University of Windsor

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Electronic Theses and Dissertations

2011

Experimental Study on Bridge Stay Cable Vibration

Mitigation Using External Viscous Damper
Le Huang
University of Windsor

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Huang, Le, "Experimental Study on Bridge Stay Cable Vibration Mitigation Using External Viscous Damper" (2011). Electronic Theses
and Dissertations. Paper 80.

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 EXPERIMENTAL STUDY ON BRIDGE STAY CABLE
VIBRATION MITIGATION USING EXTERNAL VISCOUS
DAMPER

By

Le Huang

A Thesis
Submitted to the Faculty of Graduate Studies
through the Department of Civil and Environmental Engineering
in Partial Fulfillment of the Requirements
for the Degree of Master of Applied Science at the
University of Windsor

Windsor, Ontario, Canada

2011

© 2011 Le Huang
Experimental Study on Bridge Stay Cable Vibration Mitigation Using External Viscous

Damper

by

Le Huang

APPROVED BY:

______________________________________________
J. Wu (External Reader)
Department of Electrical Engineering

______________________________________________
B. Budkowska (Department Reader)
Department of Civil and Environmental Engineering

______________________________________________
S. Cheng (Advisor)
Department of Civil and Environmental Engineering

______________________________________________
C. Lee (Chair of Defence)
Department of Civil and Environmental Engineering

04 April 2011
 Author's Declaration of Originality

I hereby certify that I am the sole author of this thesis and that no part of this
thesis has been published or submitted for publication.

I certify that, to the best of my knowledge, my thesis does not infringe upon
anyone's copyright nor violate any proprietary rights and that any ideas, techniques,
quotations, or any other material from the work of other people included in my thesis,
published or otherwise, are fully acknowledged in accordance with the standard
referencing practices. Furthermore, to the extent that I have included copyrighted
material that surpasses the bounds of fair dealing within the meaning of the Canada
Copyright Act, I certify that I have obtained a written permission from the copyright
owner(s) to include such material(s) in my thesis and have included copies of such
copyright clearances to my appendix.

I declare that this is a true copy of my thesis, including any final revisions, as
approved by my thesis committee and the Graduate Studies office, and that this thesis has
not been submitted for a higher degree to any other University or Institution.

iii
 ABSTRACT

Selecting an optimum damper size for a specific cable with designed damper

location is very important. An experimental study of cable vibration mitigation by using

external viscous damper was carried out in the present work. A linear viscous oil damper

with six adjustable damper sizes was designed and fabricated. The experimental work

mainly focused on evaluating the equivalent structural damping ratio of a cable-damper

system using forced vibration tests. General damper design curves based on the current

experimental results have been developed and compared with those in the existing

literature. In addition, the damper stiffness effect, which is ignored in the past work, has

been extensively studied. Results show that the impact of damper stiffness on its

efficiency is highly dependent on the damper location. An approximately linear

relationship is found to exist between the damper stiffness and the damping ratio. As

stiffness of the damper increases, the equivalent modal damping ratio of the system will

decrease.

iv
 ACKNOWLEDGEMENTS

The thesis would not have been possible started without the support and guidance of Dr.

Shaohong Cheng who has enriched my fundamental knowledge of structural dynamics. I

would like to especially thank Dr. Cheng for her guidance every step of the way, constant

support, advice and encouragement whenever it was needed. Secondly, I would also like

to thank my thesis proposal committee, Dr. Budkowska, and Dr. Wu for their helpful

comments on my thesis. Finally, I would like to thank laboratory technicians, Mr. Lucian

Pop, Mr. Matt St.Louis, and Mr. Pat Seguin for their help during my experimental

research.

v
 TABLE OF CONTENTS

AUTHOR'S DECLARATION OF ORIGINALITY ........................................................ III

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

ACKNOWLEDGEMENTS ............................................................................................... V

LIST OF FIGURES .......................................................................................................... IX

LIST OF TABLES ......................................................................................................... XIV

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

1.1 Background ............................................................................................................ 1

1.2 Motivation.............................................................................................................. 2

1.3 Objectives .............................................................................................................. 3

CHAPER 2 LITERATURE REVIEW .......................................................................... 4

2.1 Various Types of Cable Vibration Phenomena ..................................................... 4

2.1.1 Rain-wind induced vibration ......................................................................... 4

2.1.2 Vortex-induced vibration ............................................................................... 7

2.1.3 High-speed vortex excitation ......................................................................... 8

2.1.4 Buffeting ........................................................................................................ 9

2.1.5 Wake galloping .............................................................................................. 9

2.1.6 Galloping for dry inclined cable .................................................................. 10

vi
 2.2 Cable Vibration Mitigation Methods ................................................................... 11

2.2.1 Aerodynamic control ................................................................................... 11

2.2.2 Mechanical control ...................................................................................... 13

2.2.3 State-of-the-art on mitigating cable vibration using external damper ......... 17

CHAPTER 3 EXPERIMENTAL DETAILS .............................................................. 30

3.1 Experimental Setup.............................................................................................. 30

3.2 Damper Design and Calibration .......................................................................... 40

3.2.1 Damper Design ............................................................................................ 40

3.2.2 Damper Calibration ..................................................................................... 44

3.3 Free Vibration Tests ............................................................................................ 46

3.3.1Testing procedures ........................................................................................ 47

3.3.2 Theoretical background ............................................................................... 48

3.3.3 Potential problem found in the free vibration tests...................................... 50

3.4 Forced Vibration Tests ........................................................................................ 52

3.4.1Testing procedures ........................................................................................ 53

3.4.2 Theoretical background ............................................................................... 54

3.4.3 Advantage of forced vibration tests ............................................................. 55

CHAPTER 4 RESULTS AND DISCUSSIONS ........................................................ 57

4.1 Pre-processing of Experimental Data .................................................................. 57

4.2 Experimental Results ........................................................................................... 61

vii
 4.2.1 Damping ratio of a damped cable ................................................................ 61

4.3 Comparison with Other Studies ........................................................................... 68

4.3.1 Universal damper design curve (Pacheco et al, 1993)................................. 68

4.3.2 Damper design curves (Cheng et al, 2010).................................................. 71

4.4 Effect of Damper Stiffness .................................................................................. 77

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS ............................... 95

5.1 Conclusions for Current Study ............................................................................ 95

5.2 Recommendations for Future Work .................................................................... 96

REFERENCE.................................................................................................................... 98

APPENDIX A DAMPER PROPERTIES .................................................................. 102

APPENDIX B MATLAB CODE FOR DATA PROCESSING ................................ 109

APPENDIX C FILTERED AND CONVERTED DATA .......................................... 111

APPENDIX D DAMPER DESIGN CURVES (CHENG ET AL, 2010) ................... 121

VITA AUCTORIS .......................................................................................................... 124

viii
 LIST OF FIGURES

Figure 1-1 Sutong Bridge (Left) and Stonecutter Bridge (Right).......................................2

Figure 2-1 Variation of the angle of formation of the water rivulet with wind speed θ =

45°(Hikami and Shiraisi, 1988).........................................................................5

Figure 2-2 Three different mechanisms of rain-wind induced excitation proposed by

Verwiebe (1998)..............................................................................................6

Figure 2-3 Qualitative trend of vortex shedding frequency with wind velocity during

lock-in (Simiu, 1986).......................................................................................8

Figure 2-4 Illustration of the axial vortex and Karman vortex interaction in the wake of a

yawed inclined cable (Matsumoto et al., 1995)...............................................9

Figure 2-5 Wind attacking a bluff body (Elsa, 2007)........................................................11

Figure 2-6 Indented surface used on the Tatara Bridge (Sun et al., 2010)........................12

Figure 2-7 Illustration of different types surface protrusions............................................13

Figure 2-8 Cross-ties System in Cable-stayed Bridges......................................................14

Figure 2-9 MR damper installed on cable-stayed bridges.................................................15

Figure 2-10 A picture of Rubber damper (left) and rubber damper installed on Tatara

bridge(right)...................................................................................................16

Figure 2-11 Friction damper installed on Uddevalla bridge in Sweden............................16

Figure 2-12 Two oil dampers installed on second Nanjing Yangtze Bridge.....................17

Figure 2-13 Dynamic amplification curves (Kovàcs, 1982)..............................................19

Figure 2-14 Universal damper design curve by Pacheco et al.(1993)...............................20

Figure 2-15 Variation of the maximum damping ratio with respect to the bending

stiffness parameter (Tabatabai and Mehrabi, 2000)......................................22

ix
Figure 2-16 Schematic illustration of kinetic energy time history of the nth mode of a

damped cable (Cheng et al., 2010).................................................................23

Figure 2-17 Damping ratio of the first mode in experimental model (Sulekh and Pacheco,

1990)..............................................................................................................25

Figure 2-18 Experimental test setups by Xu et al. (1999).................................................26

Figure 2-19 Responses amplitude with and without damper (Xu et al., 1999).................27

Figure 2-20 Taut Cable with Spring-mass Damper...........................................................28

Figure 3-1 Experimental setup...........................................................................................31

Figure 3-2 Iron mass..........................................................................................................32

Figure 3-3(a) Universal Flat Load Cell Top View.............................................................33

Figure 3-3(b) Universal Flat Load Cell Front View..........................................................33

Figure 3-4 Hydraulic hand pump.......................................................................................34

Figure 3-5 Accelerometer placed in the middle of the cable.............................................35

Figure 3-6 Back View of Shaker installed on cable..........................................................36

Figure 3-7 HP 15MHz function/Arbitrary Waveform Generator......................................37

Figure 3-8 AstroDAQ Xe Data Acquisition System..........................................................38

Figure 3-9 Realtime mode before testing...........................................................................39

Figure 3-10 Capturing data in Realtime mode...................................................................39

Figure 3-11 Damper design sketch....................................................................................40

Figure 3-12 Oil damper designed in the current study......................................................41

Figure 3-13 Four Solid Aluminum Plates..........................................................................42

Figure 3-14 Three Springs from Mcmaster.com................................................................44

Figure 3-15 Damper Calibration Setup..............................................................................46

x
Figure 3-16 Free Vibration Test Setup..............................................................................47

Figure 3-17 Mass Block with wire tied on it.....................................................................48

Figure 3-18 Sample of 1st modal displacement decay curve.............................................50

Figure 3-19 free vibration tests to determine friction existed between the plastic stick and

the lid.............................................................................................................52

Figure 3-20 Forced Vibration Test Setup..........................................................................53

Figure 3-21 half power method illustration on frequency-response curve........................56

Figure 4-1 First modal displacement time-history of B_3200_20500_S05_D06_A50_7.40

Hz.....................................................................................................................62

Figure 4-2 Frequency-response curve for case B_3200_20500_S05_D06_A50..............64

Figure 4-3 Equivalent first modal damping ratio with damper installed at 4%L..............66

Figure 4-4 Equivalent first modal damping ratio with damper installed at 6%L..............67

Figure 4-5 Equivalent first modal damping ratio with damper installed at 10%L............68

Figure 4-6 Universal damper design curve proposed by Pacheco et al. (1993)................69

Figure 4-7 Comparison between the current results and the universal damper design curve

proposed by Pacheco et al. (1993).................................................................70

Figure 4-8 Layout of a cable-damper system studied by Cheng et al (2010)....................71

Figure 4-9 Schematic illustration of kinetic energy time history of the nth mode of a

damped cable (Cheng et al., 2010).................................................................72

Figure 4-10 Comparison between the current study and Cheng et al’s design curve at

4%L................................................................................................................74

Figure 4-11 Comparison between the current study and Cheng et al’s design curve at

6%L................................................................................................................74

xi
Figure 4-12 Comparison between the current study and Cheng et al’s design curve at

10%L..............................................................................................................75

Figure 4-13 Impact of damper stiffness on equivalent 1st modal cable damping ratio

(damper location 4%L)..................................................................................80

Figure 4-14 Impact of damper stiffness on equivalent 1st modal cable damping ratio

(damper location 6%L)..................................................................................80

Figure 4-15 Impact of damper stiffness on equivalent 1st modal cable damping ratio

(damper location 10%L)................................................................................81

Figure 4-16 Damping ratio vs Damper stiffness (C = 18.4 N·s/m)...................................82

Figure 4-17 Damping ratio vs Damper stiffness (C = 46.7 N·s/m)...................................82

Figure 4-18 Damping ratio vs Damper stiffness (C = 70.3 N·s/m)...................................83

Figure 4-19 Damping ratio vs Damper stiffness (C = 164.8 N·s/m).................................83

Figure 4-20 Damping ratio vs Damper stiffness (C = 275.5 N·s/m).................................84

Figure 4-21 Damping ratio vs Damper stiffness (C = 1463.8 N·s/m)...............................84

Figure 4-22 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 18.4 N·s/m.................................................................................92

Figure 4-23 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 46.7 N·s/m.................................................................................92

Figure 4-24 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 70.3 N·s/m.................................................................................93

Figure 4-25 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 164.8 N·s/m...............................................................................93

xii
Figure 4-26 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 164.8 N·s/m...............................................................................94

Figure 4-27 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 1463.8 N·s/m.............................................................................94

xiii
 LIST OF TABLES

Table 2-1 Summary of emprical damper design formulae in the existing literature:

Optimal damper coefficient and maximum damping ratio for a viscous

damper.............................................................................................................24

Table 3-1 Properties of Aluminum Plates..........................................................................42

Table 3-2 Oil Properties.....................................................................................................43

Table 3-3 Spring Properties...............................................................................................44

Table 3-4 Damper properties..............................................................................................45

Table 4-1 Summary of maximum displacements in case B_3200_20500_S05_D06

_A50 ...............................................................................................................63

Table 4-2 Cable damping ratio for different damper sizes and locations..........................65

Table 4-3 Different damper size and its corresponding nondimentional damping

parameter ψ.....................................................................................................73

Table 4-4 Damper stiffness effect on cable damping ratio (%).........................................74

Table 4-5 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 18.4 N·s/m...................................................................................86

Table 4-6 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 46.7 N·s/m...................................................................................87

Table 4-7 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 70.3 N·s/m...................................................................................88

Table 4-8 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 164.8 N·s/m.................................................................................89

xiv
Table 4-9 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 275.5 N·s/m.................................................................................90

Table 4-10 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 1463.8 N·s/m...............................................................................91

xv
 CHAPTER 1 INTRODUCTION

1.1 Background

With the development of modern structural materials and the advancement of

construction technologies, it is possible to build structures which have longer span and

taller height. This type of structures is usually flexible and thus very sensitive to

environmental dynamic effects such as wind and earthquake. Cable is a common type of

structural component. It has been widely used in various types of civil structures, such as

cable-stayed bridge, suspension bridge, mesh antenna etc. Cable-stayed bridge has

become more and more popular since 1950’s because of its ease of construction,

aesthetics, technology maturity and economy. The current longest cable-stayed bridge in

the world is the Sutong Bridge in China, which was completed in 2008. It has a main

span of 1088 m. The second longest span is the Stonecutter Bridge which is also in

China and has a main span of 1018 m. The third longest is the Tatara Bridge in Japan. Its

main span length is 890 m. Bridge stay cable is flexible and low in inherent damping. It

is prone to dynamic excitation and exhibit large amplitude of vibrations. This is more

pronounced for longer cables. The rapidly increased bridge span length in recent years

requires longer length cable to support the superstructure. This causes more frequently

observed cable vibration problem on bridge site! Cables can be excited by various

mechanism such as rain-wind induced vibration, vortex-excited vibration, high-speed

vortex excitation, galloping, buffering etc (Yamaguchi and Fujino, 1998; Jiang, 2006;

Zhou, 2005; Sun et al., 2010). Frequent and excessive cable vibrations will cause cable

fatigue failure at its anchorage. There are lots of cable fault in the past. For example, it

1
was found on Maracaibo Bridge in Venezuela in 1978 that more than 500 steel wires

were damaged. A year later, another 3 cables were totally failed (Zhou, 2005). In the case

of Jinan Yellow River Bridge in China, all the cables were replaced after 13 years in

service because of fatigue failure (Zhou, 2005). Stay cables account at least 15% of the

total cost of a cable-stayed bridge project. Therefore, reducing the probability of stay

cable failure caused by vibration is very important not only in terms of structural safety,

but for economy as well.

Figure 1-1 Sutong Bridge (Left) and Stonecutter Bridge (Right)

1.2 Motivation

External damper is the most commonly used device for controlling bridge stay

cable vibrations in field. Developments of empirical formulae for damper design were

mostly conducted using numerical simulation or analytical method. Only limited numbers

of experimental studies were performed. Besides studies on how damper stiffness could

affect the damper efficiency are scarce in existing literatures.

These facts initiated the motivations to conduct an experimental study on the

behavior of a cable-damper system, and utilized such a system to verify results reported

2
in the existing literatures. In addition, efforts will also be dedicated to investigating the

effect of damper stiffness on the efficiency of its performance.

1.3 Objectives

The objectives of the current study are proposed as follows:

1) Design and develop an experimental setup to study the behavior of a cable-

damper system. Based on the conditions of the current structures lab located in

Essex Hall B-19, design and develop the arrangements to satisfy the requirements

of the proposed experimental study.

2) Design and build a linear viscous damper which will allow having adjustable

damper capacity.

3) Develop proper methodologies and procedures to process the experimental data.

Data processing and analysis will be mainly conducted in the Matlab and

Microsoft Excel environment.

4) Conduct cable free vibration tests to obtain modal properties of the studied cable-

damper system.

5) Conduct forced vibration tests to obtain modal cable damping ratio and compare it

with existing literature.

6) Study damper stiffness effect by adding springs to the damper in the tests and

compare variation of modal cable damping ratio with and without springs.

3
 CHAPER 2 LITERATURE REVIEW

2.1 Various Types of Cable Vibration Phenomena

Cable vibrations can be classified into two main categories: wind-induced

vibrations and non-wind-induced vibrations. Wind-induced vibrations include rain-wind

induced vibration, vortex-induced vibration, high-speed vortex-excited vibration,

buffeting, galloping etc., whereas non-wind-induced vibrations include parametric

excitation and external excitation. Periodic bridge deck motion and traffic loads could

induce vibration when the motion frequency approaches to natural frequency of stay

cable. A brief review of various types of cable vibration phenomena is presented in the

following subsection.

2.1.1 Rain-wind induced vibration

Although there are many reasons to excite vibrations of stay cables, it is believed

about 95% of the reported vibration problems on cable-stayed bridges belong to rain-

wind induced oscillations (Wagner and Fuzier, 2003). In the past 30 years, numerous

cable-stayed bridges were observed exhibiting large amplitude of stay cable oscillations

under certain environmental conditions. It became evident that these vibrations occurred

when moderate rain was combined with moderate wind conditions (HNTB Corporation,

2007). The main feature of this complex phenomenon was concluded by Tanaka (2003).

He indicated that under the combined action of rain and wind, at specific angles of attack

and intensity of rainfall, water rivulets would form at the upper and lower surfaces of the

4
cable (Figure 2-1). The formation of these rivulets would cause the change of the balance

of gravitational, aerodynamic and surface capillary forces, which eventually led to

unsymmetrical cable cross-section. Therefore, under certain physical conditions, non-

zero lift force would occur to excite the cable.

Hikami and Shiraish (1988) performed a wind tunnel test to simulate the

generating mechanism of rain-wind induced vibration. In particular, the formation and

role of the water rivulet along the cable in excitation mechanism was investigated.

Results showed that a second rivulet was formed along the upper windward surface of the

cable at an angle θ equal to 45°. The formation of this upper rivulet was associated with

wind speed. As the wind speed increased to a certain value, the raindrops would

overcome the gravity and friction forces to form the upper rivulet. The relationship

between the angle of formation and wind speed is shown in Figure 2-1.

Figure 2-1 Variation of the angle of formation of the water rivulet with wind speed θ = 45°

(Hikami and Shiraisi, 1988)

5
The lower rivulet was found to have a stabilizing effect which would produce an

aerodynamic damping force to suppress cable motion; whereas the upper rivulet would

generate aerodynamic force to excite the cable motion.

Verwiebe (1998) further identified three different mechanisms of rain-wind

induced excitation, the sketch of which are illustrated in Figure 2-2. The first mechanism

occurred when wind was in the direction of cable motion. The oscillation of water rivulet

intensified the cable vibration, and it reached the maximum vibration amplitude when

wind speed is 25m/s.

(a) Wind Tunnel Setup (b) Sketch of 1st mechanism

(c) Sketch of 2nd mechanism (d) Sketch of 3rd mechanism

Figure 2-2 Three different mechanisms of rain-wind induced excitation proposed by

Verwiebe (1998)

6
The second mechanism occurred when wind velocity exceeded 18m/s and the cable

vibrated in the cross-wind direction. The vibration stimulated the anti-symmetrical

oscillation of the upper and lower rivulets. The third mechanism occurred when cable

was inclined orthogonally to the wind direction. When speed of wind was smaller than

19m/s, only the underside rivulet oscillated. However, when wind speed was above

19m/s, oscillation of both upper and downside rivulet occurred.

2.1.2 Vortex-induced vibration

Vortex shedding (Matsumoto et al., 1995) is a common excitation phenomenon of

stay cables. When flow passes cable, it will separate and form shedding vortices in the

wake region. These vortices are well known as Kárman Vortices. When alternating

shedding frequency of the vortices approaches to the natural frequency of the cable,

resonance will occur and the cable will be excited. The vortex resonance drives cable to

interact with surrounding flow even when the critical wind speed was exceeded by

certain range. This is known as lock-in phenomenon as depicted in Figure 2-3. Although

vortex-induced cable vibration is commonly observed on bridges, it was found that the

amplitude of vibration was generally very small. Davenport (1994) stated that the

amplitude of vortex-induced vibration rarely reached the size of the cable radius.

7
 Figure 2-3 Qualitative trend of vortex shedding frequency with wind velocity during

lock-in (Simiu, 1986)

2.1.3 High-speed vortex excitation

High-speed vortex excitation is a newly proposed mechanism of wind-induced

cable vibrations in recent years. It was observed on inclined cables in several full-scale

measurements in wind tunnel tests (Matsumoto, 1995), (Cheng et al., 2003), (Cheng et al.

2008). According to Matsumoto's research, it was found to be directly associated with the

axial vortex shedding and its interaction with conventional Kárman Vortices shedding in

the wake of the cable. Testing results showed a Kárman Vortex interacting with the axial

flow as the illustration in Figure 2-4. However, the characteristics of this axial vortex

have not been fully identified yet. Although this type of instability is also associated with

vortex-shedding in the wake, it occurs at a wind speed much higher than the conventional

Kárman Vortex-excited vibration discussed in section 2.1.2, it is therefore called high-

8
speed vortex excitation. This phenomenon is greatly affected by the intensity of

turbulence in the upcoming wind.

Figure 2-4 Illustration of the axial vortex and Karman vortex interaction in the wake of a

yawed inclined cable (Matsumoto, 1995)

2.1.4 Buffeting

Buffeting is a type of random forced vibration induced by the turbulent

component in the wind. It differs from other cable vibration mechanisms in that it does

not deal with an aerodynamic or resonant phenomenon. The response characteristics are

highly dependent on the turbulent nature of the wind. The vibration amplitude is

relatively small. Though no violent cable motion would occur, the frequent low-

amplitude of cable motion has the potential of inducing fatigue failure at cable anchorage.

2.1.5 Wake galloping

Wake galloping occurs when a cable locates in the wake of another cable or other

9
structural members. Aerodynamic forces acting on the cable change while wind angle of

attack changes (HNTB, 2007). The main difference between wake galloping and vortex-

induced vibration is that the former is a high amplitude vibration which occurs only at

high wind speed. It will cause fatigue failure at cable anchorage. It was found that wake

galloping could also happen and the hangers of suspension and arch bridges. However,

up to date, fatigue failure at cable anchorage due to wake galloping has never been

reported in literature for cable-stayed bridges.

2.1.6 Galloping for dry inclined cable

The divergent type of motion induced by wind for dry inclined cable was

observed in wind tunnel tests by Saito et al. (1994) and Honda et al. (1995) in the

subcritical Re regime; and Miyata et al. (1994) and Cheng et al. (2003, 2008a, 2008b) in

the transition and critical Reynolds number regime. It was also reported in the field

(Irwin et al., 1999; Virlogeux, 1998). The excitation mechanism was studied by Cheng et

al. (2008b), she proved that mechanism was similar to that for Den Hartog criterion

 dC 
galloping. Glauert-Den Hartog criterion [Simiu, 1996]  L + C D  < 0 clearly express
 dβ  β =0

galloping occurrence cases (Den Hartog, 1956). Where α is the attacking angel, C L is Lift

coefficient and C d Drag coefficient. From the criterion, we known circular cross-section

will never experience galloping if wind attacking direction is perpendicular to stay cable

since α will be zero because of symmetry of circle. Figure 2-5 shows lift and drag forces

when wind passing a bluff body with attacking angle α. Cheng et al. (2008b) also stated

that the surrounding flow field of dry inclined cable against wind has strong 3D

10
characteristics. Therefore, driving mechanism of divergent motion was complicated; the

Den Hartog criterion might only be part of the reason, and further investigation was

recommended.

Figure 2-5 Wind attacking a bluff body (Elsa, 2007)

2.2 Cable Vibration Mitigation Methods

To suppress cable vibration on cable-stayed bridges induced by various

mechanisms, many mitigation methods have been developed and applied in field with

different levels of success. These methods can generally be classified as aerodynamic

control method and mechanical control method.

2.2.1 Aerodynamic control

Aerodynamic control is a passive control method which modifies aerodynamic

feature of the cable cross-section. It is most effective in controlling rain-wind-induced

cable vibration by preventing accumulation of rain to form water stream on the surface of

11
stay cable. The surface modification should ensure no other aerodynamic instability

would be excited. This method should be carefully considered not to increase the drag

coefficient of cable cross-section. Many researchers conducted wind tunnel tests or

numerical simulation to justify the effectiveness of aerodynamic control method. There

are mainly two commonly used surface modification techniques for stay cables (Sun et al.,

2010; Saito et al., 1994; Kleissl, 2009; Flamand, 1994; Bosdogianni, 1996; Virlogeux,

1998): Cable surface indentations (Figure 2-6) and cable surface protrusions (Figure 2-7).

Aerodynamic control has the advantages that the repair and maintenance is easy and

simple and the cost is low. However, the development and construction details of

different cases need to be experimentally validated and developed. For example, the total

damping ratio of the cable before and after applying surface modification needs to be

tested based on site service conditions.

Figure 2-6 Indented surface used on the Tatara Bridge (Sun et al., 2010)

12
 Figure 2-7 Illustration of different types surface protrusions

2.2.2 Mechanical control

Mechanical control is the most popularly used mitigation method on bridge site to

control cable vibration. Cross-ties and external dampers are commonly used for this

purpose.

Cross-ties have been installed on many cable-stayed. For example, it is applied on

the Normandy Bridge in France, the Second Severn Bridge in the United Kingdom, the

Helgeland Bridge in Norway, and the Meiko Nishi Bridge in Japan (Kumarasena, 2007).

Stay cable normally has low natural frequencies and inherent damping. So it is easily

excited by dynamic loads. By the addition of cross-tie system, two or more stay cables

are interconnected. Therefore, the in-plane stiffness (natural frequencies) can be

increased. This could effectively suppress stay cables vibration in low frequencies.

13
Besides, the internal damping of the cable can also be increased if cross-tie is used.

Yamaguchi (1995) proposed in his research that if softer cross-ties were used, the cable

damping could be increased more. Figure 2-8 shows two different cross-ties system in

cable-stayed bridges.

Figure 2-8 Cross-ties System in Cable-stayed Bridges

Based on the functioning characteristics external dampers can be active damper,

semi-active damper and passive damper. Active damper control requires the input of

external power. It is not commonly used in stay cable vibration mitigation.

Semi-active dampers are becoming more popular in recent years. MR damper is

the most common type of semi-active control device in the field. MR damper is short

for magnetorheological damper, it filled with magnetorheological fluid which could

change its damper size by varying magnetic field (Zhou, 2005). Because of their

mechanical simplicity, high dynamic range, low power requirements, large force capacity,

and robustness, MR damper is considered as one of the most promising devices for

mechnical vibration control (Jung, 2005). The only problem for MR damper is that it

always requires an external power source. Figure 2-9 shows an MR damper installed on

a cable-stayed bridge.
14
 Figure 2-9 MR damper installed on cable-stayed bridges

At present, majority of the external dampers used on bridge site belong to

passive damper. It has variety of types such as rubber damper, friction damper, and oil

damper.

1. Rubber damper

Rubber damper consists of several high-damping rubber pads. The cable vibration

energy is absorbed by shear deformation of the rubber pads (VSL, 2008). Damper is

normally installed inside the cable casing between the cable and the steel tube near the

bridge deck anchorage. It retains aesthetic appearance of the bridge since the size this

rubber damper is small. Figure 2-10 shows a rubber damper installed on the Tatara bridge

in Japan (Zhou, 2005).

2. Friction damper

Frictional forces arising from the relative motion of two contacting surfaces are a

well-known source of energy dissipation. Friction damper was invented to increase the

damping of cable-damper system in a simple and cost-effective way (Lopez, 2004).

15
Friction damper is used on Uddevalla bridge in Sweden, as shown in Figure 2-11.

Figure 2-10 A picture of Rubber damper (left) and rubber damper installed on Tatara

bridge(right)

Figure 2-11 Friction damper installed on Uddevalla bridge in Sweden

3. Oil damper

Mechanism of oil damper is very simple. Oil damper consists of a cylindrical

container filled with viscous oil, and a piston which can move through the oil. oil

provides viscous resistance while piston moves in the damper cylinder. The resistance

force is the damping force. Oil dampers have many advantages such as wide range of

16
damping capacity, technology maturity, low cost etc. Its application on cable-stayed

bridges is common. Since oil damper can only provide axial damping force along its

cylindrical direction, in order to control both in-plane and out-of-plane cable vibration,

two oil dampers are required to be installed on one cable at certain inclination angle.

Figure 2-12 shows such an oil damper installation scheme on the second Nanjing

Yangtze Bridge.

Figure 2-12 Two oil dampers installed on second Nanjing Yangtze Bridge

2.2.3 State-of-the-art on mitigating cable vibration using external damper

External damper is the most widely used external device to mitigate cable

vibration because of aesthetic and practical reasons. As discussed earlier, external damper

can be classified as active, semi-active, and passive dampers. Passive dampers

generallyare more cost effective, easier to maintain, and reliable in performance. The

existing theoretical and numerical studies and field experience show that passive dampers

can effectively increase cable modal damping ratio and suppress cable vibration without

the requirement of external power source and complicated equipments. In order to

evaluate efficiency of passive dampers on suppressing stay cable vibration. the attainable

17
maximum modal damping ratio was used as an index. Tabatabai and Mehrabi (2000)

collected information of over one thousand bridge stay cables and concluded that

undamped stay cable generally exhibit damping ratio in the range of 0.05% to 0.5%. This

is not sufficient to avoid cable vibration problems on bridge site.

As mentioned in Section 2.1.1, 95% of stay cable vibrations belong to rain-wind-

induced vibration. PTI Guide Specification (2000) and Federal Highway Administration

(2007) both suggested that to avoid rain-wind-induced cable vibration, the Scruton

number S c0 of the cable should be great than 10. Scruton number S c0 is defined as S c0 =

mξ /ρD2), where m is the mass of cable per unit length (kg/m or lbf/ft), ξ is the damping

as ratio of critical damping, ρ is air density (kg/m3 or lbf/ft3), and D is cable diameter (m

or ft). By inspecting the four variables in the definition of the Scruton number, cable

mass and diameter are determined at the design stage, air density is a constant. Therefore,

to satisfy the condition of S c0 > 10, the damping ratio 𝜉 of the cable should be increased

by installing external damper or other measures.

Achieving maximum modal damping ratio 𝜉 max was studied in recent two
R

decades. A few researchers have found that damper coefficient of external damper can be

optimized to yield a maximum modal damping ratio for an applied cable.

Kovàcs (1982) first introduced dynamic amplification curve and empirically

defined the maximum attainable modal damping for the first mode. He illustrated the

maximum first modal damping ratio when damper location was fixed. It is shown in

Figure 2-13.He concluded that there is an optimal damper size for each cable. Very small

damper size would not take any effect on mitigation of cable vibration. On the other hand,

very large damper size would generate large damping force; therefore the damper would

18
act as a support at the location. He proposed that the maximum first modal damping ratio

𝜉 ≈ 0.5x c /L and the optimal damper size c opt ≈ 1 / (2π ).mLω01 / (nxc / L ) , where n is the
1T

mode number, c opt is the optimal damper size, x c is the damper location, L is the cable

length.

Figure 2-13 Dynamic amplification curves (Kovàcs, 1982)

Yoneda et al. (1989) developed an analytical model of a damped cable as a

complex eigenvalue problem. The optimal damper size and the maximum damping ratio

for different modes were calculated. He proposed that the optimal first modal damper size

could be estimated by c opt = 6.25 / (2π ) (mLω01nxc / L ) /[sin 2 (πnxc / L)] and the maximum

6.25
𝑥 𝑥
2𝜋 𝑐
first modal damper size could be determined from 𝜉 max = (0.45 + 𝐿𝑐). It was
𝐿

found that the optimum damper coefficient decreased with the increase of vibration mode.

Pacheco et al.(1993) proposed a universal curve, as shown in Figure 2-14, in

multi-modes which is very useful for selecting size and installation location of damper

during preliminary design stage. In the analysis, the cable was treated as a taut cable, i.e.

19
its bending stiffness and sag were ignored. Without conducting large amount of

numerical simulations, the universal damper design curve could be conveniently used to

find the proper damper size and location for the required amount of damping ratio of a

particular stay cable or estimate the attainable damping ratio of a particular stay cable for

the given damper size and location.

Figure 2-14 Universal damper design curve by Pacheco et al.(1993)

To verify the universal curve proposed by Pacheco et al.(1993), Krenk (2000)

performed a numerical analysis in terms of complex eigenvalue problem. An asymptotic

solution was obtained for the damping ratio of the lower modes, valid for damper

location close to one support. In his analysis, when the nondimensional damper location

x c /L is larger than 0.05, differences between the proposed asymptotic results and the

universal damper design curve by Pacheco et al. (1993) were found. The results by Krenk

led to a slightly larger optimal external damper size. He proposed that the maximum

modal damping could be estimated by 𝜉 ≈ 0.5x c /L and the attainable optimal damper

20
size by c opt ≈ √𝑇𝑚 /(𝜋n𝑥𝑐 /L), where T is the tension of modal cable, m is mass per units

length of cable, n is the mode number.

Shallow cable was studied by Cremona (1997). He extended the concept of

universal curve to sagging cable based on the formulation by Irvine (1974), of which the

effect of the inclination of the cable chord was ignored. It was found that sagging effect

would lead to reduction of attainable modal damping ratio. He proved that the damping

ratios of anti-symmetric modes of shallow cables were identical to those of taut cables.

On the contrary, the damping ratios of symmetric modes depended on the value of the

Irvine parameter. This reduction mainly affected in the first symmetric mode. As an

example, he further identified that for the longest cable of the Normandy bridge, the

damping ratio reduction was 30% when compared to the taut cable solution. The

reduction of damping ratio for the long cable was found to be significant because of the

sagging effect due to cable self- weight.

Tabatabai and Mehrabi (2000) considered cable bending stiffness and sag

extensibility in an analytical formulation by the finite difference method. The increase in

the cable bending stiffness was found to result in a decrease of the damper efficiency.

Figure 2-15 shows the variation of the maximum damping ratio with respect to the

bending stiffness parameter. The bending stiffness parameter was defined as ζ =

�𝐻𝐿2 /𝐸𝐼 , where E is the modulus of elasticity and I is the bending moment of inertia,

and H is the cable tension. Non-dimensional damper size is defined as η = c/√𝐻𝑚. The

first modal damping ratio of the cable is denoted as ξ 1 . It can be deduced that the

maximum attainable damping ratio is around 8%. This is higher than that obtained from

21
the universal damper design curve. It was also found that the bending stiffness effect was

more relevant for dampers that were located closer to the cable end.

Figure 2-15 Variation of the maximum damping ratio with respect to the bending

stiffness parameter (Tabatabai and Mehrabi, 2000)

Zhou (2005) derived a general equation for the optimal damper size associated

with different modes by using complex modal analysis. The damper stiffness was also

included in the formulation. A non-dimensional damper stiffness γ = (kx c )/H was

introduced, where k is the damper stiffness in “N/m”; x c is the distance between the

damper and the near end of the cable anchorage in “m”; and H is the tension of cable in

“N”. He concluded that the optimal damper size for different modes could be expressed

as c opt,i = 1/(𝜋 2 )mL𝜔1(1+γ)/i/(l 1 /L). This equation suggests that as the non-dimensional

damper stiffness increases, the optimal damper size will increase linearly.

Hoang et al. (2007) proposed the effect of bending stiffness on the modal

properties of a stay cable in an analytical study. An explicit asymptotic formula for the

modal damping of a cable attached to a general type damper was derived. The study

22
focused mostly on the effectiveness of maximum modal damping ratio by considering the

cable bending stiffness effect. Results showed that while the flexure property in the cable

could reduce the maximum attainable modal damping ratio by 20%, it could significantly

increase the optimal damping coefficient of the damper.

Cheng et al. (2010) derived the maximum damping ratio achievable by a cable-

damper system in numerical simulation using an energy-based approach. The cable

sagging effect and cable bending stiffness were considered in the analysis for determining

the optimal damper property and maximum attainable cable modal damping ratio. The

overall increase of the cable damping offered by the external damper was determined by

examining the time history of the kinetic energy of a damped cable. The kinetic energy

decay ratio was used to determine the efficiency of the damper. Figure 2-16 illustrates

schematically kinetic energy decay time-history of the nth mode of a damped cable.

Where, the nth kinetic energy decay ratio dn was defined as follow:

𝑗1 �𝐸𝑘𝑖,𝑛 �𝑚𝑎𝑥 −�𝐸𝑘(𝑖+1),𝑛 �

d n = ∑𝑖=1 𝑚𝑎𝑥
, where �𝐸𝑘𝑖,𝑛 �𝑚𝑎𝑥 and �𝐸𝑘(𝑖+1),𝑛 �𝑚𝑎𝑥 are the
𝑗 �𝐸𝑘𝑖,𝑛 �𝑚𝑎𝑥

maximum nth modal kinetic energy of the cable in the ith and the (i+1)th cycles,

respectively, and j is the number of cycle pairs selected in the calculation. The damping

ratio was derived to be ξ n = -ln(1-d n )/(4π). A set of damping estimation curves were

developed for the practical parameter ranges of bridge stay cables, which directly relate a

damper design with the corresponding attainable damping ratio in the damped cable. By

applying regression analysis to the numerical simulation results, it was proposed that the

maximum attainable damping ratio could be estimated by:

ξ n,max = 48.52( x c /L)1.03[L/(H/EI)1/2]0.033, and the optimal damper size by

c opt,n = 0.261e-0.00061[L/(H/EI)1/2] ( x c /L)-1.192.

23
 There are other researchers who studied the cable vibration problems, such as

Yoneda et al.(1989), Uno et al. (1991), Zhou(2005), Fujino and Hoang(2007) etc.. Table

2-1 summarizes the empirical formulae for estimating optimal damper size and mximum

modal damping ratio suggested in these studies.

Figure 2-16 Schematic illustration of kinetic energy time history of the nth mode

of a damped cable (Cheng et al., 2010)

Table 2-1 Summary of emprical damper design formulae in the existing literature:

Optimal damper coefficient and maximum damping ratio for a viscous damper

Literature copt ,n ξ n ,max

Kovàcs 1 / (2π ).mLω01 / (nxc / L ) 0.5.xc / L
(1982)
6.25 / (2π ).xc / L.
6.25 / (2π ) (mLω01nxc / L ) /[sin (πnxc / L)]
Yoneda et 2
al. (1989) [0.45 + xc / L]

6.25 / (2π ) (mLω01nxc / L ) /[ n sin (πxc / L)] 3.3 / (2π ).xc / L

Uno et 2
al.(1991)
Pacheco et 0.10. mLω01 /(nxc / L) 0.52.xc / L
al.(1993)
Cheng et al.
0.261e-0.00061[L/(H/EI)1/2] ( x c /L)-1.192 48.52( x c /L)1.03[L/(H/EI)1/2]0.033
(2010)

24
 All the literature reviewed above are based on either analytical approach or

numerical simulation. Experimental works are generally expensive and time consuming.

But it can validate the results obtained in the theoretical analysis, or even lead to new

findings. Therefore, a number of researchers conducted experiments to study cable

vibration problems.

Sulekh and Pacheco (1990) developed a small-cable model. A steel-wire was

fitted with improvised viscous damper, and the damping ratio was estimated from the

displacement time-history obtained from free vibration tests using the logarithmic

decrement method. The steel-wire was 2.08m long and the unit mass per length of the

model cable was 0.07 kg/m. The tension was set to 59.4N. In order to provide additional

damping to the cable, an oil viscous damper was built. The piston of the damper was

simulated by a light wooden square plate with 0.02m on sides. The piston was placed in

an open container with silicone oil. By attaching the damper at 6% of cable length from

one end, a relation between the kinematic viscosity of the silicone oil (simulating damper

property) and the obtained damping ratio was derived as shown in Figure 2-17.

Figure 2-17 Damping ratio of the first mode in experimental model

(Sulekh and Pacheco, 1990)

25
It was commented that the calibration of the damper was difficult because of the design.

From the figure, it can be seen that as kinematic viscosity of the silicon oil increased,

damping ratio increased and reached to certain point and then decreased. This verified

qualitatively that there existed an optimal damper size when the damper was installed at a

certain location.

Xu et al.(1999) conducted an experimental study on vibration mitigation of stay

cables in cable-stayed bridges using oil dampers. The experimental setup is shown in

Figure 2-18. The two ends of a steel cable were cast into two steel cylinders. The upper

end of the cable was then pin-connected to a smaller steel anchor that was able to slide

vertically along a column. The lower end of the cable was pin-connceted to a larger steel

anchor via a load cell. A piston and silicone oil damper was made for experimental tests.

The natural frequencies of cable measured by the free vibration tests were found in good

agreement with the theoretical estimation by Irvine (1974). A series of large-amplitude

force-controlled tests and displacement-controlled tests were also carried out. In the

former, the excitation force generated by the shaker was a constant all the time which was

5N, and in the latter, the excitation displacement remained as a constant which is

0.141mm. Non-linear characteristics of cable vibration were observed in the forced

vibration tests. A significant reduction in the cable response after attaching to a damper

confirmed the necessary of installing external damper to suppress stay cable vibration on

site. Figure 2-19 shows response amplitudes with and without damper

26
 Figure 2-18 Experimental test setups by Xu et al. (1999)

Figure 2-19 Responses amplitude with and without damper (Xu et al., 1999)

Tabatabai and Mehrabi (2000) performed an experimental study to justify their

derivation on the effect of cable bending stiffness on its vibration control. The modal

cable consisted of a bundle of seven-wires, 6.4 mm-diameter strands encased in a

27
polyvinyl chloride pipe. The length of the cable was 13.695 m, the quivalent axial rigidity

EA = 49,318 kN (including grout and cover pipe), and equivalent flexural rigidity EI =

2.28 kN m2 (including grout and cover pipe). These resulted in a bending stiffness

parameter ζ =100, and the mass per units length m = 3.6 kg/m. A small damper with two

different damper sizes 1680 N·s/m and 15130 N·s/m were made. The damper was

attached at 6% of the cable length from one end of the cable. Three cases were tested: a

free vibration test without damper, and two free vibration tests with damper. The

experimental results proved that the bending stiffness effect should be considered in the

damper design.

In majority of the existing studies, the stiffness of the external damper is

neglected when evaluating the efficiency of an external damper in controlling cable

vibration. Only very few recent research addressed this issue. Krenk and Hogsberg (2005)

used a system shown in Figure 2-20 to study the damper stiffness effect. As shown in the

figure, the studied taut cable is assumed to be attached to a damper and a spring which

are in parallel. The theoretical analysis results clearly indicated that the damper efficiency

could be increased by introducing negative stiffness component.

Figure 2-20 Taut Cable with Spring-mass Damper (Zhou, 2005)

Results by Zhou (2005) further suggested that damper stiffness and optimal

damper size approximately had a linear relationship, i.e. a larger damper stiffness would

lead to a larger optimal damper size. Zhou also proposed a maximum modal damping

ratio decay rate R max = 1/(1+γ), where γ = (kx c )/H is the non-dimensional damper

28
stiffness, k is the damper stiffness (N/m); x c is the distance between the damper and the

near end of cable anchorage(m); H is the tension of cable (N). It is worth noting that

literature related to damper stiffness and its impact on damper efficiency of controlling

cable vibration is scare. Since this parameter could directly affect the effective

performance of an external damper, a well planned systematic investigation is essential.

In the current study, the impact of damper stiffness on the effectiveness of damper

performance will be studied experimentally.

29
 CHAPTER 3 EXPERIMENTAL DETAILS

In order to experimentally study the effectiveness of an external damper in

suppressing cable vibration and better understand the behavior of a cable damper system,

the design and set up of the current experimental study will be presented in this chapter.

First, all the testing equipments and components of the cable-damper system will be

described. The design and calibration of the damper will be presented next. The

procedures of free vibration and forced vibration tests, including the methodologies to

determine damping ratio of a damped cable based on these tests will be illustrated in the

last part of this chapter.

3.1 Experimental Setup

The experimental study was performed in the Structures Lab at the University of Windsor

(Essex Hall B-19). The equipments required for this experimental study are described as

follows:

1. Model cable

A model cable was designed to simulate the behavior of a real bridge stay

cable in the lab environment. The model cable has a length of 9.33m after

mounting in place. It has 4.65mm in diameter and 0.092kg/m in unit mass. The

moment of inertia of the cable is calculated to be 15.8 mm4. The cable was

mounted horizontally onto two steel columns with one end connected with a load

cell and another end connected with a hydraulic pumper to introduce tension load.

30
 1 𝑇
The fundamental frequency of a simple cable can be estimated based on f = √ ,
2𝐿 𝑚

where L is length of the cable, T is the tension, and m is the mass per units length

of cable. The cable modes which are easily excited by dynamic forces generally

have frequencies lower than 15 Hz. Since the length of the modal cable is

constrained by the column spacing in the lab, cable pretension and unit mass

should be adjusted to render the fundamental frequency of the model cable to be

less than 15 Hz. At early stage of tests, the cable was observed to exhibit elliptical

motion when pretension was set too large. Further, if the pretension is too small,

the cable was slack. After a number of trials, it was found that a preferable

pretension range should be between 2500N to 4000N. In order to achieve the

frequency range of a real cable, supplementary iron mass blocks was installed on

cable to increase the unit mass of the modal cable. After adding the mass blocks,

the fundamental frequency of cable in lab environment can be kept below 8 Hz.

Model cable

Figure 3-1 Experimental setup

31
2. Supplementary iron mass block

The supplementary iron mass block is shown in Figure 3-2. Twenty of 50g

iron mass blocks were mounted onto the model cable with equal spacing to

increase the unit mass of the model cable, so it would yield a fundamental

frequency comparable to a full scale cable. After adding the mass blocks, the unit

mass of the model cable became 0.2 kg/m.

m = (0.092 * 9.33 + 20*50/1000)/9.33 = 0.2 (kg/m)

Iron Mass

Figure 3-2 Iron mass

3. Universal flat load cell

A universal flat load cell (Model FL25U-2SG) was used in the

experimental study to measure the pretension in the model cable. It has a

maximum capacity 25000lbs, and a resolution of 10N. It is attached to one end of

the cable. There is a piece of flat rubber placed between the load cell and the

cramp, which is used for insulation as shown in Figure 3-3.

32
 The load cell was calibrated by the universal tensile tester with a short

sample of the model cable. By monitoring the voltage changes on the data

acquisition system while increased the tension slowly, the linear relationship

between the voltage and the applied tension gave a calibration constant of 5.458

kN/mV.

Rubber

Figure 3-3(a) Universal Flat Load Cell Top View

Rubber

Figure 3-3(b) Universal Flat Load Cell Front View

33
4. Hydraulic hand pump

The hydraulic hand pump OTC 4017 (serial number HP061108) is shown in

Figure 3- 4. It was used to apply tension to the model cable to simulate pretension

in a real bridge stay cable. It has maximum pressure capacity of 10,000 psi.

Figure 3-4 Hydraulic hand pump

5. Accelerometer

An accelerometer from Dalimar Instruments Inc. (Model number 352A24) was

used to capture acceleration of the cable motion. The applicable testing freq.

range is 1 to 8000Hz, and the resolution is 9.95mV/g. It can record acceleration

along a single direction. The accelerometer was glued exactly at the top of the

cable to make sure that acceleration along vertical direction is measured. This is

very important since the accelerometer is very sensitive to motion along any

direction. The cable is push up and down during the tests to see if accelerometer is

moving vertically. In addition, the real time collected data displayed by the

Astrolink Xe controlling software was monitored to ensure the motion to be

34
 symmetric about the centerline. Since the present study is focused on testing the

first modal damping ratio of the cable, the accelerometer was placed in the middle

of the cable, as shown in Figure 3-5.

Accelerometer

Figure 3-5 Accelerometer placed in the middle of the cable

6. Electronic dynamic smart shaker

An electronic dynamic smart shaker (Modal number K2007E01) from Modalshop

Inc. is used in the current experimental study to excite the cable in the forced

vibration tests. It can provide up to 7 pounds (31 N) peak sine force with a 1/2

inch (1.27 cm) stroke and its frequency range is from 1 to 9000Hz. And it has

three vibration levels, 5db, 10db and 20db. The shaker is placed on a tripod and

installed exactly to provide vertical excitation. Improper installation will result in

looping motion, unsymmetrical motion etc. Level was used to make sure the

shaker is installed exactly in the vertical plane of the cable. A load cell is also

attached to the shaker to record loading history by the shaker at the mounting

35
 point. Figure 3-6 shows when the shaker is installed at 5% of cable length away

from one end of cable during the forced vibration test.

Load cell

Start
Button

Figure 3-6 Back View of Shaker installed on cable

7. Signal generator

The HP signal generator (Modal 33120A), as shown in Figure 3-7, was used to

generate signals of excitation force pattern for the shaker. It can simulate a

dynamic force with frequency range of 1Hz to 15MHz. It was connected to the

shaker by a cable to transfer the required signal to it. In the forced vibration test of

this experimental study, the external excitation force was assumed to be a

sinusoidal force with a frequency of 6 to 8 Hz.

36
 Figure 3-7 HP 15MHz function/Arbitrary Waveform Generator

8. Data Acquisition system

The AstroDAQ Xe data acquisition system is shown in Figure 3-8. The

accelerometer and the load cell were connected to it to record experimental data.

The system had 8 channels of inputs and was connected to the computer using a

USB 2.0 port. Channel 8 was used to capture load cell data which gave pretension

in kN (after input 5.458 kN/mV in system). Channel 3 was used for capturing

acceleration data. The accompanied data acquisition controlling software

provided a real time monitoring of the recorded data and could perform

preliminary analysis such as data filtering. The software contains Realtime mode

and Review mode. The Realtime mode provides real-time waveform monitoring

and data capture capabilities. The Review mode is used for reviewing and

analyzing saved data in the data files. When cable is vibrating, blue lines will

37
show in Waveform 3 (corresponding to Channel 3), which is the acceleration data

recorded by the accelerometer (Figure 3-9). To start a test, basic setups including

file name, sampling frequency (ranging from 1 to 200,000 Hz) etc should be set in

the controlling software. Figure 3-9 shows the interface of the software when

collecting the acceleration data.

Figure 3-8 AstroDAQ Xe Data Acquisition System

38
 Figure 3-9 Realtime mode before testing

Figure 3-10 Capturing data in Realtime mode

39
3.2 Damper Design and Calibration

3.2.1 Damper Design

Different kinds of passive dampers are reviewed in section 2.2.2. A passive

damper with adjustable damper size was required in this experimental study. In order to

fulfill the requirement, a passive oil damper with adjustable damper size was designed

and built. The mechanism of the oil damper is that the oil inside container can provide

resistance to the plate merged in the oil while it is moving in it. The sketch of the damper

design is depicted in Figure 3-11. The final finished damper is shown in Figure 3-12.

Figure 3-11 Damper design sketches

40
 Plastic
stick to Springs
connect installed
cable and on cable
plate

Oil #1

(a) Side View of Damper installed on cable

Aluminum
Hook

Pin to seal
container

(b) Top view of Damper installed on cable

Figure 3-12 Oil damper designed in the current study

41
As shown in Figure 3-12, the designed damper consists of six parts:

1. A plastic container which has a lid on top with 4 pins equally spaced to seal the

container. The container has a diameter of 100 mm.

2. An aluminum plate with thickness of 1mm. To simulate different damper size, a

total number of four plates with different diameters were made, as presented in

Table 3-1 and shown in Figure 3-13. There is a small hole on the plate which is

used for damper calibration. Further illustration will be provided in the next

section (3.2.2 Damper Calibration).

Table 3-1 Properties of Aluminum Plates

Plate No. #1 #2 #3 #4

Diameter (mm) 99 95 90 80

Weight (g) 16.63 15.45 14.05 11.15

Figure 3-13 Four Solid Aluminum Plates

42
 3. A plastic stick which is used to connect the model cable and the aluminum plate.

It has a mass of 11g.

4. Oil in the plastic container. Two types of oil of different viscosity and density

were used in the test to simulate different damper resistance. Their physical

properties are listed in Table 3-2.

Table 3-2 Oil Properties

Oil No. #1 #2

Color Yellow White

Density @ 20°C, (kg/l) 0.8434 1.25

Viscosity @ 40 oC (mm2s-1) 44.4 277.2

5. To study the damper stiffness effect, springs were installed in some cases which

connected the cable and the damper container. Springs of three different

stiffnesses were used, as shown in Figure 3-14. Their stiffness are listed in Table

3-3.

Table 3-3 Spring Properties

Spring Serial No. 9654K53 9654K812 9654K115

Stiffness k (N/cm) 0.77 1.47 2.15

Material Steel Steel Zinc Steel

43
 9654K53 9654K812 9654K115

Figure 3-14 Three Springs from Mcmaster.com

6. Two aluminum hooks on top of the lid in order to hook the springs (simulate

damper stiffness). Figure 3-12 shows the two aluminum hooks attached to the lid.

3.2.2 Damper Calibration

An LVDT was used for damper calibration. With four different size aluminum

plates and two different types of oil, it yields a total of eight combinations for the damper

capacity. Therefore, eight different damper sizes can be simulated. The weight of the

LVDT stick is 15.18g. The whole calibration setup is shown in Figure 3-15. Before

calibrating the damper, a linear relationship was found between the AstroDAQ Xe Data

Acquisition system and the LVDT, i.e. 1 volt = 2.5mm. During calibration, a mass block

was put on the top of the damper stick and subsequently released. The displacement of

44
the damper sticker was recorded by the data acquisition system. After the mass block

stopped, the collected displacement data was reviewed and the slope of displacement

time-history curve was computed for velocity. Repeat those steps by applying different

mass blocks from 0g to 400g with 50g or 100g increment case by case. Since passive oil

damper theoretically has a linear relationship between the applied force F and the

velocity v, the damper coefficient c can thus be determined based on c = F/v. Note that,

the applied force should include the weight of the plastic stick of the damper plate, the

LVDT stick, the aluminum plate and the added mass. In Appendix A, all the data

captured in damper calibration will be presented. The damping coefficients of these

dampers are listed in Table 3-4.

Table 3-4 Damper properties

Oil #1+ Oil #1 Oil #2 Oil #1 Oil #2 Oil #2

Combination
Plate #3 +Plate #2 +Plate #3 +Plate #1 +Plate #2 +Plate #1

Damping

Properties 18.4 46.7 70.3 164.8 275.5 1463.8

(N·s/m)

45
 100g mass block
LVDT
cylinder

LVDT
conditioner

LVDT
stick

Figure 3-15 Damper Calibration Setup

3.3 Free Vibration Tests

As introduced in Chapter 2, the equivalent damping ratio of cable-damper system

is a key parameter to describe the damper efficiency. In order to compute the damping

ratio from the experimental raw data, different analysis methods have been used. The

logarithmic decrement method used free vibration response data to determine the

damping ratio of the system, whereas the half-power method used the forced vibration

data instead. The theoretical background and testing procedures of free vibration test will

be presented in this section. The potential problems of using this approach and the

possible impact on the testing results will be discussed.

46
3.3.1Testing procedures

In the preliminary testing stage, efforts were dedicated to use free vibration test

and logarithmic decrement method to measure cable damping property. The test setup is

shown in Figure 3-16.

Figure 3-16 Free Vibration Test Setup

A mass block of 1kg shown in Figure 3-16 is used for exciting the model cable.

1. Set the input file name. In order to archive all the data files from the experimental

work, file name was assigning using the following pattern:

Cable_Tension_number and weight of supplementary mass _Shaker

location_Damper location_Accelerometer location_Shaker excitation Frequency.

For a file named as, B_3200_20050_S05_D06_A50_7Hz, it represents the Cable

B with pretension of 3200N is tested. There are twenty of 50g iron masses added.

For forced vibration tests, the shaker is installed at 5% L from left end of cable

support. The damper is installed at 6% L from right end of cable support. The

accelerometer is placed at the middle-span of the cable. The excitation frequency

of the shaker is 7Hz.

2. Set the sampling frequency and the sampling time in the AstroDAQ Xe

controlling software. The sampling frequency is normally set as 1000Hz, and the

47
 sampling time is set as 30 seconds. The reason to set sampling frequency as

1000Hz is to guarantee each peak of the vibration cycles was captured.

3. Tie a wire on the mass block and then attach it to the cable, Figure 3-17 shows the

mass block used in the free vibration tests.

Figure 3-17 Mass Block with wire tied on it

4. Burn wire to excite the cable and at the mean time start recording data in

AstroDAQ Xe controlling software.

5. Review captured data in AstroDAQ Xe controlling software and save them in the

excel format.

3.3.2 Theoretical background

Logarithmic decrement method is used to compute damping ratio of the damped

cable using the displacement time-history data collected in free vibration tests. The

captured data need to be pre-processed before applying logarithmic decrement method.

48
Pre-processing of experimental data will be described in Chapter 4. After pre-processing

the data, only the 1st modal response is retained in the displacement time-history for

determining 1st modal damping ratio. A sample of pre-processed free vibration

displacement time-history is shown in Figure 3-18. Logarithmic decrement is the natural

logarithmic value of the ratio of two adjacent peak values of displacement time-history in

free decay vibration. The damping ratio is computed by

1 𝑌1
𝛿= 𝑛
𝑙𝑛 �𝑌 � (3.1)
𝑛+1

𝛿
and 𝜉= (3.2)
�4𝜋2 +𝜉 2

where 𝛿 is logarithmic decay; 𝜉is the damping ratio of the modal cable; n is the number

of cycle; Y 1 and Y n+1 are the displacement at the 1st and the (n+1)th peak of the

displacement time-history curve.

49
 4

3 Peak values

1
Displacement(cm)

-1

-2

-3
Sample of 1st modal displacement
decay curve
-4

-5
0 0.5 1 1.5 2 2.5 3
Time(Second)

Figure 3-18 Sample of 1st modal displacement decay curve

3.3.3 Potential problem found in the free vibration tests

Although how to use free vibration test results to determine damping existed in

the structure is mature in theories, after many times of trial tests, the following

disadvantages are found, some of which could significantly affect the results.

1. Duration of cable vibration is very short. Due to the presence of damper, the cable

vibration usually stops within 10 seconds. It is thus difficult to choose the useful part

of data.

50
2. It is hard to control the cable to vibrate in the vertical direction. If the mass block was

not tied to pass the mass center of the cable, tensional motion of the model cable will

be excited.

3. Strange noise from damper was heard during early stages of testing. It was found to

be induced by the friction between the plastic stick and the lid. To verify the

significance of friction in contributing to suppress cable motion, a set of comparison

test was done. The cable responses were measured when the damper was removed

and when the damper was installed without the aluminum plate on the stick.

Theoretically, these two sets of results should be very close. However, a considerable

difference was detected as shown in Figure 3-19, where the grey line represents the

former case and the black line represents the latter one. From the figure, it can be seen

that the fiction between the stick and the lid was acting like a damping force which

helped to control cable vibration. Since the friction gernerated was not stable and

could not be calibrated, in order to make sure the damper only have viscous damping

by the oil, a thinner plastic stick replaced the previous one. Note that, The old plastic

stick has a diameter of 9.5mm, and the new one has a diameter of 6.3mm.

4. Another problem associated with using free vibration tests is that the damping ratio

determined by the logarithmic decrement method is amplitude dependent. For

example, for the displacement time-history shown in Figure 3- 18, if different peaks

are selected for Eq. (3.2), it will yield different damping ratios as:

First and second Peak: 𝜉 = ln(3.926/3.516)/(2*π*1) = 1.76%;

First and fourth Peak: 𝜉 = ln(3.926/2.793)/(2*π*3) = 1.81%;

First and sixth Peak: 𝜉 = ln(3.926/2.082)/(2*π*5) = 2.02%;

51
 First and ninth Peak: 𝜉 = ln(3.926/1.388)/(2*π*8) = 2.07%.

Therefore, it will not be an appropriate index if comparisons with other

studies needs to be made.

4
Black is free vibration with damper(No plate) Red is just free vibration

-1

-2

-3

-4

-5

-6
0 5 10 15 20 25
Time

Figure 3-19 Free vibration responses to determine friction between the plastic stick and

the lid

3.4 Forced Vibration Tests

Based on the above reasons, it was decided to use the forced vibration tests for the

purpose of the current study. With everything remains the same as the free vibration tests,

instead of a lumped mass block, a smart shaker was used to excite the cable continuously.

The testing setup is illustrated in Figure 3-20. To investigate the effect of damper

stiffness on damping ratio, springs were installed on model cable in some cases.

52
 Figure 3-20 Forced Vibration Test Setup

3.4.1Testing procedures

A. Tests not considering damper stiffness (springs are not installed)

1. Set up input file name. (Same as free vibration test).

2. Set sampling frequency and sampling time. (Same as free vibration test).

3. Connect the signal generator to the smart shaker.

4. Start the shaker and set the vibration amplitude to be 20db. Start the signal

generator and adjust the excitation frequency to check the maximum displacement

of the model cable and its corresponding excitation frequency. The maximum

displacement of the modal cable can be observed when the excitation frequency by

the signal generator equals to the natural frequency of the cable, i.e. when

resonance occurs. The excitation frequency is normally around 7Hz based on the

present condition.

5. Start the excitation frequency from (Natural frequency – 0.5 Hz). For example, if

the natural frequency is 7Hz, the starting excitation frequency of the signal

generator will be set at 6.5 Hz.

53
 6. Review the recorded data in AstroDAQ Xe controlling software and save captured

data in the excel format.

7. Increase the excitation frequency by 0.05 Hz, and then repeat steps 6 and 7. A total

number of twenty runs should be done.

B. Tests considering damper stiffness (springs are installed)

1. Attach two springs on the lid of the damper and then mount them on the cable. (As

shown in Figure 3.12)

2. Repeat steps 5 to 7 in part A.

3. Change a new pair of springs, and then repeat steps 1 and 2. Altogether 3 different

types of springs (9654K53, 9654K812, 9654K115)

3.4.2 Theoretical background

Half-power method was used to find damping ratio from forced vibration response data.

It is different from the logarithmic decrement method in that the former is a time domain

analysis, where as this is a frequency domain analysis. The model cable vibration

amplitude corresponding to each external excitation frequency was determined first based

on the recorded displacement time-history at that particular excitation frequency. A

frequency-response curve can be plotted, as shown in Figure 3-21, which describes the

relation between the external excitation frequency and the maximum displacement

response of the cable. The standard procedures of the half-power method are as follows:

1. Identify the peak response of the frequency-response curve. For example, point A

in Figure 3-21, this has a maximum displacement D.

54
 2. Obtain the displacement corresponding to half-power points by D/√2.

3. Draw a horizontal line at D/√2 and intersect with the frequency-response curve at

two points (B and C). These two points are the half-power points in Figure 3-21.

4. Draw two vertical lines from the half-power points. The intersections with the

horizontal axis (R 1 and R 2 in Figure 3-21) are the excitation frequency or

frequency ratio (if the horizontal axis has a dimensionless form by normalizing

with the cable natural frequency) corresponding to the half-power points.

𝑅 −𝑅
5. Compute damping ratio of the cable-damper system using 𝜉= 𝑅2 +𝑅1.
2 1

3.4.3 Advantage of forced vibration tests

1. It is easy to capture model cable response at each excitation frequency.

2. The smart shaker can be installed in a way to guarantee the excitation of in-plane

motion of the model cable.

3. The signal generator can control the vibration frequency. Repeatability of the

testing conditions is good, which allows studying the damper stiffness effect under

the same basic setup.

4. Damping ratio result obtained from the half-power method is less sensitive to the

vibration amplitude.

55
 2
Maximum Displacement for each excitation frequency

1.8

A
D
1.6

1.4

1.2
B C
D/(21/2)
1

0.8

0.94 0.96 R1 0.98 1 1.02 R2 1.04 1.06

R(Excitation frequency/Natural frequency)

Figure 3-21 half power method illustration on frequency-response curve

56
 Chapter 4 Results and Discussions

Based on the comparison between the free vibration and the forced vibration

methods presented in the previous chapter, it was decided to use the latter to measure the

cable damping ratio in the current study. Analysis results in terms of damping ratio of the

damped cable corresponding to different damper sizes and damper locations, impact of

damper stiffness on its suppression effect will be presented in this chapter. In addition,

comparison of these results with the existing literature data will be made.

4.1 Pre-processing of Experimental Data

Data pre-processing consists of two steps: i) data filtering; 2) converting

acceleration time-history to displacement time-history. When the cable was excited to

vibrate, acceleration of cable motion at a specific point was recorded by an accelerometer.

In general, the recorded data (raw data) contains contributions from different cable modes

and noise from various environmental sources. Filter was applied to the raw data to

extract the response of a certain mode of interest. For example, if it is aimed to determine

the 2nd modal damping ratio of the damped cable, then the filter will be designed in a way

which only allows the 2nd modal response (acceleration in this study) to be retained. In

order to determine the damping ratio using either the logarithmic decrement method (free

vibration) or the half-power method (forced vibration), displacement time history of cable

response is required, which can be obtained by double integration or Fourier Transform

of the filtered acceleration time history.

57
Step 1: Data filtering

Since the current study is focused on the 1st modal damping ratio of the damped

cable, the Butterworth Filter is used to filter higher modes and extract the acceleration

response associated with the first mode. The Butterworth filter is a type of signal

processing filter designed to have as flat a frequency response as possible in the pass

band so that it is also termed as a maximally flat magnitude filter

(http://wikipedia.org/wiki/Butterworth_filter). The frequency pass band is set as [1st

modal frequency – 0.6 Hz, 1st modal frequency + 0.6 Hz]. The order of the filter is

normally set as 2.

Step 2: Compute displacement time-history

Two approaches of converting acceleration time-history to displacement time-

history were used. The first one is the Double Integration method, and the second one is

the Fourier Transform method.

The double integration method is based on the mathematical relation among

displacement, velocity and acceleration. The acceleration time-history is integrated with

respect to time to obtain the velocity time-history, and then integrate again to obtain the

displacement time-history. A Matlab code was developed to perform the calculation by

using trapezoidal rule. After a few sample calculations, it was found that there were some

disadvantages of using the double integration method. First of all, although the

trapezoidal rule is a well defined method to conduct integration “graphically”, it is still an

approximation method which contains numerical error. In the dynamic analysis,

especially for the current case, even the 4th digit after the decimal point is still important.

58
Therefore, the accumulated numerical error, which could be considerable, would lead to

incorrect results. Secondly, the computation time is relatively long. One set of

acceleration time-history contains more than 15,000 data, and based on double

integration, it results in billions of operations to process one set of data.

The acceleration time-history obtained from experiment is the representation of

modal cable vibration signal in the time domain, whereas Fourier transform is the

representation of the modal cable vibration signal in the frequency domain. The

definition of Fourier transform is an integral function shown below:

𝑓̂ (𝑓𝑛 ) = ∫ 𝑒 (−2𝜋𝑖𝑓𝑛𝑡) 𝑓(𝑡)𝑑𝑡 (4.1)

where 𝑓(𝑡) represents the response time-history, t represents time (Second) and 𝑓𝑛 is the

natural frequency (Hz).

The inverse of Fourier transform is

𝑓(𝑡) = ∫ 𝑒 2𝜋𝑖𝑓𝑛𝑡 𝑓̂ (𝑓𝑛 )𝑑𝑓𝑛 (4.2)

By applying Fourier transform and its inverse function, the time domain data and the

corresponding frequency domain data could be transferred back and forth. In its

application to the current study, the displacement time-history will be much easier to get

if all the time domain data are transferred to the frequency domain. The detailed

procedures are given as follows:

1. Apply Equations (4.1) and (4.2) to the current study. The circular frequency 𝜔𝑛 =

2𝜋𝑓𝑛 :

�̈ (𝜔 ) = ∫ 𝑒 (−𝑖𝜔𝑛𝑡) 𝐷̈(𝑡)𝑑𝑡
𝐷 (4.3)
𝑛

�̈ (𝜔 )𝑑𝜔
𝐷̈(𝑡) = ∫ 𝑒 𝑖𝜔𝑛𝑡 𝐷 (4.4)
𝑛 𝑛

59
 where D̈ (t) represents the acceleration time-history (cm/s2), t represents the time

(Second), and fn is the natural frequency (Hz).

The same for the velocity time-history and the displacement time-history:

�̇ (ω ) = ∫ e(−iωn t) Ḋ (t)dt
D (4.5)
n

�̇ (ω )dω
Ḋ (t) = ∫ eiωn t D (4.6)
n n

where Ḋ (t) is velocity time-history, and

� (ωn ) = ∫ e(−iωn t) D(t)dt

D (4.7)

� (ωn )dωn
D(t) = ∫ eiωn t D (4.8)

where D(t) is displacement time-history.

d
2. Since D̈ (t) = dt Ḋ (t), by substituting equation (4.6) into it, yields:

d �̇ (ω )dω ]
D̈ (t) = dt [∫ eiωn t D n n

d �̇ (ω )dω
= ∫(dt eiωn t ) D n n

�̇ (ω )dω
= ωn i ∫ eiωn t D (4.9)
n n

Combining Equation (4.4) and (4.9), gives:

�̈ (ω ) =iω D
D �̇
n n (ωn ) (4.10)

From Equation (4.10), it can be seen that in the frequency domain, the acceleration

data could be easily converted to velocity data by dividing ωn i. The same procedure

can be derived for converting velocity data to displacement data in the frequency

domain, i.e.:

�̇ (ω ) =i ω �D(ω )
D (4.11)
n n n

Combining Equations (4. 10) and (4.11), gives:

�̈ (ω )/ ω 2
�D(ωn ) = −D (4.12)
n n

60
 Substitute Equation (4.12) into Equation (4.8), yields

�̈ (ω )dω ]/ (- ω 2)
D(t) = [∫ eωn ti D or D(t) = D̈ (t)/ (- ωn 2) (4.13)
n n n

It can be seen from Equation (4.13) that compared to the double integration method, it is

more convenient to use Fourier transform to convert acceleration time-history to

displacement time-history. In a sample run, by using Fourier transform, the computing

time has been shortened to 10 seconds for processing one set of data. Therefore, it is

decided to use Fourier transform to compute displacement time-history in the remaining

analysis. The Matlab code for data processing is presented in Appendix B.

4.2 Experimental Results

4.2.1 Damping ratio of a damped cable

As illustrated in the previous sections, in order to obtain the first modal damping

ratio of a damped cable, the recorded cable acceleration time-history raw data needs to be

filtered first and then converted to the first modal displacement time-history. Next, the

frequency-response curve is generated, based on which the damping ratio of a cable-

damper system can be determined by applying the half-power method.

A sample analysis of a testing case is presented as follows:

In this testing case, twenty of 50g supplementary mass blocks were attached to the model

cable. The damper was installed at 6% of cable length away from the right support of the

modal cable. The smart shaker was installed at 5% of cable length away from the left

support of the modal cable. The accelerometer was placed in the middle of the cable to

capture acceleration data. As described in Section 3.3.1, based on these testing conditions,

61
the data file was named as B_3200_20050_S05_D06_A50_XXHz, where XX represents

the excitation frequency from the signal generator which was adjustable. The test started

by setting the initial excitation frequency at 7.00 Hz, with the increment of 0.05Hz, and

finished at 7.90Hz. At each excitation frequency, the sampling rate was 1000Hz, and the

sampling time was 15 seconds. The acceleration time-histories corresponding to each

excitation frequencies were saved as excel files. The first modal displacement time-

histories corresponding to each excitation frequencies were computed by applying the

Butterworth filter to extract the response of the first mode and then converted to

displacement time-history using Fourier transform analysis. For example, Figure 4-1

shows the filtered first modal displacement time-history of the studied cable at excitation

frequency of 7.40Hz.
2

X: 9.333
1.5 Y: 1.722

0.5
Displacement(cm)

-0.5

-1

-1.5

-2
0 5 10 15
Time(Second)

Figure 4-1 First modal displacement time-history of case

B_3200_20050_S05_D06_A50_7.40Hz

62
 The complete set of displacement time-histories corresponding to each excitation

frequency in this testing case is included in Appendix C. The maximum cable

displacements at each excitation frequency are summarized in Table 4-1.

Table 4-1 Summary of maximum displacements in case B_3200_20050_S05_D06_A50

Excitation Frequency *Frequency Ratio R Maximum Displacement

(Hz) (cm)
7.00 0.946 0.5871
7.05 0.953 0.6798
7.10 0.959 0.7916
7.15 0.966 0.9228
7.20 0.973 1.0780
7.25 0.980 1.2530
7.30 0.986 1.4440
7.35 0.993 1.6240
7.40 1.000 1.7220
7.45 1.007 1.7110
7.50 1.014 1.5860
7.55 1.020 1.4120
7.60 1.027 1.2350
7.65 1.034 1.0790
7.70 1.041 0.9433
7.75 1.047 0.8322
7.80 1.054 0.7380
7.85 1.061 0.6566
7.90 1.068 0.5890
*R = excitation frequency/natural frequency.

Figure 4-2 shows the frequency response curve plotted based on data in Table 4-1.

The cubic spline function was applied for curve fitting. Following the procedures of the

half-power method illustrated in Section 3.4.2, in Figure 4-2, the peak displacement of

the frequency response curve is 1.722 cm, so the corresponding to half power points is
1
1.722× = 1.218 cm. The frequencies corresponding to the two half power points can
√2

thus be found from the figure, which are R 1 = 0.97875 and R 2 = 1.02770, respectively.

63
 𝑅 −𝑅
Thus the damping ratio of the damped cable can be determined as 𝜉= 𝑅2 +𝑅1 = 0.0244=
2 1

2.44%.

2
D Max. Displacement for each excitation frequency CM

Fit by cubic spline

1.8 Current testing data
1.722

1.6

1.4

1.224
1.2

0.8

0.6

0.4
0.97875 1.02770
0.94 0.96 0.98 1 1.02 1.04 1.06 1.08
Frequency ratio R(Excitation Frequency/Natural Frequency)

Figure 4-2 Frequency-response curve for case B_3200_20050_S05_D06_A50

Following the same procedure, the equivalent damping ratio of a cable attached to

damper of different sizes and installed at different locations are listed in Table 4-2 with

the following testing conditions:

• Cable tension T = 3200 N

• Length of cable L = 9.33 m

• Unit mass of cable = 0.2kg/ m

64
 Table 4-2 Equivalent cable damping ratio for different damper sizes and locations

Oil #1+ Oil #1 Oil #2 Oil #1 Oil #2 Oil #2

Cases
Plate #3 +Plate #2 +Plate #3 +Plate #1 +Plate #2 +Plate #1

Damping

Properties 18.4 46.7 70.3 164.8 275.5 1463.8

(N·s/m)

Damper
1.53 2.49 2.80 2.31 2.08 1.15
location 4% L

Damper
2.38 3.80 4.27 2.44 2.29 1.12
location 6% L

Damper
2.46 5.73 5.40 2.82 2.33 1.24
location 10% L

Damper design curve plotted for different damper locations based on the current testing

results are shown in Figures 4-3 to 4-5. Two facts can be observed from the figures:

1) As the damper moves away from the cable end, the optimum damper size decreases

and the corresponding maximum damping ratio increases. When the damper installed

at 4%L, the optimum damper size is 75 N·s/m and the maximum damping ratio is

2.73%. When the damper is installed at 6%L, the optimum damper size is 66 N·s/m

and the maximum damping ratio is 4.20%. When the damper moves to 10%L, the

optimum damper size is 55 N·s/m and the maximum damping ratio is 5.85%.

65
2) As the damper moves away from the cable end, the damper size has a greater effect

on the equivalent cable damping ratio. For the three studied damper location 4%L,

6%L, and 10%L, if double the optimum damper size, and compare the associated

equivalent 1st modal cable damping ratio with that corresponding to the optimum

damper size, it is found that the equivalent 1st modal cable damping ratio is reduced

by 8.4%, 28.6%, and 34.2%, respectively. For instance, when the damper is installed

at 6%L, the equivalent 1st modal cable damping ratio corresponding to the optimum

damper size of 66 N·s/m is 4.27%, whereas by doubling the damper size to 132

N·s/m, it decreases to 3.00%. The reduction is 28.6%.

3
Spline fitted curve
2.75 Experimental data
Equivalent 1st modal damping ratio ( ξ ) (%)

2.5

2.25

1.75

1.5

1.25

0.75

0.5

0.25

0
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Damper Size C (Ns/m)

Figure 4-3 Equivalent first modal damping ratio with damper installed at 4%L

66
 4.5
Spline fitted curve
4.25
Experimental data
4
Equivalent 1st modal damping ratio ( ξ ) (%)
3.75

3.5

3.25

2.75

2.5

2.25

1.75

1.5

1.25

1
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Damper Size C (Ns/m)

Figure 4-4 Equivalent first modal damping ratio with damper installed at 6%L

6
5.75
5.5
Spline fitted curve
5.25
Experimental data
Equivalent 1st modal damping ratio ξ( ) (%)

5
4.75
4.5
4.25
4
3.75
3.5
3.25
3
2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Damper Size c (Ns/m)

Figure 4-5 Equivalent first modal damping ratio with damper installed at 10%L

67
4.3 Comparison with Other Studies

4.3.1 Universal damper design curve (Pacheco et al, 1993)

As reviewed in Chapter 2, install external damper to suppress stay cable

vibrations is one of the most commonly used methods on bridge site. In designing the

viscous damper, an estimation of the additional damping offered by the damper for the

various natural modes of the cable is usually obtained by eigenvalue analysis. Such an

analysis is usually very time consuming. Pacheco et al. (1993) proposed a universal

damper design curve which is very useful for selecting damper size and installation

location during the preliminary design stage. Without large amount of numerical

simulation effort, the universal damper design curve could be used to find the proper

damper size and location for the required amount of damping ratio of a particular stay

cable or estimate the equivalent damping ratio of a particular stay cable for a given

damper size and location. This universal curve was developed by performing eigenvalue

analysis of a taut cable attached with a linear viscous damper. In the analysis, the cable

sag and bending stiffness were neglected. The cable structural inherent damping was

assumed to be zero. The universal damper design curve proposed by Pacheco et al. (1993)

is shown in Figure 4-6. The plot is produced by using two non-dimensional parameters,

i.e. ξ i /(x c /L) in the vertical axis, and c/(mLω 01 )ix c /L in the horizontal axis, where “m” is

the mass per unit length, “L” is the total length of cable, “ω 01 ” is the first mode angular

natural frequency of cable, “i” is the mode number which is 1 for our case. “x c /L” is the

location damper installed.

68
 Figure 4-6 Universal damper design curve proposed by Pacheco et al. (1993)

Figure 4-7 shows the comparison between the current results and the universal

damper design curve by plotting them in the same figure. It can be seen from the figure

that although the pattern of the current results seem to be lifted and shifted towards left,

the overall pattern of the current set agrees with the universal curve. During the tests, it

was observed from time to time that if the cable vibration amplitude was relatively large,

the contact between the plastic stick of the damper and the lid would occur, particularly

when the excitation frequency of the signal generator approaches to the natural frequency

of the cable. This type of contact would induce friction force, which would directly

contribute to the total damping force offered by the damper. Therefore, the actual

damping provided by the damper is larger than its calibrated value shown in Table 3-4.

Considering this fact, if the damping ratio results are plotted using the “real” damping

value, the present set of results would be shifted toward right, i.e. has a better agreement

with the universal curve. In addition, the derivation of the universal curve is based on an

69
idealized taut cable, i.e. the bending stiffness and sag of the cable are neglected, which

are in presence in the physical tests and would have an impact on the experimental results.

1
Fitted curve from Experiment data
0.9 Universal curve
Experimental results

0.8

0.7

0.6
ξ /(x /L)

0.5
i c

0.4

0.3

0.2

0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(c/mLω 01)(xc/L)i

Figure 4-7 Comparison between the current results and the universal damper design curve

proposed by Pacheco et al. (1993)

It can be seen from the figure 4-7 that test on the present study, the point

corresponding to the optimum damper design has coordinates of (0.05, 0.63). It’s a very

useful reference for damper design. For example, if we have a cable has unit weight “m”

of 98 kg/m and 215 meters in length “L”. The tension “T” applied on cable is 3000kN, so

𝜋 𝑇
that its first modal circular natural frequency is ω 01 = 𝐿 �𝑚 = 2.557 rad/s, if the required

damper should be installed at 5% of cable length towards cable-deck anchorage, then the

required damper size for the 1st cable mode should equal to c = 0.05 mLω 01 /5% = 53876

N·s/m and its corresponding damping ratio is ξ 1 = 0.63*0.05 = 0.0315 = 3.15%. For the

model cable tested in the current study, which has cable tension “T” of 3200N, unit mass

70
“m” of 0.2kg/m, length “L” equal 9.33m and its first modal circular frequency “ω 01 ” is

42.59 rad/s. If the damper installs at 5% of cable length, the optimal damper size “c” is

79.5 N·s/m.

4.3.2 Damper design curves (Cheng et al, 2010)

In an independent study, Cheng et al (2010) developed a group of damper design

curves using energy-based approach. In the derivation, both the bending stiffness and

sagging effect were taken into account. The cable-damper system studied by Cheng et al

(2010) is shown in Figure 4-8.

Figure 4-8 Layout of a cable-damper system studied by Cheng et al (2010)

The cable with length L is laid out in the horizontal direction with a pretension H, a mass

per units span m, finite flexural rigidity EI, and a damper with damping coefficient c

attached transversely a distance L d from one cable end. All those parameters were

combined into non-dimensional forms as:

• Damper location parameter 𝛤 d = Ld /L.

R

• Bending stiffness parameter ζ = L/(H/EI)1/2 = 297

• Nondimensional Damping parameter ψ = (πc)/(mLω 1s ).

where the 1st mode frequency ω 1s = (π/L)(H/m)1/2. In the current study, ω 1s = 42.59 rad/s.

71
The free vibration response of a cable-damper system is obtained from finite element

simulation using ABAQUS, and the kinetic energy decay ratio and the corresponding

structural equivalent damping ratio of the damped cable was determined. The kinetic

energy decay ratio d n was defined as follow (Cheng et al, 2010):

1 𝑗 �𝐸𝑘𝑖,𝑛 �𝑚𝑎𝑥 −�𝐸𝑘(𝑖+1),𝑛 �

d n = 𝑗 ∑𝑖=1 �𝐸𝑘𝑖,𝑛 �𝑚𝑎𝑥
𝑚𝑎𝑥
(4.14)

where �𝐸𝑘𝑖,𝑛 � is the kinetic energy of the cable in the ith cycle of the nth mode, which is

shown in Figure 4-9. The equivalent damping ratio of the cable was derived as ξ n = -ln(1-

d n )/(4π). Design curves were produced by varying the three non-dimensional parameters

in the practical length, tension, bending stiffness of real bridge stay cables. They are

attached in Appendix C.

Figure 4-9 Schematic illustration of kinetic energy time history of the nth mode of

a damped cable (Cheng et al., 2010)

Figures 4-10, 4-11, and 4-12 show the comparison between the current

experimental results and those by Cheng et al (2010). The X-axis represents the

nondimentional damping parameter ψ, and Y-axis represents the equivalent first modal

72
 damping ratio ξ (%). The nondimentional damping parameter ψ corresponding to the six

damper sizes used in the current study are listed in Table 4-3.

Table 4-3 Different damper size and its corresponding nondimentional damping

parameter ψ

Oil #1+ Oil #1 Oil #2 Oil #1 Oil #2 Oil #2

Combination
Plate #3 +Plate #2 +Plate #3 +Plate #1 +Plate #2 +Plate #1

Damping Properties
18.4 46.7 70.3 164.8 275.5 1463.8
(N·s/m)

Nondimentional
0.7 1.9 2.8 6.5 10.9 57.8
damping parameter ψ

73
 3
Spline fitted for current study
2.75 Current experimental data
Cheng et al's Curve
2.5

Equivalent 1st modal damping ratio ξ (%) 2.25

1.75

1.5

1.25

0.75

0.5

0.25

0
0 5 10 15 20 25 30 35 40 45 50 55 60
Nondimensional damping parameter

Figure 4-10 Comparison between the current study and Cheng et al (2010)’s design curve

at 4%L

4.5
4.25 Spline fitted for current study
Current experimental data
4
Cheng et al's curve
Equivalent 1st modal damping ratio ξ (%)

3.75
3.5
3.25
3
2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0 5 10 15 20 25 30 35 40 45 50 55 60
Nondimensional damping parameter

Figure 4-11 Comparison between the current study and Cheng et al (2010)’s design curve

at 6%L
74
 6
5.75 Spline fitted for current study
5.5 Current experimental data
5.25 Cheng et al's curve
Equivalent 1st modal damping ratio ξ (%)
5
4.75
4.5
4.25
4
3.75
3.5
3.25
3
2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0 5 10 15 20 25 30 35 40 45 50 55 60
Nondimensional damping parameter

Figure 4-12 Comparison between the current study and Cheng et al (2010)’s design curve

at 10%L

Figures 4-10 to 4-12 are shown that the current experimental curves are lifted and

shifted to the left from those by Cheng et al (2010). As explained in Section 4.3.1, this

could be resulted from the friction generated between the lid and the damper stick which

contribute additional damping to the damper. It is worth mentioning that when the

damper is installed closer to the cable anchorage, the cable attachment point to the

damper has smaller vibration amplitude. Therefore, the damping force provided by the

damper, which is proportional to the response of the cable at the attachment point, is

lower. Since in the experiment, the actual damping provided by the damper includes that

due to oil viscosity (which we designed for) and that due to friction between the damper

lid and the piston (observed during tests), therefore, the latter has more weight when

75
damper location is closer to the cable end. This could be the reason why in Figure 4-10

(damper location 4%L), the current damper design curve shifts more towards the left of

the one proposed by Cheng et al (2010), and Figure 4-12 (damper location 10%L) shows

better agreement.

In general, the optimal damper sizes from the current experimental results are

smaller than those predicted by Cheng et al (2010). The efficiency of the damper in terms

of damping ratio corresponding to the optimal damper size at 4%, 6% and 10% L are

higher than Cheng et al’s predicted results. There are a number of reasons which could

cause this discrepancy. First of all, it could be resulted from the non-linearity of the

damper property due to additional friction between the lid and the damper stick near

resonance response of the cable. Since the half-power method used in the current study

for determining the damping ratio of the damped cable utilizes the cable response data in

the neighborhood of the resonance point, this friction-related nonlinearity which is not

included in the formulation by Cheng et al (2010), could lead to a lower damping ratio, as

compared to the current set. Secondly, it could be related to the supplementary mass cell

added on the cable model. There are twenty 50g mass cells installed evenly on the cable

to simulate the behavior of a real stay cable. These supplementary mass might not only

increase the mass property of the cable, but also have some contributions to the bending

stiffness of the cable. Previous studies (eg. Cheng et al. 2010, Hoang et al, 2007) showed

that cable bending stiffness is an important parameter in determining equivalent damping

ratio of a damped cable. The current design curve, which is plotted based on the bending

stiffness of the cable itself, could deviate from the actual one which considers the

stiffness contribution from the supplementary mass.

76
 The results from the current experimental tests show reasonable trend of viscous

damper performance at different locations with different sizes. The fact that the optimal

damper size is existed and attainable for the damper design of cable-stayed bridge has

been confirmed.

4.4 Effect of Damper Stiffness

Possible impact of damper stiffness on its effectiveness to control cable vibrations

was noticed by Zhou (2005) and Krenk and Hogsberg (2005). However, no extensive

study on this issue is available in the literature. To understand the effect of damper

stiffness on the performance of a damper, a set of experimental tests have been designed.

Two parallel springs with the same spring constant connecting the damper lid and the

cable are used to simulate the damper stiffness effect (Figure 3-12). Since the two springs

are parallel to each other, the equivalent spring constant k eq = 2k, where k is the stiffness

constant of each spring. As described in Chapter 3, three types of spring with different

spring constants were used in the current tests. For example, if two 9654K53 springs with

spring constant k = 0.77 N/cm were attached to the cable, k eq would be 1.54 N/cm.

Experimental results of the 1st modal damping ratio of the modal cable with dampers

installed at 4%L, 6%L and 10%L with different springs are summarized in Table 4-4.

From Table 4-4, it can be seen that for the same damper installed at the same

location, if the damper stiffness increases, the equivalent cable damping ratio will

decrease. From Figures 4-13 to 4-15, it clearly shows that damper stiffness have greater

effect on the equivalent cable damping ratio when the damper is installed closer to the

cable end. For instance, when the damper is installed at 4%L, the equivalent cable

damping ratio ignoring damper stiffness (k =0) is equal to 2.8%; whereas the equivalent
77
cable damping ratio considering damper stiffness (k = 4.3 N/cm) is equal to 2.17%,

reduced by 22.5%.

It can be seen from Figures 4-13 to 4-15 that the existence of damper stiffness

will have a negative impact on damper efficiency. Compared to the damping force

generated by the damper itself, the elastic force resulted from the damper stiffness has a

much smaller magnitude. Compared to a damper location more towards the cable mid-

point, when the damper is installed closer to the cable end, it is less effective in

controlling cable vibration, i.e. the damping force contributed by the damper is smaller

when the damper location is closer to the cable anchorage. Therefore, the reduction effect

induced by the damper stiffness will decrease when moving the damper away from the

cable end.

78
 Table 4-4 Damper stiffness effect on cable damping ratio (%)

Damper Oil #1 Plate #3 Oil #1 Plate #2 Oil #2 Plate #3 Oil #1 Plate #1 Oil #2 Plate #2 Oil #2 Plate #1
Size (18.4 N·s/m) (46.7 N·s/m) (70.3 (N·s/m) (164.8 (N·s/m) (275.5 N·s/m) (1463.8 N·s/m)

Damper
4 6 10 4 6 10 4 6 10 4 6 10 4 6 10 4 6 10
location (%)

Without
1.53 2.38 2.46 2.49 3.80 5.73 2.80 4.27 5.40 2.31 2.44 2.82 2.08 2.29 2.33 1.15 1.12 1.24
spring

With
9654k53
1.43 2.35 2.40 2.31 3.58 5.56 2.60 4.12 5.37 2.32 2.34 2.80 1.78 2.17 2.24 1.12 1.05 1.24
(k eq =1.54
N/cm)
With
9654k812
1.36 2.30 2.36 2.19 3.42 5.36 2.36 3.99 5.29 2.27 2.29 2.73 1.32 2.06 2.10 1.06 1.02 1.16
(k eq = 2.94
N/cm)
With
9654k115
1.26 2.25 2.29 2.13 3.21 4.88 2.17 3.70 5.17 2.20 2.26 2.70 1.12 2.01 2.03 1.05 0.99 1.15
(k eq = 4.30
N/cm)

79
 3

k =0
k = 1.54 N/cm
k = 1.94 N/cm
k = 4.30 N/cm
2.5
Damping ratioξ (%)

1.5

1
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Damper Size c (Ns/m)

Figure 4-13 Impact of damper stiffness on equivalent 1st modal cable damping ratio

(damper location 4%L)

4.5

k = 0
4 k = 1.54 N/cm
k = 2.94 N/cm
k = 4.30 N/cm
3.5
Damping ratio ξ (%)

2.5

1.5

0.5
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Damper Size (C) Ns/m

Figure 4-14 Impact of damper stiffness on equivalent 1st modal cable damping ratio

(damper location 6%L)

80
 6

5.5 k = 0
k = 1.54 N/cm
k = 2.94 N/cm
5
k = 4.30 N/cm
Damping ratio ξ (%)

4.5

3.5

2.5

1.5

1
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Damper Size (C) Ns/m

Figure 4-15 Impact of damper stiffness on equivalent 1st modal cable damping ratio

(damper location 10%L)

In Figures 4-16 to 4-21, the impact of damper stiffness on the same damper installed at

different locations are shown more clearly. It can be observed that an approximately

linear relationship exists between the damper stiffness and the damping ratio. When the

damper size is smaller than twice of the optimum value. When the damper size is larger

than twice of the optimum one, the stiffness effect becomes unstable. In these cases, since

the damper is very strong, it will actually act more like an additional support at that

location.

81
 4%L

2.6 6%L
10%L

2.4

2.2
Damping ratio ξ

1.8

1.6

1.4

1.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Damper stiffness K (N/cm)

Figure 4-16 Damping ratio vs Damper stiffness (C = 18.4 N·s/m)

6
5.8 4%L
5.6 6%L
5.4 10%L

5.2
5
4.8
4.6
Damping ratio ξ

4.4
4.2
4
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Damping stiffness K (N/cm)

Figure 4-17 Damping ratio vs Damper stiffness (C = 46.7 N·s/m)

82
 5.5
5.4
5.2
5
4%L
4.8
6%L
4.6
10%L
4.4
Damping ratio ξ

4.2
4
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Damping stiffness K (N/cm)

Figure 4-18 Damping ratio vs Damper stiffness (C = 70.3 N·s/m)

4%L
6%L
10%L
2.8
Damping ratio ξ

2.6

2.4

2.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Damping stiffness k (N/cm)

Figure 4-19 Damping ratio vs Damper stiffness (C = 164.8 N·s/m)

83
 2.5
4%L
2.4
6%L
10%L

2.2

2
Damping ratio ξ

1.8

1.6

1.4

1.2

1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Damping stiffness k (N/cm)

Figure 4-20 Damping ratio vs Damper stiffness (C = 275.5 N·s/m)

1.25

4%L
6%L
1.2 10%L

1.15
Damping ratio ξ

1.1

1.05

0.95
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Damping stiffness K (N/cm)

Figure 4-21 Damping ratio vs Damper stiffness (C = 1463.8 N·s/m)

84
 Zhou (2005) studied the damper stiffness effect by proposing a nondimensional

parameter, the maximum modal damping ratio decay rate R max = 1/(1+ 𝛾), where 𝛾 is the

non-dimensional damper stiffness. 𝛾 = kx c /H. k is stiffness of damper (N/m); x c is

damper location from near end of anchor (m); and H is the tension of Cable (N). This

parameter is defined as the difference between the modal damping ratio with and without

considering damper stiffness normalized by the modal damping ratio without considering

damper stiffness. It clearly indicates the impact of damper stiffness on the modal

damping ratio of a damped cable.

Tables 4-5 to 4-10 present a comparison of damping ratio decay rate between the

current set and Zhou’s work. The two sets of results are found to agree well with each

other with error to be less than 5% when the damper size is not closed to the optimum

one. Damper stiffness obviously has a greater effect on the damping ratio when the

damper size is closed to optimum value. Thus maximum modal damping ratio decay rate

R max becomes smaller near that range. This set of comparison is also portrayed in Figures

4-21 to 4-27 for a more convenient comparison. It can be observed from the figures that

in terms of the reduction rate of maximum modal damping in the cable due to the

existence of damper stiffness, the two sets of results agree well. However, the current set

is more sensitive to the damper location. It is worth noting that the maximum modal

damping ratio decay rate R max = 1/(1+ 𝛾) = H/(H+kx c ). Compared to cable pretension H,

the product of damper stiffness k and damper location x c is very small. Thus, either

varying k or x c will slight affect the magnitude of R max . This could be the reason why the

damper stiffness effect derived from Zhou’s formula for different damper locations all

collapse on one curve.

85
 Table 4-5 Damping ratio decay rate of experimental results and Zhou’s formula with damper size 18.4 N·s/m

Damper location
Equivalent
Spring 4% L 6%L 10%L
stiffness k eq
No. R max R max R max
(N/cm) 𝛾 𝛾 𝛾
Current Zhou(2005) Current Zhou(2005) Current Zhou(2005)

9654K53 1.54 0.018 93.5% 98.2% 0.026 98.7% 97.7% 0.046 97.6% 95.6%

9654K812 2.94 0.034 88.9% 96.7% 0.051 96.6% 95.2% 0.086 95.9% 92.1%

9654K115 4.3 0.050 82.4% 95.2% 0.075 94.5% 93.0% 0.126 93.1% 88.8%

86
 Table 4-6 Damping ratio decay rate of experimental results and Zhou’s formula with damper size 46.7 N·s/m

Damper location
Equivalent
Spring 4% L 6%L 10%L
stiffness k eq
No. R max R max R max
(N/cm) 𝛾 𝛾 𝛾
Current Zhou(2005) Current Zhou(2005) Current Zhou(2005)

9654K53 1.54 0.018 92.8% 98.2% 0.026 94.2% 97.7% 0.046 97.0% 95.6%

9654K812 2.94 0.034 88.0% 96.7% 0.051 90.0% 95.2% 0.086 93.5% 92.1%

9654K115 4.3 0.050 85.5% 95.2% 0.075 84.5% 93.0% 0.126 85.2% 88.8%

87
 Table 4-7 Damping ratio decay rate of experimental results and Zhou’s formula with damper size 70.3 N·s/m

Damper location
Equivalent
Spring 4% L 6%L 10%L
stiffness k eq
No. R max R max R max
(N/cm) γ γ γ
Current Zhou(2005) Current Zhou(2005) Current Zhou(2005)

9654K53 1.54 0.018 92.9% 98.2% 0.026 96.5% 97.7% 0.046 99.4% 95.6%

9654K812 2.94 0.034 84.3% 96.7% 0.051 93.4% 95.2% 0.086 98.0% 92.1%

9654K115 4.3 0.050 77.5% 95.2% 0.075 86.7% 93.0% 0.126 95.7% 88.8%

88
 Table 4-8 Damping ratio decay rate of experimental results and Zhou’s formula with damper size 164.8 N·s/m

Damper location
Equivalent
Spring 4% L 6%L 10%L
stiffness k eq
No. R max R max R max
(N/cm) 𝛾 𝛾 𝛾
Current Zhou(2005) Current Zhou(2005) Current Zhou(2005)

9654K53 1.54 0.018 92.8% 98.2% 0.026 94.2% 97.7% 0.046 97.0% 95.6%

9654K812 2.94 0.034 88.0% 96.7% 0.051 90.0% 95.2% 0.086 93.5% 92.1%

9654K115 4.3 0.050 85.5% 95.2% 0.075 84.5% 93.0% 0.126 85.2% 88.8%

89
 Table 4-9 Damping ratio decay rate of experimental results and Zhou’s formula with damper size 275.5 N·s/m

Damper location
Equivalent
Spring 4% L 6%L 10%L
stiffness k eq
No. R max R max R max
(N/cm) 𝛾 𝛾 𝛾
Current Zhou(2005) Current Zhou(2005) Current Zhou(2005)

9654K53 1.54 0.018 85.6% 98.2% 0.026 94.8% 97.7% 0.046 96.1% 95.6%

9654K812 2.94 0.034 63.5% 96.7% 0.051 90.0% 95.2% 0.086 90.1% 92.1%

9654K115 4.3 0.050 53.8% 95.2% 0.075 87.8% 93.0% 0.126 87.1% 88.8%

90
 Table 4-10 Damping ratio decay rate of experimental results and Zhou’s formula with damper size 1463.8 Ns/m

Damper location
Equivalent
Spring 4% L 6%L 10%L
stiffness k eq
No. R max R max R max
(N/cm) 𝜸 𝜸 𝜸
Current Zhou(2005) Current Zhou(2005) Current Zhou(2005)

9654K53 1.54 0.018 97.4% 98.2% 0.026 93.8% 97.7% 0.046 100% 95.6%

9654K812 2.94 0.034 92.2% 96.7% 0.051 91.1% 95.2% 0.086 93.5% 92.1%

9654K115 4.3 0.050 91.3% 95.2% 0.075 88.4% 93.0% 0.126 92.7% 88.8%

91
 100

95

)
max
90
Maximum modal damping ratio decay rate (R

85

80

75

70

65 Current study 4%L

Zhou's 4%L
60 Current study 6%L
Zhou's 6%L
55 Current study 10%L
Zhou's 10%L
50
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Nondimentional damper stiffness

Figure 4-22 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 18.4 N·s/m

100

95
)
max

90
Maximum modal damping ratio dacay rate (R

85

80

75

70

65 Current study 4%L

Zhou's 4%L
60 Current study 6%L
Zhou's 6%L
55 Current study 10%L
Zhou's 10%L
50
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Nondimensional damper stiffness

Figure 4-23 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 46.7 N·s/m

92
 100

95

)
max
90
Maximum modal damping ratio decay rate (R

85

80

75

70

65 Current study 4%L

Zhou's 4%L
60 Current study 6%L
Zhou's 6%L
55 Current study 10%L
Zhou's 10%L
50
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Nondimensional damper stiffness

Figure 4-24 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 70.3 N·s/m

100

95
)
max

90
Maximum modal damping ratio decay rate (R

85

80

75

70

65 Current study 4%L

Zhou's work 4%L
60 Current study 6%L
Zhou's work 6%L
55 Current study 10%L
Zhou's work 10%L
50
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Nondimensional damper stiffness

Figure 4-25 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 164.8 N·s/m

93
 100

95

)
max 90
Maximum modal damping ratio decay rate (R

85

80

75

70

65 Current study 4%L

Zhou's 4%L
60 Current study 6%L
Zhou's 6%L
55 Current study 10%L
Zhou's 10%L
50
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Nondimensional damper stiffness

Figure 4-26 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 275.5 N·s/m

100

95
)
max

90
Maximum modal damping ratio decay rate (R

85

80

75

70

65
Current study 4%L
60 Zhou's 4%L
Current study 6%L
Zhou's 6%L
55
Current study 10%L
Zhou's 10%L
50
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Nondimensional damper stiffness

Figure 4-27 Damping ratio decay rate of experimental results and Zhou’s formula with

damper size 1463.8 N·s/m

94
 Chapter 5 CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions for Current Study

In the current research, an experimental study was performed to investigate the

effectiveness of a linear viscous oil damper in controlling bridge stay cable vibration. A

series of experimental tests have been conducted, which include free vibration tests with

and without damper, forced vibration tests with damper considering or not considering

damper stiffness. The half-power method was utilized for analyzing all forced vibration

tests data. General damper design curves have been proposed based on experimental

results. The damper stiffness effect on efficiency of a damper, which is rarely reported in

the literature, has been extensively examined in the current study. The main conclusions

are summarized as follows:

1. A linear viscous damper which has six adjustable damper sizes have been

designed and built.

2. A procedure to preprocess the experimental raw data has been developed. The

Butterworth filter was used for filtering the experimental raw data and the Fourier

transform theorem was applied to convert acceleration time-history to

displacement time-history.

3. Free vibration tests have been performed to study dynamic properties of the

model cable and justify the reliability of the model system such as natural

frequencies, vibration modes etc.

4. Forced vibration tests have been conducted to produce general damper design

curves. Comparisons with the existing literature show that the general pattern of

95
 the proposed damper design curves agreed well with other studies. The existence

of the optimum damper size has been confirmed.

5. Damper stiffness effect has been extensively studied. An approximately linear

relationship is found to exist between the damper stiffness and the damping ratio.

As the damper stiffness increases, damping ratio will decrease. The damping ratio

decay rate is highly dependent on the damper installation location.

The main contributions of this study are that general damper design curves based on

the experimental study have been developed. To the knowledge of the author, this is the

first time that such curves are developed based on pure experimental study. In addition,

the damper stiffness effect, which is usually ignored in the past studies, has been

investigated in details.

5.2 Recommendations for Future Work

Based on the experience gained from the present study, some recommendations

are made below to be considered for future research on this topic:

1. Refinement of damper design could be done since it will justify the difference

between previous researches and current study. The friction force generated in a

viscous damper will always exist. From the current study, it was observed that

friction could stop the damper at small vibration amplitude. The friction is not only

generated from the contact between the lid and the stick, but also between the oil and

the plate when the plate is intended to start moving from the still position. This non-

linear behavior should be considered for future study.

96
2. Study the effect of damper support stiffness on the damper efficiency. From the

current study, it can be seen that as damper moves away from the cable end, a better

damping ratio will yield. However, on real bridges, the damper could not reach too

far away from the cable end due to size limitation. If it requires installing damper

beyond its conventional location, a support will be needed to connect the damper and

the bridge deck. The stiffness of the support could also have an impact on the damper

efficiency.

97
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13. Elsa, C de Sá. (2007). Structural Engineering Documents - Cable Vibrations in Cable-
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101
 Appendix A Damper Properties

With Oil #1 calibration results are shown below:

Plate #1 first calibration:

Mass velocity
(kg) Force(N) (volts/s) Velocity(m/s)
0 0 0 0
0.04281 0.4199661 0.36282 0.00091
0.04281 0.4199661 0.340306 0.00085
0.09281 0.9104661 1.86334 0.00466
0.09281 0.9104661 1.82184 0.00455
0.14281 1.4009661 3.18011 0.00795
0.14281 1.4009661 3.20155 0.00800
0.19281 1.8914661 4.33894 0.01085
0.19281 1.8914661 4.5418 0.01135
0.24281 2.3819661 5.48554 0.01371
0.24281 2.3819661 5.52327 0.01381
0.34281 3.3629661 7.68069 0.01920
0.34281 3.3629661 7.71736 0.01929

Oil #1 Plate #1 first calibration

C= 163.86 Ns/m
4
3.5 y = 163.86x + 0.1437
3 R² = 0.994
2.5
Force (N)

2 Damper property
1.5
1 Linear (Damper
0.5 property)

0
0 0.005 0.01 0.015 0.02 0.025
Velocity (m/s)

102
 Plate #1 Second Calibration:

Mass (kg) Force(N) velocity (volts/s) Velocity(m/s)

0 0 0 0
0.04281 0.4199661 0.725 0.00181
0.04281 0.4199661 0.72 0.00180
0.04281 0.4199661 0.747 0.00187
0.09281 0.9104661 2.03 0.00508
0.09281 0.9104661 2.06 0.00515
0.09281 0.9104661 2.08 0.00520
0.14281 1.4009661 3.31 0.00828
0.14281 1.4009661 3.33 0.00833
0.14281 1.4009661 3.31 0.00828
0.19281 1.8914661 4.63 0.01158
0.19281 1.8914661 4.59 0.01148
0.19281 1.8914661 4.58 0.01145
0.24281 2.3819661 5.41 0.01353
0.24281 2.3819661 5.65 0.01413
0.24281 2.3819661 5.63 0.01408
0.34281 3.3629661 7.95 0.01988
0.34281 3.3629661 7.91 0.01978

Oil #1 Plate #1 Second calibration

C= 164.78 Ns/m
4
3.5 y = 164.78x + 0.0613
3 R² = 0.9976
2.5 New Damper property
Force

2 2nd calibration
1.5
1 Linear (New Damper
0.5 property 2nd
0 calibration)
0 0.005 0.01 0.015 0.02 0.025
Velocity

Average of two calibration C= 164.3 N·s/m

103
Plate #2 Calibration:

Mass (kg) Force(N) velocity (volts/s) Velocity(m/s)

0 0 0 0
0.04163 0.4083903 3.69 0.00923
0.04163 0.4083903 3.69 0.00923
0.09163 0.8988903 9.35 0.02338
0.09163 0.8988903 9.36 0.02340
0.14163 1.3893903 13.78 0.03445
0.14163 1.3893903 13.45 0.03363
0.19163 1.8798903 17.41 0.04353
0.19163 1.8798903 16.49 0.04123
0.24163 2.3703903 19.42 0.04855
0.24163 2.3703903 20.33 0.05083

Oil #1 Plate #2 calibration

C= 46.722 Ns/m
2.5
y = 46.722x - 0.0852
2 R² = 0.9826
1.5
Damper properites
Force

0.5 Linear (Damper

properites)
0
0 0.02 0.04 0.06
-0.5
Velocity

104
Plate #3 Calibration:

Mass (kg) Force(N) velocity (volts/s) Velocity(m/s)

0 0 0 0
0.04023 0.3946563 10.34 0.02585
0.04023 0.3946563 9.69 0.02423
0.09023 0.8851563 21.09 0.05273
0.09023 0.8851563 21.09 0.05273
0.14023 1.3756563 30.33 0.07583
0.14023 1.3756563 30.03 0.07508
0.19023 1.8661563 41.78 0.10445
0.19023 1.8661563 36.77 0.09193
0.24023 2.3566563 42.72 0.10680

Oil #1 Plate #3 calibration

C= 18.356 Ns/m
2
y = 18.356x - 0.0456
1.5 R² = 0.9968

1 Damper proerties
Force

0.5 Linear (Damper

proerties)
0
0 0.05 0.1 0.15
-0.5
Velocity

105
With Oil #2 calibration results are shown below:

Plate #1 calibration:

Mass (kg) Force(N) velocity (volts/s) Velocity(m/s)

0 0 0 0
0.04281 0.4199661 0.0529171 0.00013
0.04281 0.4199661 0.0536366 0.00013
0.09281 0.9104661 0.1873 0.00047
0.09281 0.9104661 0.181515 0.00045
0.14281 1.4009661 0.341727 0.00085
0.14281 1.4009661 0.338904 0.00085
0.19281 1.8914661 0.51657 0.00129
0.19281 1.8914661 0.504878 0.00126
0.24281 2.3819661 0.59101 0.00148
0.24281 2.3819661 0.589863 0.00147
0.29281 2.8724661 0.726229 0.00182
0.29281 2.8724661 0.741984 0.00185
0.34281 3.3629661 0.873696 0.00218
0.34281 3.3629661 0.874107 0.00219

Oil #2 Plate #1
C= 1463.8 Ns/m
4
3.5 y = 1463.8x + 0.1614
3 R² = 0.9943
2.5
Force

2
1.5
1
0.5
0
0 0.0005 0.001 0.0015 0.002 0.0025
Velocity

106
Plate #2 calibration:

Mass (kg) Force(N) velocity (volts/s) Velocity(m/s)

0 0 0
0.04163 0.4083903 0.447455 0.00112
0.04163 0.4083903 0.422917 0.00106
0.09163 0.8988903 1.22646 0.00307
0.09163 0.8988903 1.22476 0.00306
0.14163 1.3893903 1.95663 0.00489
0.14163 1.3893903 1.99883 0.00500
0.19163 1.8798903 2.76645 0.00692
0.19163 1.8798903 2.759 0.00690
0.24163 2.3703903 3.3376 0.00834
0.24163 2.3703903 3.31824 0.00830
0.29163 2.8608903 4.1325 0.01033
0.29163 2.8608903 4.1925 0.01048
0.34163 3.3513903 4.67651 0.01169
0.34163 3.3513903 4.71936 0.01180

Oil#2 plate#2 275.52 Ns/m

4
3.5 y = 275.52x + 0.0473
3 R² = 0.9976
2.5
Force

2 Damper proerties
1.5
1 Linear (Damper
proerties)
0.5
0
0 0.005 0.01 0.015
Velocity

107
Plate #3 calibration:

velocity
Mass (kg) Force(N) (volts/s) Velocity(m/s)
0 0 0
0.04023 0.3946563 1.29429 0.00324
0.04023 0.3946563 1.28662 0.00322
0.09023 0.8851563 4.28083 0.01070
0.09023 0.8851563 4.2681 0.01067
0.14023 1.3756563 7.14481 0.01786
0.14023 1.3756563 7.1028 0.01776
0.19023 1.8661563 9.67055 0.02418
0.19023 1.8661563 9.85063 0.02463
0.24023 2.3566563 12.6537 0.03163
0.24023 2.3566563 12.522 0.03131
0.29023 2.8471563 14.754 0.03689
0.29023 2.8471563 14.9834 0.03746
0.34023 3.3376563 18.8875 0.04722
0.34023 3.3376563 18.7 0.04675

Oil #2 Plate #3 calibration

C= 70.332 Ns/m
4
3.5 y = 70.332x + 0.1311
3 R² = 0.9962
2.5
Force

2 Damper properites
1.5
1 Linear (Damper
0.5 properites)
0
0 0.01 0.02 0.03 0.04 0.05
Velocity

108
 Appendix B Matlab code for data processing

Filter design

It could be set up in filter design and analysis tool by type in “fdatool” or by follow code

function Hd = filter %FILTER Returns a discrete-time filter object

% M-File generated for filtering instead of filter design and analysis tool

% Butterworth Bandpass filter designed using FDESIGN.BANDPASS.

% All frequency values are in Hz.

Fs = 100-; % Sampling Frequency is 1000 Hz

N = 2; % Order

% Fc1 and Fc2 need to change case by case. I normally set Fc1 as natural frequency + 0.6

Hz and Fc2 as natural frequency – 0.6 Hz.

Fc1 = 4; % First Cutoff Frequency

Fc2 = 5.5; % Second Cutoff Frequency

% Construct an FDESIGN object and call its BUTTER method.

h = fdesign.bandpass('N,Fc1,Fc2', N, Fc1, Fc2, Fs);

Hd = butter(h);

% End of program and please look at workspace and filter object “Hd” was generated.

Natural frequency analysis

pwelch(b,[],[],2048,1000,'one sided') % find out fundamental frequency

%pwelch is defined as the best power spectrum analysis tool in Matlab, and Psd is no

longer recommended.

109
Transfer from acceleration data to displacement data (Fourier transform)

%Signal processing using Fourier Transform(Time domain to Frequency domain)

a=b*100.513 % convert data units to m/s^2, 100.513 is conversion factor which

calibrated in advance.

Af = filter(Hd,a) % apply designed filter “Hd” generated from previous code.

F = fft (Af) %convert all filtered data set from time domain to frequency domain by

fourier transform

Df = F/(2*pi*7.30)^2 % convert acceleration to displacement, 7.30 is the fundamental

frequency

D=ifft (Df)*100 % convert displacement data from frequency to time domain by inverse

fourier transform; and 100 is convert meter to CM

T=0:1/1000:length(Af)/1000-1/1000 % Create time increament

plot(T,D)

xlabel('Time(Second)')

ylabel('Displacement(cm)')

title('Displacement vs Time')

grid on

110
 Appendix C filtered and converted data

3200_20500_S05_D06_A50 without spring

B_3200_20050_S05_D06_A50_7.00 Hz
Displacement vs Time
0.6
X: 13.59
Y: 0.5871

0.4

0.2
Displacement(cm)

-0.2

-0.4

-0.6

-0.8
0 2 4 6 8 10 12 14
Time(Second)

B_3200_20050_S05_D06_A50_7.05 Hz
Displacement vs Time
0.8

0.6 X: 8.27
Y: 0.6798

0.4

0.2
Displacement(cm)

-0.2

-0.4

-0.6

-0.8
0 5 10 15
Time(Second)

111
B_3200_20050_S05_D06_A50_7.10 Hz
Displacement vs Time
0.8
X: 13.12
Y: 0.7916
0.6

0.4

0.2
Displacement(cm)

-0.2

-0.4

-0.6

-0.8
0 2 4 6 8 10 12 14
Time(Second)

B_3200_20050_S05_D06_A50_7.15 Hz
Displacement vs Time
1

X: 6.475
0.8 Y: 0.9228

0.6

0.4
Displacement(cm)

0.2

-0.2

-0.4

-0.6

-0.8

-1
0 5 10 15
Time(Second)

112
B_3200_20050_S05_D06_A50_7.20 Hz
Displacement vs Time
1.5

1 X: 8.124
Y: 1.078

0.5
Displacement(cm)

-0.5

-1

-1.5
0 5 10 15
Time(Second)

B_3200_20050_S05_D06_A50_7.25 Hz
Displacement vs Time
1.5

X: 12.93
Y: 1.253
1

0.5
Displacement(cm)

-0.5

-1

-1.5
0 5 10 15
Time(Second)

113
B_3200_20050_S05_D06_A50_7.30 Hz
Displacement vs Time
1.5
X: 12.92
Y: 1.444

0.5
Displacement(cm)

-0.5

-1

-1.5
0 5 10 15
Time(Second)

B_3200_20050_S05_D06_A50_7.35 Hz
Displacement vs Time
2

1.5 X: 9.12
Y: 1.624

0.5
Displacement(cm)

-0.5

-1

-1.5

-2
0 2 4 6 8 10 12 14 16 18
Time(Second)

114
B_3200_20050_S05_D06_A50_7.40 Hz
Displacement vs Time
2

X: 9.333
1.5 Y: 1.722

0.5
Displacement(cm)

-0.5

-1

-1.5

-2
0 5 10 15
Time(Second)

B_3200_20050_S05_D06_A50_7.45 Hz
Displacement vs Time
2

X: 6.141
1.5
Y: 1.711

0.5
Displacement(cm)

-0.5

-1

-1.5

-2
0 5 10 15
Time(Second)

115
B_3200_20050_S05_D06_A50_7.50 Hz
Displacement vs Time
2

1.5
X: 13.05
Y: 1.586

0.5
Displacement(cm)

-0.5

-1

-1.5

-2
0 2 4 6 8 10 12 14 16 18
Time(Second)

B_3200_20050_S05_D06_A50_7.55 Hz
Displacement vs Time
1.5

X: 13.21
Y: 1.412

0.5
Displacement(cm)

-0.5

-1

-1.5
0 5 10 15
Time(Second)

116
B_3200_20050_S05_D06_A50_7.60 Hz
Displacement vs Time
1.5

X: 12.81
1 Y: 1.235

0.5
Displacement(cm)

-0.5

-1

-1.5
0 5 10 15
Time(Second)

B_3200_20050_S05_D06_A50_7.65 Hz
Displacement vs Time
1.5

1 X: 13.34
Y: 1.079

0.5
Displacement(cm)

-0.5

-1

-1.5
0 5 10 15
Time(Second)

117
B_3200_20050_S05_D06_A50_7.70 Hz
Displacement vs Time
1

X: 13.44
0.8 Y: 0.9433

0.6

0.4
Displacement(cm)

0.2

-0.2

-0.4

-0.6

-0.8

-1
0 5 10 15
Time(Second)

B_3200_20050_S05_D06_A50_7.75 Hz
Displacement vs Time
1

0.8
X: 12.54
Y: 0.8322
0.6

0.4
Displacement(cm)

0.2

-0.2

-0.4

-0.6

-0.8

-1
0 5 10 15
Time(Second)

118
B_3200_20050_S05_D06_A50_7.80 Hz
Displacement vs Time
0.8

X: 12.47
Y: 0.738
0.6

0.4

0.2
Displacement(cm)

-0.2

-0.4

-0.6

-0.8
0 2 4 6 8 10 12 14 16
Time(Second)

B_3200_20050_S05_D06_A50_7.85 Hz
Displacement vs Time
0.8

0.6 X: 10.33
Y: 0.6566

0.4

0.2
Displacement(cm)

-0.2

-0.4

-0.6

-0.8
0 5 10 15
Time(Second)

119
B_3200_20050_S05_D06_A50_7.90 Hz
Displacement vs Time
0.8

0.6
X: 9.069
Y: 0.589
0.4

0.2
Displacement(cm)

-0.2

-0.4

-0.6

-0.8
0 5 10 15
Time(Second)

120
 Appendix D Damper design curves (Cheng et al, 2010)

Figure D-1 Equivalent 1st modal damping ratio of a damped cable when damper at 2% L

Figure D-2 Equivalent 1st modal damping ratio of a damped cable when damper at 4% L

121
Figure D-3 Equivalent 1st modal damping ratio of a damped cable when damper at 6% L

Figure D-4 Equivalent 1st modal damping ratio of a damped cable when damper at 10% L

122
Figure D-5 Equivalent 1st modal damping ratio of a damped cable when damper at 15% L

123
 VITA AUCTORIS

NAME: Le Huang

PLACE OF BIRTH: Nanjing, Jiangsu, China

YEAR OF BIRTH: 1983

EDUCATION: 2004- 2009 B.A.Sc University of Windsor, Windsor, Ontario

2009- 2011 M.A.Sc University of Windsor, Windsor, Ontario

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