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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
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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.
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Keywords: Multiple tuned mass damper, Vortex induced vibration, cable-
stayed bridge, field monitoring, mitigation,
Student Number: 2013-23857
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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
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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
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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
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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
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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.
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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
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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
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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.
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- 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
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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
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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.
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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
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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%
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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)
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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)
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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
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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
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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
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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
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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
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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)
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( ) 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
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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
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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
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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
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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.
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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
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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
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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.
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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
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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)
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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%
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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
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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
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