추정

제 2 진도대교에서 간섭현상으로 인해 발생한 와류진동에 대하여, 풍동 실험을

통하여 낮은 구조 감쇠비가 와류진동의 원인으로 지목 되었고, 그에 따라 감쇠비

증가를 위해 Multiple Tuned Mass Damper (MTMD) 가 설치되었다. 본 연구는

계측 데이터를 활용하여 감쇠비를 추정하고 풍동 실험의 결과를 규명하고자 수행

되었다. 시간 영역에서의 구조물 시스템 규명 방법인 NExT-ERA 를 활용하여

제 2 진도대교의 1 차 모드 감쇠비를 추정한 결과 평균 0.3%로 나타났다. 이는

이전으이 풍동 실험의 결과와 잘 일치하며, 케이블강교량설계지침에서 요구하는

사장교의 구조 감쇠비보다 낮은 값이다. 제 1 진도대교의 평균 감쇠비인 0.5%로

나타났다. MTMD 가 설치된 이후 제 2 진도대교는 감쇠비 증가를 보였는데

특별히 와류진동이 발생하는 바람 조건-10m/s 이상의 북서풍- 에서는 약

3.4%의 평균 감쇠비를 나타내었으며 이 때의 연직 응답 역시 MTMD 설치

이전에 비해 감소하였다. 이상의 결과를 통하여 제 2 진도대교에 설치된

MTMD 가 1차 모드 진동인 와류진동 제진에 대해 매우 효과적이라 할 수 있다.

주요어: 사장교, 감쇠비, 시스템 식별, 와류진동

학번: 2011-24090