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i

공학석사학위논문

Application of a multiple tuned mass damper

for mitigating wind-induced vibration of a

parallel-stayed bridge

병렬사장교의 풍진동 저감을 위한 다중동조

질량감쇠기의 적용

2015년 2월

서울대학교 대학원

건설환경공학부

Ouahidi Ayoub

ii

iii

iv

ABSTRACT

Vortex induced vibration (VIV) was observed in the second Jindo Bridge in

South Korea, and a Multiple Tuned Mass Damper (MTMD) was designed and

installed by a private design company to increase the mechanical damping and

reduce the vibration.

Due to overdesigning of the actual damping system, an alternative design of the

MTMD is proposed. The design procedure was decided based on wind tunnel tests,

field monitoring, and numerical simulation.

The design theory of the TMD and MTMD was studied, and the design

properties of the single TMD are used for the preliminary design of the MTMD.

Design criteria were considered and MTMD design procedure was established based

on the defined criteria.

The procedure was applied to the second Jindo Bridge. Then an alternative

design was proposed. Satisfying the design criteria and presenting good properties

for mitigating the VIV, this procedure is sufficiently safe, based on the performed

numerical simulation. Moreover, it is not overdesigned for the required performance.

After understanding and applying the MTMD design procedure, the current

MTMD performance is evaluated based on field monitoring and the general

properties of the MTMD and VIV are verified.

v

Keywords: Multiple tuned mass damper, Vortex induced vibration, cable-

stayed bridge, field monitoring, mitigation,

Student Number: 2013-23857

vi

TABLE OF CONTENTS

ABSTRACT ................................................................................................. IV

TABLE OF CONTENTS ............................................................................. VI

LIST OF FIGURES................................................................................... VIII

LIST OF TABLES ....................................................................................... IX

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

1.1 RESEARCH CONTEXT AND PLAN ....................................................................... 1

1.2 BACKGROUND RESEARCH ................................................................................ 3

CHAPTER 2 RESEARCH CONTEXT ......................................................... 5

2.1 STRUCTURAL DESIGN LABORATORY ................................................................ 5

2.1.1 Research fields .......................................................................................... 5

2.1.2 Experimental facility ................................................................................. 6

2.1.3 Laboratory description............................................................................... 7

2.1.4 Current research issues and motivations ................................................... 7

2.2 INVESTIGATED BRIDGE: SECOND JINDO BRIDGE.............................................. 8

2.3 CURRENT MTMD DESIGN AND PROBLEM DEFINITION ..................................... 9

CHAPTER 3 THEORY OF THE MULTIPLE TUNED MASS DAMPER 13

3.1 SINGLE TUNED MASS DAMPER........................................................................ 13

3.1.1 Definition and function............................................................................ 13

3.1.2 Characteristics: design of the TMD ......................................................... 13

3.1.3 Theory of the TMD: Derivation .............................................................. 14

vii

3.2 MULTIPLE TUNED MASS DAMPER ................................................................... 16

3.2.1 Introduction ............................................................................................. 16

3.2.2 Design of MTMD .................................................................................... 16

CHAPTER 4 APPLICATION: DESIGN OF THE MTMD ......................... 21

4.1 DESIGN CRITERIA ........................................................................................... 21

4.2 PROCEDURE OVERVIEW .................................................................................. 22

4.3 ALTERNATIVE DESIGN RESULTS ...................................................................... 24

4.3.1 Determination of the mass ratio .............................................................. 24

4.3.2 Calculating the optimal bandwidth .......................................................... 26

4.3.3 Deciding on the TMDs damping ratio ..................................................... 26

4.3.4 Checking the performance of the MTMD ............................................... 27

4.3.5 Summary of the design of the MTMD .................................................... 29

4.4 THE EFFECT OF THE BANDWIDTH ON MITIGATION .......................................... 30

4.5 COMPARISON WITH THE CURRENT MTMD DESIGN........................................ 31

CHAPTER 5 PERFORMANCE EVALUATION OF THE CURRENT

MTMD ......................................................................................................... 33

5.1 GENERAL DEVELOPMENT OF THE VIV ........................................................... 34

5.2 ACCELERATION AND POWER SPECTRAL DENSITY .......................................... 35

5.3 TMD RESPONSE: PHASE LAG BETWEEN THE MTMD AND THE BRIDGE ......... 37

CHAPTER 6 CONCLUSIONS ................................................................... 40

REFERENCES ............................................................................................. 42

FRENCH EXTENDED ABSTRACT .......................................................... 44

viii

LIST OF FIGURES

FIGURE 1 WIND TUNNEL FACILITY............................................................................................. 7

FIGURE 2 PARALLEL TWIN JINDO BRIDGE .................................................................................. 9

FIGURE 3 DISPOSITION OF THE TWO BRIDGE DECKS .................................................................. 9

FIGURE 4 CURRENT MTMD CONFIGURATION .......................................................................... 10

FIGURE 5 BRIDGE FREQUENCY VARIATION FOR THE OBSERVED PERIOD ................................. 11

FIGURE 6 EFFECT OF THE DAMPING RATIO ON THE DISPLACEMENT OF THE BRIDGE ................. 12

FIGURE 7 MTMD FREQUENCY DISTRIBUTION MODEL ............................................................. 17

FIGURE 8 FREE BODY DIAGRAM OF THE BRIDGE (LEFT), AND TMDS (RIGHT) .......................... 18

FIGURE 9 DESIGN PROCEDURE OF THE MTMD ....................................................................... 23

FIGURE 10 TOTAL EQUIVALENT DAMPING RATIO ..................................................................... 24

FIGURE 11 DISPLACEMENT OF THE TMD (STROKE) ................................................................. 25

FIGURE 12 EFFECT OF THE TMD DAMPING RATIO ON THE EQUIVALENT DAMPING ................... 27

FIGURE 13 TIME HISTORY OF THE DISPLACEMENT OF THE CENTRAL TMD............................... 28

FIGURE 14 ACCELERATION OF THE BRIDGE WITH AND WITHOUT TMD .................................... 29

FIGURE 15 EFFECT OF THE FREQUENCY CHANGE ON THE RESPONSE FOR DIFFERENT MASS RATIO

AND CURRENT DESIGN ...................................................................................................... 30

FIGURE 16 FIELD MONITORING SETTING FOR THE SECOND JINDO BRIDGE ............................... 33

FIGURE 17 DEVELOPMENT OF THE VIV FOR UNCONTROLLED BRIDGE ..................................... 34

FIGURE 18 DEVELOPMENT OF THE MTMD AFTER THE INSTALLATION OF THE MTMD ............ 35

FIGURE 19 PHASE LAG BETWEEN THE TMD AND BRIDGE FOR THE 'SAFE CASE', V=3.4M/S ...... 38

FIGURE 20 PHASE LAG BETWEEN THE TMD AND BRIDGE FOR THE 'VIV CASE', V=10.55M/S .. 38

file:///C:/Documents%20and%20Settings/Administrator/My%20Documents/Ayoub/thesis/thesis%20SNU.docx%23_Toc408845903file:///C:/Documents%20and%20Settings/Administrator/My%20Documents/Ayoub/thesis/thesis%20SNU.docx%23_Toc408845905file:///C:/Documents%20and%20Settings/Administrator/My%20Documents/Ayoub/thesis/thesis%20SNU.docx%23_Toc408845907

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LIST OF TABLES

TABLE 1 STMD PARAMETERS ................................................................................................. 15

TABLE 2 TMD PARAMETERS WHEN INCREASING THE MASS RATIO ........................................... 25

TABLE 3 FINAL DESIGN PROPERTIES OF THE MTMD ............................................................... 29

TABLE 4 COMPARISON BETWEEN THE CURRENT AND ALTERNATIVE DESIGN ............................. 31

TABLE 5 COMPARISON OF THE PSD AND ACCELERATION WITH AND WITHOUT MTMD ............ 36

1

CHAPTER 1

INTRODUCTION

1.1 Research context and plan

In the view of the recent development of super-structures such as the

suspension and cable stayed bridges, a growing interest for increasing the safety of

these structures has been observed. Structural control solutions are more and more

used in order to be conforming to the performance and safety specifications.

Wind induced vibration is one of the main concerns when dealing with long

span cable supported bridges. It is considered to be the most aggressive external

excitation. In regions where the seismic activity is low, the wind induced vibration

is the primary source of damage for structures.

In this context, the mitigation of wind induced vibration has become of main

concern. There are various methods to mitigate the wind induced vibration:

structural modification, aerodynamic measures and mechanical measures.

Structural modification method involves modifying the properties of the bridge,

such as the Mass, damping or stiffness matrices. Aerodynamic measures are made

by modifying wind flow around the bridge components in order to have a better

aerodynamic performance. This is done by installing aerodynamic devices like

fairings, or by modifying the aerodynamic shape. The third method is the

mechanical measure: it requires the installation of passive or active control devices

2

such as dampers. Passive devices do not need external energy and are mainly

aimed to increase the total damping of the structure and consequently increase the

energy dissipation. Active devices, on the contrary, need an external source of

energy and can adjust and apply forces to the bridge in a controlled manner.

The second Jindo Bridge which is the subject of this research was equipped

with a Multiple Tuned Mass Damper (MTMD in the following) which performance

was evaluated.

In this research, the choice was set on a passive control device, namely a

multiple tuned mass damper, in order to mitigate the wind-induced vibration in the

second Jindo Bridge. An alternative design is proposed with a goal to achieve the

target design values and offer a design that is conforming to the specifications. The

current design of the bridge has been overdesigned; the causes of this are discussed

and an alternative design is proposed to overcome this weakness.

A definition of the research context is given in the following part, defining the

background support research, the bridge specifications, current MTMD design and

the problem definition, followed by a background theory for the design of the

MTMD. Application of the design is then carried on following the theory, and

following our design procedure, the design is checked by numerical simulation and

final design parameters are obtained. After producing an alternative design of the

MTMD, the following part evaluates the performance of the current damper based

on field monitoring results.

3

1.2 Background research

The Tuned Mass Damper was first proposed in 1919 and its theory developed

in Den Hartog’s book on mechanical vibrations (1940). Other researchers such as

Randall et al. (1981), Tsai and Lin (1993) extended the theory for a damped Single

TMD. It was found that if the frequency of the TMD is tuned close to the natural

frequency of the main structure, a large reduction of motion can be observed.

However the TMD was found to be very sensitive to changes in the natural

frequency of the main structure. Considering fabrication errors, imprecise

estimation of the natural frequency, and change of the properties due to weather

conditions, this is clearly a disadvantage for the TMD. To overcome this, multiple

tuned mass damper system design was proposed, from which Abe and Fujino (1994)

who proposed a characterization of the MTMD and some design formulas for a

large number of TMDs. The use of the MTMD for mitigating wind vibration was

explored and it was found to be effective for harmonic excitation which can

represent the vortex induced vibration.

The second Jindo Bridge was subjected to Vortex induced vibration (VIV) on

few occasions and on April 19, 2011, a high VIV was observed for a duration of

two hours. The acceleration of the bridge at that time exceeded 1.5𝑚/𝑠2 which is

3 times higher than the allowable serviceability limit (Korean Society of Civil

Engineers (KSCE), 2006)

A series of wind tunnel tests were performed (Seo et al. 2013) to evaluate and

identify the reason of this VIV and the low damping ratio of the bridge was pointed

4

out to be the main cause of the vibration. Identification of the damping ratio from

field monitoring was done and the damping ratio was found to be 0.29% (Kim and

al. 2013), lower than the recommended 0.4% value.

The installation of a damping system was recommended, and consequently a

multiple tuned mass damper system was designed and installed by another design

company in the second Jindo Bridge in 2012 (TE Solution, 2012).

The damping ratio of the bridge was then estimated by field monitoring after

the installation of the MTMD. It was found that the damping ratio increases when

the wind velocity is around the critical velocity for VIV (Calmer, 2013). The study

of the design of the current TMD (TE Solution) will be shown below

5

CHAPTER 2

RESEARCH CONTEXT

2.1 Structural design laboratory

2.1.1 Research fields

This research was done in the Structural Design Laboratory, supervised by

Prof Kim Ho-Kyung. The structural design laboratory focuses on technology

development for long span cable-stayed bridges. The laboratory performs study and

analysis in different domains, and develops analysis-design program for suspension

and cable stayed bridges. There are four major fields of research in the laboratory.

- Wind engineering: this is the main focus of the laboratory, and the

researches performed in this domain are numerous. From which we can

list non exhaustively :

Frequency-time domain buffeting analysis

Study on the extraction of aeroelastic parameters

Flow measurement by Particle Image Velocimetry (PIV)

Active turbulence generator for simulating wind

Estimation of vehicle runnability on a bridge

- Planning and design of cable supported bridges: it studies and develops

analysis and design methods for cable supported and cable-stayed bridges.

Design philosophy and guidelines are developed, and code calibration is

performed based on reliability analysis.

6

- Structural health monitoring, maintenance and evaluation: it concerns the

evaluation of the performance of bridges by field monitoring and

comparing it with the expected performance. Design feedback and

aerodynamic characteristics of bridges can be obtained by field monitoring.

The monitoring of structures has a crucial role since it estimates the future

lifetime and maintenance cost and tracks fatigue related problems.

- Extreme engineering: this field is still in the planning phase and it will be

added to the laboratory facilities. Its main focus is about structures in

extreme conditions of pressure, temperature or load. The main purpose of

this research field is to develop new high performance materials for

extreme conditions, analysis theory and new design codes. In addition, it

aims to propose efficient structure system.

2.1.2 Experimental facility

The laboratory has a wind tunnel for simulating the effect of wind on bridge

models. The wind tunnel has a section of 1m x 1.5m for a length of 4m. The

maximum wind speed is 23m/s. The wind tunnel can be equipped with a turbulence

generator, which simulates the vertical and longitudinal turbulence. A particle

Image Velocimetry (PIV) system is also available for visualization of the wind flow

field. The wind tunnel is shown in the figure below

7

Figure 1 Wind tunnel facility

2.1.3 Laboratory description

The laboratory is similar to a small company, headed by a professor who

manages the projects and guides students in their research. The laboratory is

composed of 18 members, half of which are master students. The working time is

from 10am to 6pm, and every student is required to be in the laboratory during this

time.

All students are supposed to participate in the research activity of the

laboratory and to conduct their own research projects. The student’s research is

supervised by the professor and by the senior students but autonomous work is

required to conduct the research in these working conditions.

2.1.4 Current research issues and motivations

The laboratory has conducted previous researches on the second Jindo Bridge,

and the related results concluded that there is a need for installing a mitigation

system on the bridge to increase the damping ratio. However, the damping system

8

was installed by another design company, which means that the laboratory did not

have the chance to acquire detailed knowledge about the design of the dampers.

Hence, a theoretical study was conducted in this research. The understanding of the

TMD behavior was also deepened by the results from field monitoring.

2.2 Investigated bridge: Second Jindo Bridge

The Jindo Bridge is composed of two parallel cable-stayed bridges. The first

one was opened to traffic in1984 while the second one was built later in 2005. The

main span of the bridges is 340m and the two decks are separated with a distance

of 10m. The two bridges have similar cross sections. The Jindo bridges have a two

side spans of 70m which makes a total length of 484m for 11.7m wide for the 1st

Jindo Bridge, and 12.5m for the second one.

The second Jindo Bridge is instrumented with wireless sensors to monitor its

condition. It was equipped in 2012 by an MTMD to mitigate the vortex induced

vibration. In this research, we focus on the second Jindo Bridge for the design of

the MTMD and performance evaluation by field monitoring.

9

Figure 2 Parallel twin Jindo Bridge

Figure 3 Disposition of the two bridge decks

2.3 Current MTMD design and problem definition

As said before, in order to increase the damping of the bridge and mitigate the

VIV, Multiple tuned mass damper was designed by ‘TE Solution’. The design

methodology is briefly exposed in the following.

22.25m

0.089m

9.9m

10

Figure 4 current MTMD configuration

The reduction factor is calculated from field monitoring. The data collected

from April 2011 was considered and the acceleration reached 1.5m/s2 for a target

acceleration of 0.5m/s2 . This gives a reduction factor of 33%, which was

expressed in the current design as following:

0 . 3saeq

R

which gives

2

seq

aR

Where ξs is the bridge damping. The equivalent damping ξeq is then

obtained and its calculated value is equal to ξs = 0.29%.

The mass ratio is obtained by the equivalent damping ratio. By applying a

safety factor of 1.2 on the mass ratio, and calculating the equivalent damping ratio ,

the obtained value is ξeq = 3.97%.

The expected reduction factor is finally R = √ξbridge

ξeq= 27.4%

11

Design problem

Large bandwidth (frequency range of the dampers) :

For the current design, the bandwidth is decided based on field monitoring

results. From field monitoring results for over one year, the variation of the

frequency was found to be 0.022Hz

The bandwidth is calculated from the field monitored bandwidth by

considering a bandwidth coefficient of 2 (2 times the field monitored bandwidth).

02 0.022 0.044

bB f Hz Hz

This bandwidth is large and is decided without any theoretical background.

That is why an alternative design would find the appropriate bandwidth coefficient

based on mitigation effect and robustness.

High damping ratio:

As seen above, the reduction factor is calculated as a square root of the

damping ratios, which supposes that the acceleration and inverse of square root of

00.022f Hz

Temperature (C)

Brid

ge n

atural freq

uen

cy

Figure 5 Bridge frequency variation for the observed period

(TE solution report)

12

the damping ratio are proportional and gives a non-needed high equivalent

damping ratio. The required damping was estimated by Seo et al. (2013) in the

following figure

Figure 6 Effect of the damping ratio on the displacement of the bridge

It is seen that a damping ratio of 0.4% is sufficient to stay below the allowable

limit in terms of displacement. It is then clearly seen that the equivalent damping

ratio of the current design is overdesigned. In this research the alternative design

considers wind tunnel tests, theoretical formulation and numerical simulation, with

a more reasonable target design value for the damping ratio (a value of 1% will be

considered for safety reasons)

13

CHAPTER 3

THEORY OF THE MULTIPLE TUNED MASS

DAMPER

3.1 Single tuned mass damper

3.1.1 Definition and function

A tuned mass damper is a device composed of a mass, a spring and a damper,

attached to structures in order to reduce the dynamic response. The following

figure shows a Tuned mass damper

Figure 3-1 Single TMD attached to a damped main structure

Tuned mass dampers (TMDs) have been used to control the vibration of tall

buildings, bridges, steel structures and other constructions. The principle of a TMD

is that it is attached to a structure and dissipates energy by moving out of phase of

the main structure. The TMD frequency is tuned to the natural frequency of the

bridge

3.1.2 Characteristics: design of the TMD

Designing a TMD means finding the optimal parameters that reduce the most

14

the motion of the main structure. The optimal parameters are defined for each of

the components: the mass, the damper and the spring.

The parameters to determine are

Mass of the TMD : m

The damping coefficient cd

The stiffness coefficient kd

The optimal parameters are well known, and were found in previous research.

The TMD theory, developed by Den Hartog, is introduced in the following part.

3.1.3 Theory of the TMD: Derivation

The considered model for the Bridge and TMD system is a two degree of

freedom system, as shown in the figure below

In this model, since the bridge damping is very low (0.3%), we approximate

the system with 2 degree of freedom system, with a damped TMD and undamped

primary mass.

Figure 3-2 Bridge-TMD considered model

15

a. Parameters:

In order to simplify the derivation, the following parameters are defined

Table 1 STMD parameters

Bridge TMD

Natural frequency

Damping ratio

frequency ratio

Mass ratio

Tuning frequency df

b. Equation of motion

The equation of motion for the primary mass and the secondary mass is

expressed as following:

Primary mass

Secondary mass

In our case, we neglect the ground acceleration and we consider only the wind

excitation which will be considered as a periodic excitation

The solutions are written as and

Replacing in the equation of motion we obtain

u dd d d gm ku k u c u p ma

u ud dd d d d d d gm k u c u m m a

0

i tp p e

2 2( ) u 0dd d d dm k ic u m

2

0( ) ( ) dd dm k u k ic u p

i tu u e

i t

ddu u e

16

Solving for u the following form is found :

0 ipu Hek

in which H is a coefficient called amplification factor. The

expression of H is given in function of the parameters defined earlier in the table 1.

Minimization of H gives the optimal design parameters and finally gives all the

parameters in function of the mass ratio. From previous studies, considering the

damping of the bridge neglected, the obtained formulas are as following:

1 0.5

1opt

mf

m

(3 0.5 )

8(1 )(1 0.5 )opt

m m

m m

and

1

0.5

mH

m

3.2 Multiple tuned mass damper

3.2.1 Introduction

A multiple tuned mass damper (MTMD) consists of a number of single tuned

mass dampers with natural frequencies around the natural frequency of the target

controlled mode of the main structure.

The main reason of the use of the MTMD is to have a better robustness of the

mitigation. The single TMD is vulnerable to the variation of the parameters of the

bridge. Indeed, changes in the ambient temperature cause variation of the natural

frequency of the bridge. As a result, the TMD is no more tuned and the mitigation

effect is reduced. The MTMD, tuned to more than one frequency, has a more stable

behavior to the change of the bridge’s natural frequency.

3.2.2 Design of MTMD

Until now, unlike the single TMD, there is no unique method to design the

17

multiple tuned mass damper. The considered design for this thesis is based on

Fujino et al (1994). The following assumptions and properties are considered: and

odd number of TMDs is considered in order to have a more symmetric behavior of

the TMDs around the natural frequency of the bridge. In this thesis, the chosen

number of TMDs was 5. The mass m and damping ξt of the TMDs are equal for

each of the TMDs. The main structure’s properties are indexed with the index ‘S’

and the damper’s from –n to n. The natural frequencies are equally spaced with a

total range of Bf called the frequency bandwidth. The frequency bandwidth

observed from field monitoring is named Δf. The central frequency of the MTMD

is given by 01

s

total

. This choice is made in order to simplify the

following derivation, and since the MTMD is not sensitive to mistuning this value

can be considered without a loss of generality. The figure below shows the

frequency distribution. We define a bandwidth coefficient γb from the frequency

bandwidth obtained from field monitoring by 𝐵 = 𝛾𝑏Δ𝑓

fB

0f

1f

2f

1f

2f

sf

Frequency f

Figure 7 MTMD frequency distribution model

18

a- Equation of Motion

The equation of motion is expressed as following: X X X M C K F

We consider a 6 dof system consisting of the bridge modeled as a single

degree of freedom system and the 5 TMDs. The equation of motion is obtained

from the free body diagram below and the following derivation. The displacement

of the bridge is relative to the ground, while the displacements of the TMDs are

relative to the bridge.

Figure 8 free body diagram of the bridge (left), and TMDs (right)

Equations of motion for the bridge and TMD are expressed as following

2

2

( )s s s s s i i i ii

c u M u k u c u k u p

( ) 0s i i i i im u u c u k u

Replacing ( )i i i i s ic u k u m u u , the final equations are given as

2

2

( 5 )s s s s s ii

c u M m u k u mu p

(1)

19

( ) 0s i i i i im u u c u k u (2)

From these equations, the mass, damping and stiffness matrices are found as :

5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

s

M m m m m m m

m m

m mM

m m

m m

m m

2 2 2 2 2 2

2 1 0 1 2( , ,m ,m ,m ,m )sK Diag M m

2 1 0 1 2(2M ,2m ,2m ,2m ,2m ,2m )s s T T T T TC Diag

2 1 0 1 2, , , , ,T

sX x x x x x x

b- Frequency response of displacement

Now that we found the MCK matrices of the system, the frequency response

function is derived in the following. For a harmonic force (t) exp(i t)f we

consider a response vector function ( )H . The equation of motion is expressed

as following (Fujino and Abe 1994):

2 ( )e 1,0,0,0,0,0 eTi t i tM i C K H

with

2

12

0

1

2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

G L L L L L

L g

L gM i C K

L g

L g

L g

20

With 2 25 2 s s sG M m iM M , 2L m and

2 22i T j jg m im m

The first column of the matrix is interesting because it corresponds to the

response function of the bridge which we want to calculate.

12( )H M i C K F

. The maximum value of the response function

occurs in the central frequency of the MTMD. Its value is given by

The frequency response function is given as a function of the TMD damping,

the safety factor and the mass ratio.

In order to estimate the effect of the MTMD on the increase of the total

damping, the equivalent additional damping is calculated as follows

2 max

1

2eq s

s sM H

The final expression of the equivalent additional damping is given by

4

002 2 2

0 0

1(2i )

2 2

n

eq s s s

j ns j T ji

max0

40

0 220 0

0 0 0 0

1( ) ( , , )

(2i )

22 2

s s t bn

s s

b bj nT

H H f

Mf f

j i j

21

CHAPTER 4

APPLICATION: DESIGN OF THE MTMD

4.1 Design criteria

In order to come up with a design procedure for the MTMD, few design

criteria were chosen, and the criteria were mainly decided either by field

monitoring or by wind tunnel tests. The criteria are listed below :

The total equivalent damping ratio 𝜉𝑒𝑞. This criterion is important for the

reduction of motion and its design value can be found in the specifications. The

chosen criterion for the equivalent damping is, however, decided from the wind

tunnel test results (Seo et al. 2013) as seen before, considering a safety margin as

1%.

The reduction factor: it is defined as the ratio between the maximum

displacement of the controlled and uncontrolled system 𝑅𝐹 =𝑢𝑐𝑜𝑛𝑡

𝑢𝑢𝑛𝑐𝑜𝑛𝑡. Its design

value is decided from the field monitoring results. The target reduction is set to be

lower than 33%.

The stroke of the TMDs : it is defined as the total displacement of the TMDs.

Its target value is determined from the bridge deck geometrical properties. The

design value is set to be lower than 1m

The TMD damping : the maximum value of the TMDs damping was also fixed

for cost reasons. A high damping requires high number of oil dampers which

22

increases the cost of the TMD. A value of 6% was fixed for the TMDs damping.

4.2 Procedure overview

The overall design overview is described by the flow chart below.

The first step is to decide on the mass ratio: the mass ratio is calculated based

on the single tuned mass damper model. The safety factor for bandwidth is

determined next based on the critical bandwidth formulation. The TMD damping is

calculated from the target equivalent damping. Once the three parameters are

resolved, the next step is the verification. The stroke of the TMD is calculated and

if its value is over the specified limit, the mass ratio is increased until the stroke

conforms to the criteria.

The reduction factor is then checked and the verification is done for the design

criteria. The mass ratio is increased again when the criteria are not verified.

23

\

Higher

mass ratio

Yes

Start

Fix the Mass ratio

Calculate TMD damping

T for eqDesign

Enter Safety factor γ

FieldMonitoringf SF f

Calculate TMD

stroke

Calculate reduction

factor

RF>33%

Calculate the MTMD

parameters

No

End

( )eq Tf

Stroke

24

4.3 Alternative design results

The previous design procedure is applied for the second Jindo Bridge. As

previously mentioned, the number of TMDs was set to five and the design

methodology and results are shown step by step in the following:

4.3.1 Determination of the mass ratio

The first step of the design is the determination of the mass ratio. To do this, a

STMD system was considered. The mass ratio is calculated by the target equivalent

damping ratio, using the damping equation:

Figure 10 Total equivalent damping ratio

0.5

2(1 )eq

25

Figure 11 Displacement of the TMD (stroke)

For the target equivalent damping of 1%, the obtained mass ratio is 0.08%.

However, the stroke of the TMD exceeds the fixed limit of 1m.

The mass ratio is then increased and the same procedure is repeated. The result

is shown in the table below:

Table 2 TMD parameters when increasing the mass ratio

Mass ratio (%) ξeq (%) Stroke (m)

0.08 1 2.2

0.2 1.5 1.1

0.3 1.9 0.75

The mass ratio of 0.3% gives a satisfying equivalent damping and stroke. This

mass will be considered for the initial design of the MTMD.

26

4.3.2 Calculating the optimal bandwidth

The next step in the design is deciding on the appropriate bandwidth. The

bandwidth definition was given earlier as a bandwidth coefficient of the field

monitored bandwidth. To decide on the bandwidth coefficient, the equation given

by Fujino and Abe (1994) was used:

For the chosen mass ratio of 0.3, the bandwidth is calculated as 0.024Hz, and

knowing that the field monitored frequency bandwidth is 0.022Hzm the bandwidth

coefficient is calculated as 𝛾𝑏 = 1.11

4.3.3 Deciding on the TMDs damping ratio

Once the mass ratio and the frequency bandwidth are calculated, the

determination of the TMD damping ratio is straightforward. The equivalent

damping ratio is plotted as a function of the TMD damping ratio. From the target

equivalent damping ratio of 1%, the damping ratio of the TMD can be calculated as

depicted in the figure below

40

1

1/ 2

2f total

j

Bj

40

02 2 20 0

1(2i )

2 2

n

eq s s s

s j jj n i

Tξ

27

Figure 12 Effect of the TMD damping ratio on the equivalent damping

For the target equivalent damping of 1%, the TMD damping is over 10%

which exceeds the 4th criteria limit of 6%. Since a high damping value implies a

high cost, the maximum allowable TMD damping ratio of 6% is considered.

4.3.4 Checking the performance of the MTMD

To assess the performance of the MTMD, the stroke and reduction factor are

checked. The verification is done by numerical analysis performed by a 6DOF

dynamic analysis using the Newmark beta method. The applied load is given such

as it simulates the largest vibration observed by field monitoring before the

installation of the current MTMD. The acceleration at that time reached 𝑎 =

1.5𝑚/𝑠2 , so the corresponding harmonic excitation is calculated as 𝑝 =

2𝜉𝑠𝑘𝑠𝑎

𝜔𝑠2 sin(𝜔𝑠𝑡)

28

Figure 13 Time history of the displacement of the central TMD

The stroke of the TMD is 68.8cm, which is in accordance with the geometry

constraint of the second Jindo Bridge. The reduction factor of the bridge is

obtained similarly by calculating the ratio between the bridge acceleration without

and with TMD. The reduction factor is calculated as 𝑅𝐹 =max(𝑎𝑀𝑇𝑀𝐷)

max(𝑎𝑢𝑛𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑑)=

0.312

1.5= 20.7%

29

Figure 14 acceleration of the bridge with and without TMD

4.3.5 Summary of the design of the MTMD

The design criteria were satisfied for the mass ratio of 0.3%, so this mass is

considered as a final design value. The properties of the MTMD can be calculated

easily from the three parameters which are the TMD damping ratio, mass ratio and

bandwidth coefficient. They are summarized in the following table

Table 3 Final design properties of the MTMD

TMD1 TMD2 TMD3 TMD4 TMD5

Mass (kg) 613.8 613.8 613.8 613.8 613.8

Damping ratio 6% 6% 6% 6% 6%

Frequency(Hz) 0.425 0.431 0.437 0.443 0.449

30

The expected performance of the bridge based on the defined criteria is an

equivalent damping of 1.43%, a reduction factor of 20.7% and a stroke of 68cm.

4.4 The effect of the bandwidth on mitigation

The interest of the MTMD is that it allows having a good robustness; which

means the same mitigation even if the bridge natural frequency is changing. A

higher bandwidth insures a higher robustness up to certain level, so to check the

robustness of our design, the effect of the increase of the bandwidth was studied.

Figure 15 Effect of the frequency change on the response for different mass ratio and

current design

The natural frequency of the bridge was varied in a range corresponding to a

bandwidth coefficient of 2, and the maximum acceleration of the bridge was

calculated for each case. The result shows that a higher bandwidth effectively gives

31

a more robust design, with a lower variation of the acceleration, but with a loss of

mitigation effect. The alternative design does not exceed the acceleration limit of

0.5𝑚/𝑠2 for the considered frequency variation, so additional robustness is not

needed and the current bandwidth is considered.

4.5 Comparison with the current MTMD design

The current design acceleration was also plotted in the same figure above and

it is seen that the reduction is very high compared to the required reduction. Other

parameters are compared in the table below

Table 4 comparison between the current and alternative design

Parameters Current design Alternative design

Mass ratio (%) 1 0.3

TMD damping (%) 3.45 6

Equivalent damping (%) 3.87 1.43

Bandwidth coefficient 2 1.1

Stroke (cm) 36 68

Expected reduction factor (%) 30 20.7

Actual reduction factor (%) 8.5 20.8

By comparing the current and alternative design, it can be seen that the

alternative design has a smaller mass which implies a lower cost. The TMD

damping However is higher than the current design which may induce additional

cost. The equivalent damping is coherent with the design value, while the current

32

MTMD is overdesigned for the equivalent damping ratio. The same problem

appears for the reduction factor which is overdesigned. The stroke of the alternative

design is higher than the current design but still satisfies the geometry

consideration. The alternative design is coherent with the target design and is not

overdesigned for the robustness and mitigation

33

CHAPTER 5

Performance evaluation of the current MTMD

After understanding the design of the MTMD and applying our design

procedure on the second Jindo Bridge to obtain a design in pair with the

specifications and avoid additional costs related to overdesigning, the current

chapter is about the performance evaluation of the current bridge design. The

objective is to obtain a feedback from field monitoring and to verify the vortex

induced vibration mechanism and bridge response.

The field monitoring setting is composed of two accelerometers in the center

of the bridge to measure the bridge acceleration, one anemometer for measuring

the wind velocity, and 4 accelerometers for each of the TMDs as depicted below

Figure 16 Field monitoring setting for the second Jindo Bridge

Accelerometer

anemometer

34

5.1 General development of the VIV

The general development of the vortex induced vibration was observed using

field monitoring. Two cases were considered in order to estimate the effect of the

MTMD: the first one is before the installation of the MTMD and the later one after.

The two cases were taken from the collected data sets with similar wind conditions

around the triggering wind velocity of 10m/s, and the acceleration of the bridge

was observed in both cases.

Figure 17 Development of the VIV for uncontrolled bridge

35

Figure 18 Development of the MTMD after the installation of the MTMD

From the figures above, we can clearly observe the effect of the MTMD on the

bridge acceleration. In the first case where there was no MTMD, the acceleration

of the bridge is gradually increasing after the VIV is triggered, while in the second

case, the VIV is not developed and the vibration of the bridge decreases after the

TMD starts operating. This shows how the TMD works to mitigate the Vortex

induced vibration

5.2 Acceleration and Power spectral density

The performance evaluation of the MTMD is further explored by comparing

the acceleration and PSD of the bridge before and after installing the device. In

order to estimate the importance of the VIV on the observed vibration, the

acceleration is filtered by a low pass filter of 1Hz to consider only the effect of the

VIV and this acceleration is compared with the raw acceleration. The PSD of the

36

bridge acceleration is also compared and the 1st mode vertical mode (VIV mode)

intensity.

Table 5 Comparison of the PSD and acceleration with and without MTMD

Without TMD (wind velocity 10.32m/s) With TMD (wind velocitt 10.55m/s)

First mode dominant, high intensity

First mode still dominant, but low intensity

Raw and filtered data similar VIV

dominant

Filtered data very low VIV mitigated

From the figures above, two observations are important to notice :

- Before the installation of the MTMD, the first vertical mode

corresponding to the VIV is dominant with a high intensity, while for the

37

latter case it is dominant but with a reduced intensity. The higher modes,

corresponding to traffic induced vibration, remain similar in both cases.

This shows that the MTMD works well to mitigate only the first vertical

mode which represents the VIV.

- The filtered acceleration in the first case is almost similar with the raw

acceleration, which means that the vortex induced vibration is the

dominant component. However, after installing the MTMD, the

acceleration representing VIV is very small compared to the raw

acceleration. This means that the VIV was mitigated and the vibration is

mainly due to higher modes caused by traffic.

5.3 TMD response: phase lag between the MTMD and the bridge

From the theoretical research on the TMD and by energy considerations, the

energy dissipation is optimal when the phase lag between the TMD and the bridge

is 90 ˚.

In order to verify the performance of the TMD, this we considered the

acceleration of the bridge and TMD for 2 wind velocity cases. The first case called

‘safe case’, where the wind velocity is lower than the critical velocity, and the

second case is the ‘VIV case’, where the wind velocity is around the critical

velocity of 10m/s.

38

Figure 19 Phase lag between the TMD and bridge for the 'safe case', V=3.4m/s

Figure 20 Phase lag between the TMD and bridge for the 'VIV case', V=10.55m/s

It can be seen that in the ‘safe case’, the bridge and TMD accelerations are in

phase (the phase lag is almost 0), and the TMD follows totally the motion of the

39

bridge, which implies no energy dissipation. The ‘VIV case’ however, shows a

phase lag around 90 ˚ and MTMD acting as a damping force on the girder.

40

CHAPTER 6

CONCLUSIONS

In order to mitigate the bridge vortex induced vibration, the current design of

the Multiple Tuned Mass Damper (MTMD) was investigated. It was found to be

overdesigned for the damping ratio and for the bandwidth.

The theory of the MTMD was studied and a design procedure was defined,

based of determined design parameters, target criteria, and definite methodology.

The design considers the wind tunnel tests results, the field monitoring as well as

the results obtained by numerical simulation.

The design procedure was adjusted and applied for the Second Jindo Bridge.

The results were satisfying in terms of reduction, stroke and damping ratio.

Moreover, the alternative design is not overdesigned for the damping ratio or the

bandwidth. The numerical simulation, finally, confirms the performance of the

MTMD which is similar to the expected one

The effect of the bandwidth was studied by numerical simulation. A high

bandwidth giving a good robustness is not required for the current bridge. The

response of the bridge remains under the acceptable levels for a variation of the

natural frequency over twice the actual variation.

The general properties of the Vortex Induced Vibration were explored and

verified by field` monitoring for the current MTMD. The performance of the

MTMD was validated by field monitoring data.

41

For further research works, a more comprehensive and cost effective design

can be obtained. The response for buffeting can be calculated using the full bridge

model. Finally, a precise estimation of the cost of each component, and the effect of

each parameter on the total cost need to be established.

42

REFERENCES

[1] Connor, J. J. Introduction to Structural Motion Control. Upper Saddle River, NJ:

Prentice Hall Pearson Education, 2003.

[2] Igusa, T., and K. Xu. "Vibration Control Using Multiple Tuned Mass

Dampers." Journal of Sound and Vibration 175.4 (1994): 491-503. Web.

[3] Main, J. A., and N. P. Jones. "Evaluation of Viscous Dampers for Stay-Cable

Vibration Mitigation." Journal of Bridge Engineering 6.6 (2001): 385. Web.

[4] Maurer Sohne. "Tuned Mass and Vicious Dampers, Technocal Information and

Products."

[5] Kareem, Ahsan, and Samuel Kline. "Performance of Multiple Mass Dampers

under Random Loading." Journal of Structural Engineering121.2 (1995): 348.

Web.

[6] Kim, Sun-Joong, Ho-Kyung Kim, Radiance Calmer, Jin Park, Gyu Seon Kim,

and Deok Keun Lee. "Operational Field Monitoring of Interactive Vortex-

induced Vibrations between Two Parallel Cable-stayed Bridges." Journal of

Wind Engineering and Industrial Aerodynamics123 (2013): 143-54. Web.

[7] Calmer, Radiance. "Estimation of Damping Ratio from Operational Monitoring

of Cable-stayed Bridge." (n.d.): n. pag. Web.

[8] Lee, Chien-Liang, Yung-Tsang Chen, Lap-Loi Chung, and Yen-Po Wang.

"Optimal Design Theories and Applications of Tuned Mass

Dampers." Engineering Structures 28.1 (2006): 43-53. Web.

43

[9] Simiu, E and Scalan, R.H. (1996) “Wind effects on structures – 3rd Ed.”, John

Wiley & Sons, Inc., New York.

[11] Strømmen, E.N. (2010) “Theory of bridge aerodynamics”, Springer, Berlin.

[12] Abé, Masato, and Yozo Fujino. "Dynamic Characterization of Multiple Tuned

Mass Dampers and Some Design Formulas." Earthquake Engineering &

Structural Dynamics 23.8 (1994): 813-35.

[13 ]Seo, Ju-Won, Ho-Kyung Kim, Jin Park, Kwon-Taek Kim, and Gi-Nam Kim.

"Interference Effect on Vortex-induced Vibration in a Parallel Twin Cable-

stayed Bridge." Journal of Wind Engineering and Industrial

Aerodynamics 116 (2013): 7-20. Web

[14] Fujino, Yozo, and Yoshitaka Yoshida. "Wind-Induced Vibration and Control of

Trans-Tokyo Bay Crossing Bridge." Journal of Structural Engineering 128.8

(2002): 1012.

[15] P., Den Hartog J. Mechanical Vibrations. New York: McGraw-Hill, 1956.

[16] Hyundai Construction. 제 2진도 대교 주형 연직 와류 진동, 제 진도 대

책 보고서. Rep. 2012.

44

FRENCH EXTENDED ABSTRACT

1- Introduction et contexte

Pour les superstructures à grande longueur ou portée, l’excitation provoquée

par le vent constitue la principale force subie. De ce fait, plusieurs structures

connaissent des dégâts provoqués par les phénomènes de battements, d’assaut de

vent, ou de vibrations induites par vortex. Cette dernière cause fera l’objet de

notre recherche.

En effet, en 2011, le deuxième pont de Jindo a connu un phénomène de vortex

qui a duré plus de deux heures, pour une vitesse du vent aux alentours de 10m/s.

L’accélération du pont a dépassé 1.5m/s2, soit plus de 3 fois la limite de service.

Plusieurs recherches ont été menées afin de déterminer la cause du niveau

élevé de vibration, et suite à des séries de tests à la soufflerie, des résultats obtenus

par Seo et al (2013) montrent que ceci est dû au niveau bas du taux

d’amortissement du pont. Ce dernier a ensuite été calculé par Kim et al.(2013) en

utilisant des contrôles sur le terrain. La valeur obtenue était inférieure à la valeur

requise par le code de construction coréen. Un système de mitigation des vibrations

induites par vortex (VIV) a donc été mis en place par une entreprise de conception,

qui a installé un amortisseur harmonique multiple.

Etant donné que l’amortisseur a été conçu par une tierce partie, il nous a paru

intéressant d’étudier la théorie des amortisseurs multiples et de parvenir à trouver

un modèle qui puisse satisfaire les critères et codes de constructions et d’éviter

45

ainsi le surdimensionnement. En effet, l’amortisseur actuel dispose d’un taux

d’amortissement de 3.97%, alors que la valeur requise pour rester dans la limite de

service est seulement de 0.4%. La méthodologie de conception ‘alternative’ sera

donc basée sur les résultats des tests de soufflerie, des données obtenues sur le

terrain, ainsi que des simulations numériques.

2- Théorie de l’amortisseur multiple

La première partie présente la méthodologie de conception de l’amortisseur.

Afin de comprendre le fonctionnement de l’amortisseur multiple, il est nécessaire

tout d’abord de passer par l’amortisseur simple. La théorie relative à ce dernier est

déjà établie et les paramètres optimaux sont donnés par des formules précises.

Cependant, pour l’amortisseur multiple, il n’existe pas de conception optimale

unique, et c’est pour cela que nous avons choisi des critères de conception et une

méthodologie basée sur la détermination des paramètres étape par étape.

Les paramètres considérés sont le rapport de masse amortisseur/pont, le taux

d’amortissement et la gamme de fréquence de l’amortisseur. Ces critères ont été

choisis sur la base des tests de soufflerie et des contrôles sur le terrain. Le taux

d’amortissement équivalent, le taux de réduction de l’accélération, la course de

l’amortisseur sont les critères de conception principaux.

La conception passe a priori par la détermination du rapport de masse, puis par

le calcul de la gamme de fréquence et du taux d’amortissement du système en fin

de compte. La vérification intervient ensuite à travers la simulation numérique par

l’analyse dynamique de l’accélération du pont et de l’amortisseur.

46

3- Résultats de la conception

La méthodologie de conception est appliquée au deuxième pont de Jindo. Les

résultats obtenus sont ainsi comparés à ceux de l’amortisseur actuel. La conception

préliminaire de l’amortisseur simple donne un rapport de masse de 0.3%, pour une

course conforme à la limitation géométrique du tablier du pont. Ce même rapport

de masse est considéré pour la conception de l’amortisseur multiple. La gamme de

fréquence est calculée avec un coefficient presque deux fois plus petit que l’actuel.

de même, le taux d’amortissement de l’amortisseur a été calculé et les vérifications

de la conformité de la conception ont été effectuées. La course de l’amortisseur se

trouve ainsi réduite, et le taux de réduction de l’accélération conforme à la limite

fixée.

En comparant les deux résultats de conception, notre amortisseur propose un

rapport de masse réduit, ce qui permet de diminuer les coûts de construction et

d’installation. De plus, il ne présente pas un problème de surdimensionnement et

est conforme aux critères fixés.

4- Evaluation de la performance du dispositif actuel

Après avoir mis en place une procédure de conception de l’amortisseur

multiple, il a été juge nécessaire d’avoir un retour d’information sur le dispositif

mis en place. En effet, le deuxième pont Jindo est équipé de plusieurs capteurs sans

fils qui permettent de suivre l’état du pont, de le contrôler, et d’en faciliter la

maintenance.

Le dispositif d’acquisition des données qui nous intéresse se compose de deux

47

accéléromètres pour l’accélération du pont, d’un anémomètre pour la vitesse du

vent, et de 4 accéléromètres pour chaque amortisseur.

Trois points sont vérifiés dans cette partie :

- Le développement de la vibration induite par vortex : avant l’installation

du dispositif d’amortissement, la vibration se propageait et augmentait

jusqu’à une certaine limite. Cependant, après l’installation des

amortisseurs, la vibration est atténuée du fait du mouvement du dispositif.

- L’accélération et la densité spectrale : il est observé que la densité

spectrale du premier mode qui correspond à la VIV est réduite après

l’installation du système, et que l’accélération causée par le vortex est

grandement réduite.

- La phase entre l’amortisseur et le pont : en théorie, cette phase doit être de

90° pour dissiper l’énergie. Deux cas sont illustrés : occurrence ou non de

la VIV. Dans le cas où on a une vitesse faible (Pas de VIV), on remarque

que le système d’amortisseur suit seulement le mouvement du pont et ne

dissipe pas d’énergie, alors que dans le cas opposé, l’amortisseur réagit en

retard de phase de 90° et dissipe l’énergie totale.

5- Conclusions

Au terme de ce travail de recherche, sur la base de l’expérimentation menée et

des tests effectués, nous avons constaté que le dispositif d’amortissement déjà mis

en place est surdimensionné, avec un taux d’amortissement supérieur de dix (10)

fois le taux d’amortissement voulu. La procédure de conception proposée a permis

48

de pallier le problème de surdimensionnement et satisfait les critères ainsi que le

code de conception cible. Il n’est pas nécessaire d’avoir une large gamme de

fréquences puisque la limite de service est satisfaite pour une gamme de fréquences

réduite.

La performance du dispositif actuel est ainsi évaluée et les propriétés générales

de la vibration induite par vortex, ainsi que le comportement des amortisseurs ont

été confirmés par les contrôles sur le terrain.

CHAPTER 1 INTRODUCTION 1.1 RESEARCH CONTEXT AND PLAN 1.2 BACKGROUND RESEARCH

CHAPTER 2 RESEARCH CONTEXT 2.1 STRUCTURAL DESIGN LABORATORY 2.1.1 Research fields 2.1.2 Experimental facility 2.1.3 Laboratory description 2.1.4 Current research issues and motivations

2.2 INVESTIGATED BRIDGE: SECOND JINDO BRIDGE 2.3 CURRENT MTMD DESIGN AND PROBLEM DEFINITION

CHAPTER 3 THEORY OF THE MULTIPLE TUNED MASS DAMPER 3.1 SINGLE TUNED MASS DAMPER 3.1.1 Definition and function 3.1.2 Characteristics: design of the TMD 3.1.3 Theory of the TMD: Derivation

3.2 MULTIPLE TUNED MASS DAMPER 3.2.1 Introduction 3.2.2 Design of MTMD

CHAPTER 4 APPLICATION: DESIGN OF THE MTMD 4.1 DESIGN CRITERIA 4.2 PROCEDURE OVERVIEW 4.3 ALTERNATIVE DESIGN RESULTS 4.3.1 Determination of the mass ratio 4.3.2 Calculating the optimal bandwidth 4.3.3 Deciding on the TMDs damping ratio 4.3.4 Checking the performance of the MTMD 4.3.5 Summary of the design of the MTMD

4.4 THE EFFECT OF THE BANDWIDTH ON MITIGATION 4.5 COMPARISON WITH THE CURRENT MTMD DESIGN

CHAPTER 5 PERFORMANCE EVALUATION OF THE CURRENT MTMD 5.1 GENERAL DEVELOPMENT OF THE VIV 5.2 ACCELERATION AND POWER SPECTRAL DENSITY 5.3 TMD RESPONSE: PHASE LAG BETWEEN THE MTMD AND THE BRIDGE

CHAPTER 6 CONCLUSIONS REFERENCES FRENCH EXTENDED ABSTRACT

11CHAPTER 1 INTRODUCTION 1 1.1 RESEARCH CONTEXT AND PLAN 1 1.2 BACKGROUND RESEARCH 3CHAPTER 2 RESEARCH CONTEXT 5 2.1 STRUCTURAL DESIGN LABORATORY 5 2.1.1 Research fields 5 2.1.2 Experimental facility 6 2.1.3 Laboratory description 7 2.1.4 Current research issues and motivations 7 2.2 INVESTIGATED BRIDGE: SECOND JINDO BRIDGE 8 2.3 CURRENT MTMD DESIGN AND PROBLEM DEFINITION 9CHAPTER 3 THEORY OF THE MULTIPLE TUNED MASS DAMPER 13 3.1 SINGLE TUNED MASS DAMPER 13 3.1.1 Definition and function 13 3.1.2 Characteristics: design of the TMD 13 3.1.3 Theory of the TMD: Derivation 14 3.2 MULTIPLE TUNED MASS DAMPER 16 3.2.1 Introduction 16 3.2.2 Design of MTMD 16CHAPTER 4 APPLICATION: DESIGN OF THE MTMD 21 4.1 DESIGN CRITERIA 21 4.2 PROCEDURE OVERVIEW 22 4.3 ALTERNATIVE DESIGN RESULTS 24 4.3.1 Determination of the mass ratio 24 4.3.2 Calculating the optimal bandwidth 26 4.3.3 Deciding on the TMDs damping ratio 26 4.3.4 Checking the performance of the MTMD 27 4.3.5 Summary of the design of the MTMD 29 4.4 THE EFFECT OF THE BANDWIDTH ON MITIGATION 30 4.5 COMPARISON WITH THE CURRENT MTMD DESIGN 31CHAPTER 5 PERFORMANCE EVALUATION OF THE CURRENT MTMD 33 5.1 GENERAL DEVELOPMENT OF THE VIV 34 5.2 ACCELERATION AND POWER SPECTRAL DENSITY 35 5.3 TMD RESPONSE: PHASE LAG BETWEEN THE MTMD AND THE BRIDGE 37CHAPTER 6 CONCLUSIONS 40REFERENCES 42FRENCH EXTENDED ABSTRACT 44